mit_ocw_statistical_physics_2_8_08.jsonl

{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0001", "text": "# I Canonical ensemble\n\n①\n|sys| ↔ |heat\nreservoir\n(bath)\ncanonical\nensemble\nmicrocanonical\nensemble\nWhat is the probability\nfor the system to be\nin a state with\nenergy En?\n\nIdeal heat bath: T ind. of energy\n\n$$\nF _ {\\mathrm{from}} \\frac {1}{T} = \\frac {\\partial S (E)}{\\partial E}\n$$\n\n$$\n\\Rightarrow \\sqrt {S _ {\\mathrm{both}} (E)} = \\frac {E}{T} \\quad E / m\n$$\n\n$$\nn \\left| \\# \\text {   of   states   } \\Gamma_ {\\mathrm{bath}} (E) = e ^ {- i \\beta B} \\right|\n$$\n\n② Boltzmann distribution\n\nOur system can have many energy levels\n\n$$\nE _ {n}, \\quad n = i, 2, \\dots\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0002", "text": "2\n\nPicture of microcanonical ensemble: sys+heat bath\n\nTotal energy   $E_{tot}$  is fixed (isolated, microcanonical)\n\n# of states when sys is in the nth state\n\n$$\n\\begin{array}{l} = 1 \\cdot \\Gamma_ {\\mathrm{bath}} (E _ {t o t} - E _ {n}) \\\\ = e ^ {(E _ {t o t} - E _ {n}) / k _ {B} T} \\\\ \\end{array}\n$$\n\nProb. for the sys be in the  $n^{th}$ state\n\n$$\n\\propto e ^ {- E _ {n} / k _ {B} T}\n$$\n\n③ The partition function - the total prob.\n\n$$\nQ (T, V) \\equiv \\sum_ {n} e ^ {- E _ {n} / k _ {B} T} \\quad (\\text { quantum })\n$$\n\n$Q = \\int dE \\sum_{n} S(E - E_n) e^{-E / k_B T}$\n\n$$\n\\overline {{\\Gamma_ {s y s} (E)}} = \\# \\text {   of   states   of   sys   with   energy   E.   }\n$$\n\n$$\n= \\int d E \\Gamma_ {s y s} (E) e ^ {- E / k _ {B} T}\n$$\n\n$$\n= \\int d E e ^ {- (E / k _ {B} T) + (S (E) / k _ {B})}\n$$\n\n$$\n= \\int d E e ^ {- \\frac {1}{k _ {B} T} (E - S (E) T)}\n$$\n\n$$\n\\rightarrow \\infty \\text { prob.   for   iys   to   have   energy } E\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0003", "text": "Sys. has an energy that maximize\n\n$$\nA = E - S (E) T\n$$\n\nmax value\n\n$$\nA _ {\\max} = \\overline {{E}} - S (\\overline {{E}}) T\n$$\n\nwhere   $\\overline{E}$  satisfies   $\\frac{1}{T} = \\frac{dS(\\overline{E})}{dE}$\n\nE is the energy of sys   $\\frac{1}{T_{bath}} = \\frac{1}{T_{cath}}$\n\nin equilibrium state   $T_{sys} = T_{bath}$\n\nA = E - ST is the Free energy\n\n(Helmholtg energy)\n\n①   $e^{-\\frac{1}{k_{B}T}}A$  ∝ prob. distribution\n\nif canonical ensemble\n\nSys. wants to minimize the free energy (which mean to reach equilibrium state)\n\n⑥   $e^{-\\frac{1}{k_{B}T}}A$  is sharply peaked\n\nAs a result\n\n$$\nQ = \\int d E e ^ {- \\frac {1}{E _ {B} T}} A\n$$\n\n$\\sim e^{-\\frac{1}{kBT} A_{min}}$ free energy of equilibrium state.\n\n$$\nA = - k _ {B} T \\ln Q\n$$\n\nΔE\nE", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0004", "text": "4\n\nCompare with microcanonical ensemble\n\nmicro canonical\n\n$$\n\\begin{array}{l} \\mathrm{Prob.}: \\Gamma (E) = e ^ {S / k _ {B}} \\quad \\Gamma (E) e ^ {- \\frac {E}{k _ {B} T}} = e ^ {- \\frac {(E - S T)}{k _ {B} T}} \\\\ = e ^ {- A / k _ {5} T} \\\\ \\end{array}\n$$\n\n④ Thermodynamical relation & quantities\n\nLet  $\\tilde{A}(E,T)=E-S(E)T$\n\nFree energy is obtained by minimize   $\\tilde{A}$\n\nrespect to E:\n\n$$\nA (T) = \\tilde {A} (\\overline {{E}} (t), T), \\quad \\frac {\\partial \\tilde {A} (E , T)}{\\partial E} | _ {E = \\overline {{E}}} = 0\n$$\n\n② Entropy: Calculate\n\n$$\n\\frac {\\partial A (T)}{\\partial T} = \\underbrace {\\left. \\frac {\\partial \\tilde {A}}{\\partial E} \\right| _ {E = \\overline {{E}}}} _ {= 0} \\frac {\\partial \\overline {{E}} (T)}{\\partial T} + \\left. \\frac {\\partial \\tilde {A}}{\\partial T} \\right| _ {E = \\overline {{E}}}\n$$\n\n$$\n\\boxed {\\frac {\\partial A}{\\partial T} = - S} \\quad \\boxed {\\frac {H e a t}{\\Delta T} = \\boxed {T \\frac {d S}{d T} = C}}\n$$\n\n⑥ Energy:   $\\overline{E} = A + S(\\overline{E})T$\n\n$$\n= A - T \\frac {\\partial}{\\partial T} A\n$$\n\nSecond way  $\\overline{E}=\\angle E>average=\\frac{\\sqrt{n}}{n}E_{n}e^{-\\beta E_{m}}/\\frac{\\sqrt{n}}{n}e^{-\\beta E_{n}}$\n\n$$\n= - \\frac {\\partial}{\\partial \\beta} \\ln Q = (k _ {B} T ^ {2} \\frac {\\partial}{\\partial T}) \\frac {- 1}{k _ {B} T} A\n$$\n\n$$\n= A - T \\frac {2}{3 T} A\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0005", "text": "⑤ Energy cost of information\n\nFlexible chain:\n\n$$\n\\begin{array}{l} \\text { Length } = (+ 1) + (+ 1) + (- 1) + (+ 1) + (+ 1) \\\\ R R L R R \\\\ = N _ {R} - N _ {L} \\\\ \\end{array}\n$$\n\nNo internal energy\n\n$$\n\\text { Total   number   of   states } = 2 ^ {N} \\quad (N = N _ {R} + N _ {h})\n$$\n\n$$\n\\text { number   of   states } w / N _ {R} - N _ {L} = 0\n$$\n\n$$\n= C _ {N} ^ {N / 2} = \\frac {N !}{(N / 2) !}\n$$\n\n$$\nT _ {0} + 1 \\dots e n t r o p y S _ {t o t} = k _ {B} \\ln 2 ^ {N} = N k _ {B} \\ln 2\n$$\n\n$$\nE n + t _ {2} o p g \\quad w i t h \\quad L e n g t h = 0 \\quad S _ {0} = k _ {B} \\ln C _ {N} ^ {N / 2}\n$$\n\n$$\n\\approx k _ {B} N \\ln N - 2 (\\frac {N}{2}) \\ln (\\frac {N}{2}\n$$\n\n$$\n= N k _ {B} \\ln 2 \\doteq S t. t\n$$\n\n$$\n\\text { Entropy   with   Length } = N \\quad S _ {1} = k _ {B} \\ln 1 = 0\n$$\n\n$$\nE n e g y \\quad \\text { required } \\quad \\text { to } \\quad \\text { stretch } \\quad \\text { the }\n$$\n\n$$\n\\text { chain   from   length } = 0 \\quad \\text { to   length } = N?\n$$\n\n$$\n\\mathrm{No} \\quad \\text { internal   energy } \\Rightarrow \\mathrm{No} \\quad \\text { energy   cost } X\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0006", "text": "S0 = NkB ln2\n→\nS1 = 0\n\nchain entropy ↓ heat bath entropy ↑\n\n$\\Delta S$ for the heat bath =  $N k_{B} \\ln 2$.\n\nEntropy of heat bath  ${S}_{\\text{bath }} = \\frac{E_{\\text{bath }}}{T}$\n\n$S_{bath} \\uparrow$ can only be achieved through  $E_{both}$\n\n$$\n\\Delta E _ {\\text { bath }} = T \\Delta S = k _ {B} T N \\ln 2\n$$\n\n$$\n= \\text { energy   cost } \\quad \\text { to   stretch } \\quad \\text { the   chain }\n$$\n\n$$\n= A _ {\\text { final }} - A _ {\\text { init }} = (- S _ {\\text { final }} T) - (- S _ {\\text { init }} T) = \\Delta S T\n$$\n\nWhat about information\n\n$$\nE _ {n + 2 0 p y} = k _ {B} N l n 2\n$$\n\n$$\nE _ {n + 2 0 p y} = 0\n$$\n\nTo encode\n\none bit of\n\nin formation,\n\nwe need\n\n${k}_{15}T\\ln 2$\n\namount of energy\n\nTotal entropy can only increase", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0007", "text": "* Ideal memory chip:\n\nto program 1G bytes of RAM\n\nat room: temperature\n\nwe need energy =   $k_{B}T(\\ln2)\\times8\\times10^{9}$\n\n$$\n= 2. 3 \\times 1 0 ^ {- 4} e ^ {i g}\n$$\n\n* Ideal bus : 1G bits / second\n\n$$\n\\text { power } = 3 \\times 1 0 ^ {- 5} \\text { eY / s } = 3 \\times 1 0 ^ {- 1 2} \\text { Wati }\n$$\n\n$$\n\\text {   computer   } \\sim 1 0 0 \\text {   Watt   }\n$$\n\nHow to measure the amount if information\n\n$$\n\\text {   in   formation   } = \\text {   Const.   } - \\text {   entropy   } \\cdot k _ {\\overline {{B}}} ^ {- 1}\n$$\n\n$$\n\\text { information } \\quad \\text { entropy / kg }\n$$\n\n0 1\n\n50% 50%\n\nO\n\nln 2\n\n0 1\n\nln2\n\n0\n\n100% 0%\n\n$$\n\\text { information } = \\ln \\Gamma - \\sum \\rho_ {i} \\ln \\frac {1}{\\rho_ {i}}\n$$\n\n$$\nI _ {\\# o f s t a t e} \\hat {L} p r o b. f u s t a t e i\n$$\n\n$$\n\\text { information } = 0 \\quad \\text { if } \\quad \\rho_ {i} = \\frac {1}{\\pi}\n$$\n\n$$\n\\text {   in   formation   } = \\ln \\gamma \\quad \\text {   if   only   one   } \\beta_ {i} = 1\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0008", "text": "⑥ Maxwell distribution and equipatition of energy.\n\n$$\n\\mathrm{Prob.} \\quad \\mathrm{to} \\quad \\mathrm{find} \\quad \\vec {p} \\quad \\mathrm{in} \\quad \\mathrm{volum} d \\vec {p}: \\quad C e ^ {- \\beta \\vec {p} ^ {2} / 2 m} d ^ {3} \\vec {p}\n$$\n\n$$\n\\text { Prob. } \\quad x _ {0} \\text { find } \\quad v \\text { between } \\quad v \\text { and } \\quad v + d v: \\quad \\tilde {C} e ^ {- \\beta \\frac {m v ^ {2}}{2}} v ^ {2} d i\n$$\n\nMaxwell distribution\n\n$$\n\\text { Kinetic   energy   of } x \\text {-motion}\n$$\n\n$$\n<   \\frac {p _ {x} ^ {2}}{2 m} > = \\frac {\\int d ^ {3} \\vec {p} \\cdot \\frac {p _ {x} ^ {2}}{2 m} e ^ {- \\beta \\vec {p} ^ {2} / 2 m}}{\\int d ^ {3} \\vec {p} \\cdot e ^ {- \\beta \\vec {p} ^ {2} / 2 m}}\n$$\n\n| v    | P(v) |\n| ---- | ---- |\n| 0.0  | 0.0  |\n| 0.25 | 0.8  |\n| 0.5  | 1.0  |\n| 0.75 | 0.6  |\n| 1.0  | 0.2  |\n\n$$\n= \\frac {\\int d p _ {x} \\frac {p _ {x} ^ {2}}{2 m} e ^ {- \\beta p _ {x} ^ {2} / 2 m}}{\\int d p _ {x} e ^ {- \\beta p _ {x} ^ {2} / 2 m}}\n$$\n\n$$\n\\int d x e ^ {- x ^ {2}} = \\sqrt {4}\n$$\n\n$$\n= - \\frac {d}{d \\beta} \\ln \\underbrace {\\int d p _ {x} e ^ {- \\beta p _ {x} ^ {2} / 2 m}} _ {= \\sqrt {\\pi} \\sqrt {\\frac {2 m}{\\beta}}}\n$$\n\n$$\n= \\frac {1}{2} \\frac {1}{\\beta} = \\frac {k _ {B} T}{2}\n$$\n\n$$\n<   \\frac {p _ {x} ^ {2}}{1 m} > = <   \\frac {p _ {y} ^ {2}}{2 m} > = <   \\frac {p _ {z} ^ {2}}{2 m} > = \\frac {k _ {B} T}{2}\n$$\n\n$$\n= <   \\frac {P _ {X} ^ {2}}{2 m ^ {3}} > \\leftarrow \\begin{array}{l l} \\text { For   a   mixture   of   particles } \\\\ \\text { of   mass   m   and   m' } \\end{array}\n$$\n\n$$\n\\text { Total   energy } E = \\frac {N k _ {B} T}{2} \\times 3\n$$\n\n$$\n\\text { heat   capacity }: C _ {V} = \\frac {\\partial E}{\\partial T} = \\frac {3}{2} N k _ {B}\n$$\n\n$$\n\\text { mixed   gas } \\quad C _ {V} = \\frac {3}{2} k _ {0} (N _ {1} + N _ {2} \\dots)\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0009", "text": "But for   $H_{2}$\n\n| Time (tion) | CV/kBN |\n|---|---|\n| 100 | 3/2 |\n| 500 | 5/2 |\n| 1000 | 5/2 |\n| 1000 | vibration |\n| 1000 | translation |\n\nRotation around z-direction\n\n$$\nE _ {m} = \\frac {L _ {3} ^ {2}}{2 I} = E _ {0} m ^ {2} \\quad m = 0, \\pm 1, \\pm 2 \\dots\n$$\n\n$$\n\\text { average   energy } <   E > = \\frac {\\sum E _ {m} e ^ {- \\beta E _ {m}}}{\\sum_ {m} e ^ {- \\beta E _ {m}}}\n$$\n\n$$\n\\begin{array}{r l} {\\mathrm{if}} & {E _ {0} \\beta \\ll 1:} \\\\ & {\\langle E \\rangle = \\frac {\\int d m \\frac {1}{2} E _ {0} m ^ {2} e ^ {- \\beta E _ {0} m ^ {2}}}{\\int d m e ^ {- \\beta E _ {0} m ^ {2}}} = \\frac {1}{2} k _ {B} T} \\end{array}\n$$\n\n$$\n\\text { if } E _ {0} P > > 1:\n$$\n\n$$\n\\langle E \\rangle = \\therefore E _ {0} e ^ {- E _ {0} / k _ {B} T}\n$$\n\n| T       | <E>     |\n| ------- | ------- |\n| 0       | 0       |\n| E₀/k_B  | 1/2 k_B T |\n| T       | >1/2 k_B T |\n| Tnot   | T       |\n\n$$\n\\begin{array}{l} T _ {2 0 t} \\quad T _ {v i k} \\\\ H _ {2} 8 5. 4 6 1 0 0 \\\\ N _ {2} \\quad 2. 8 6 \\quad 3 3 4 0 \\\\ O _ {2} = 2. 0 7 \\quad 2 2 3 0 \\\\ \\end{array}\n$$\n\n$$\nL f _ {n} a i n C _ {V} \\approx \\frac {7}{2} k _ {B} (N _ {O _ {2}} + N _ {N _ {2}})\n$$\n\n$$\na + 2 0 0 m \\text { Temperature }\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0010", "text": "⑦ Free energy of classical ideal gas\n\n$$\n\\text { partition   function }\n$$\n\n$$\nQ (T, V) = \\int \\frac {d ^ {3} N p}{N !} \\frac {d ^ {3} N q}{h ^ {3} N} e ^ {- \\beta (p _ {1} ^ {2} + \\dots + p _ {N} ^ {2}) / 2 m}\n$$\n\n$$\n= \\frac {1}{N !} (\\frac {V}{\\lambda^ {3}}) ^ {N}\n$$\n\n$$\nA (V, T) = - k _ {B} T \\ln Q\n$$\n\n$$\n= N k _ {B} T [ \\ln (n \\lambda^ {3}) - 1 ]\n$$\n\n$$\nE n t o p y\n$$\n\n$$\n\\beta = \\frac {1}{k _ {B} T}\n$$\n\n$$\n\\int \\frac {d p}{h} e ^ {- t _ {0} P ^ {2} / 2 m} = \\frac {1}{\\lambda}\n$$\n\n$$\n\\lambda = \\sqrt {2 \\pi k ^ {2} / m k T}\n$$\n\n$$\n\\sim \\frac {k}{\\sqrt {m k T}} = \\frac {k}{\\sqrt {2 m k}}\n$$\n\n$$\nK: \\text { average   kinetic   energy }\n$$\n\n$$\nS = - \\left(\\frac {\\partial A}{\\partial T}\\right) _ {V} = N k _ {B} \\left(\\frac {5}{2} + \\ln \\frac {U}{\\lambda^ {3}}\\right)\n$$\n\n$$\nv = \\frac {V}{N}\n$$\n\n$$\n\\frac {v}{\\lambda^ {3}} = \\# o f s t a t e s p e r p a i t i c l e s\n$$\n\n$$\n\\left. \\frac {\\partial A}{\\partial V} \\right| _ {T} = - k T \\frac {\\frac {\\partial}{\\partial V} Q}{Q}\n$$\n\n$$\n= - k T \\frac {\\sum_ {n} - \\beta \\frac {\\partial E _ {n}}{\\partial V} e ^ {- \\beta E _ {n}}}{\\sum e ^ {- \\beta E _ {n}}}\n$$\n\n$$\n= \\frac {\\sum_ {n} \\frac {d ^ {n} E _ {n}}{d V} e ^ {- \\beta E _ {n}}}{\\sum_ {n} e ^ {- \\beta E _ {n}}}\n$$\n\n$$\n= - \\langle P \\rangle \\leftarrow \\text { average   pressure. }\n$$\n\n$$\n\\frac {d E _ {n}}{d V}: \\text { change   of }\n$$\n\n$$\n\\text { energy   for   isolated }\n$$\n\n$$\n\\text { system } \\therefore \\frac {d E _ {u}}{d V} = p r e s s u 2\n$$\n\n$$\n\\text { for   the } n ^ {+ h} \\text { state }\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 9, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 10"], "page_start": 10, "page_end": 10, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0001-i-canonical-ensemble-027b94eb-ea783a0a:page-0011", "text": "12\n\n$$\nA (V, T) = N k _ {B} T \\left[ l n (\\frac {N A ^ {3}}{V}) - 1 \\right]\n$$\n\n$$\nP = - \\frac {\\partial A}{\\partial V} \\bigg | _ {T N} = N k _ {B} T / V \\quad \\text {   eqn.   of   state   }\n$$", "source": "mit-ocw", "source_doc_id": "0001-i-canonical-ensemble-027b94eb-ea783a0a", "source_title": "I Canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 10, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["I Canonical ensemble", "Page 11"], "page_start": 11, "page_end": 11, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0001", "text": "Ⅱ Interactions classical gas & van der Waals Eq.\n① partition function: of state\n\n$$\nQ = \\frac {1}{N !} \\int_ {\\frac {1}{(2 \\pi) ^ {3 N}}} ^ {\\pi d ^ {3} k _ {i} d ^ {3} f _ {i}} e ^ {- \\beta [ 2 \\epsilon_ {k _ {i}} + V (f _ {i}) ]}\n$$\n\n$$\n\\frac {1}{N !} \\int \\frac {\\pi d ^ {3} p _ {i}}{h ^ {3 N}}\n$$\n\n$$\n\\text { replace } \\quad V (y _ {i}) \\quad b _ {y} \\quad <   V (y _ {i}) > \\quad \\text { mean - field }\n$$\n\n$$\nV (q _ {i}) = \\frac {1}{2} \\sum_ {i, j} \\quad \\sigma (q _ {i} - q _ {j}) = \\frac {1}{2} \\int d ^ {3} q d ^ {3} q ^ {\\prime} n ^ {2} \\sigma (q _ {i} - q ^ {\\prime})\n$$\n\n$$\n= \\frac {1}{2} N \\int d ^ {3} q U (q) = \\frac {1}{2} N \\overline {{u}} n\n$$\n\n$$\n\\text { average   potential }: \\overline {{U}} = \\int d ^ {3} q U (q)\n$$\n\n$$\nQ = \\frac {1}{N !} \\int_ {\\frac {i}{(2 \\pi) ^ {3 N}}} ^ {\\pi d ^ {3} k; d ^ {2} q;} e ^ {- \\beta [ 2 \\epsilon_ {k} + \\frac {1}{2} N n \\overline {{v}} ]}\n$$\n\n$$\n\\frac {1}{N !} \\left[ \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- \\beta \\epsilon_ {k}} \\right] ^ {N} e ^ {- \\frac {\\beta}{1 2} N u v}\n$$\n\n$$\n= \\frac {1}{N !} (\\frac {V - b}{\\lambda^ {3}}) ^ {N} e ^ {- \\frac {\\beta}{2} N n \\overline {{v}}}\n$$\n\n$$\nA = - k _ {B} T \\ln Q\n$$\n\n$$\n= N k _ {B} T \\left(\\ln \\frac {N \\lambda^ {3}}{V - b} - 1\\right) + \\frac {1}{2} N n \\overline {{\\sigma}}\n$$\n\n$$\n= \\sqrt {k _ {B}} T (\\ln \\frac {N \\lambda^ {3}}{V - N v _ {0}} - 1) + \\frac {1}{2} \\frac {N ^ {2} \\overline {{v}}}{V}\n$$\n\n$$\n(V - b) ^ {N}\n$$\n\n$$\nb = \\text { excluded }   \\text { volume }\n$$\n\n$$\nb \\sim N \\frac {4 \\pi}{3} r _ {0} ^ {3} = N v _ {0}\n$$\n\n$$\n\\text { for   hand   balls }\n$$", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0002", "text": "② Eqn. f stat\n\n$$\nP = - \\frac {\\partial A}{\\partial V} = \\frac {N k _ {B} T}{V - b} + \\frac {1}{2} N ^ {2} \\overline {{v}} / v ^ {2}\n$$\n\n$$\n(V - b) (P - \\frac {N ^ {2} \\overline {{v}} / 2}{V ^ {2}}) = N k _ {B} T\n$$\n\nvan der Waals equ. of st\n\n$$\n(V - b) (P + \\frac {a}{v ^ {2}}) = R T\n$$\n\n$$\nA = N k _ {B} T (\\ln \\frac {N \\lambda^ {3}}{V - b} - 1) = \\frac {a}{V}\n$$\n\nV initial expansion\n\n$$\nP = P (V, T)\n$$\n\n$$\n\\frac {P V}{R T} = 1 + \\frac {C _ {2} (T)}{V} + \\frac {C _ {3} (T)}{V ^ {3}}.\n$$\n\n$$\nc _ {2} = b - \\frac {a}{R T} c _ {3} = b ^ {2} \\dots\n$$\n\n$$\n\\text { if } \\quad \\text { we   ignore   a   then } \\quad c _ {2} = \\text { excluded   volume }\n$$\n\n$$\nF _ {o n} \\quad F e r m i g a s \\quad C _ {2} = N \\frac {\\lambda^ {3}}{2 ^ {5 / 2}}\n$$\n\n$$\nF _ {0 2} \\quad h o n d b a l l c _ {2} = N \\frac {4 \\pi}{3} r _ {0} ^ {3} \\quad h o n d b a l l w / v _ {0} \\sim \\lambda\n$$", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0003", "text": "③ phase transition\n\n∴ Knowing the free energy A(T,V)\n\nallows us to know every thing\n\n? Is there any phase transition?\n\nIs there any singularity in  $A(T,V)$ as\n\na function of T and V.?\n\n$$\n\\text { singularity } = \\text { phase   Transition }\n$$\n\nWe find on singularity in A(T,V)\n\nseem no phase transition\n\nInstead of examining phase Transition using\n\nthe free energy, let us examine it\n\nusing Gibbs potential G(T, P)", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0004", "text": "④ Gibbs potential\n\nT\nV\nsping\nsping energy\n\nGibles potential\n\nSpring energy\n\n$$\n= \\text { free   energy   of   combined   system }\n$$\n\n$= {P.V} + {const}.$\n\n$$\n\\rightarrow \\text {   sys   } + \\text {   spring   }\n$$\n\n$$\n\\text { Partition   function } \\quad Q _ {\\text { com }} = \\sum_ {n} e ^ {- \\beta (E _ {n} + V _ {n} P)} = Q _ {\\text { sys }} e ^ {- \\beta \\overline {{V}} P}\n$$\n\n$$\n\\text { free   energy   of   combined   system: } A _ {\\mathrm{com}} = - k _ {B} T \\ln Q _ {\\mathrm{com}}\n$$\n\n$$\n\\text { is   the   Gibbs   potential }\n$$\n\n$$\n= - k _ {B} T \\ln Q _ {s y s} + \\overline {{V}} P\n$$\n\n$$\n= A _ {s y s} + \\overline {{V}} P = G _ {s y s}\n$$\n\n$$\nQ _ {c o m} = \\int d V \\sum_ {n} \\delta (V - V _ {n}) e ^ {- \\beta (E _ {n} + V P)}\n$$\n\n$$\n= \\int d V e ^ {- \\beta (A (V, T) + V P)} \\approx e ^ {- \\beta [ A (\\overline {{V}}, T) + \\overline {{V}} P ]}\n$$\n\n$$\n\\overline {{V}} \\text {   minimize   this   }\n$$\n\n$$\nk > 2 5 \\text { Prob.   distribution } f o r V\n$$\n\n$$\n\\begin{array}{l l l l l} {\\mathrm{For}} & {\\mathrm{fixed}} & {T, V} & {\\mathrm{system}} & {\\mathrm{wants~tominimize}} \\\\ {\\mathrm{For}} & {\\mathrm{fixed}} & {T, P} & {\\mathrm{system}} & {\\mathrm{wants~to~minimize}} \\end{array} \\quad \\begin{array}{l l l l l} {\\mathrm{free}} & {\\mathrm{energy}} & {A} \\\\ {\\mathrm{Gibbs~potential}} & {G} \\end{array}\n$$\n\n$$\nF _ {\\text { from   the   relation }} G = A + V P\n$$\n\n$$\n\\begin{array}{l} \\Rightarrow d G = d A + d V P = - S d T = P d V + P d V + V d P \\\\ = - S d T + V d P \\\\ \\end{array}\n$$\n\n$$\n\\Rightarrow V = \\frac {\\partial G (T , P)}{\\partial P}, S = - \\frac {\\partial G (T , P)}{\\partial T}\n$$", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0005", "text": "⑤ Phase Transition in gas of constant pressure\n\nA gas with pressure P: what is V?\n\nV is obtained by minimize Gibbs energy\n\n$$\nG = A + P V\n$$\n\nvan den Waals model\n\n$$\n\\begin{array}{l} G = N k _ {B} T \\left(\\ln \\frac {N \\lambda^ {3}}{V - b} - 1\\right) - \\frac {a}{V} + P V \\\\ = - N k _ {B} T \\ln (V - b) - \\frac {a}{V} + P V + t e r m s i n d. o f V \\\\ \\end{array}\n$$\n\nThree behaviors\n\n$$\nE _ {f} \\cdot f: \\text {   stat   :   } \\sqrt {V ^ {3} - (b + \\frac {R T}{P}) V ^ {2} + \\frac {a}{P} V - \\frac {a b}{P}} = 0 = (V - V _ {c}) ^ {3}\n$$\n\n$$\n\\text { at   critical   point. } \\quad 3 V _ {c} = b + \\frac {R T _ {c}}{P _ {c}}, \\quad 3 V _ {c} ^ {2} = \\% P _ {c}, \\quad V _ {c} ^ {3} = \\frac {a b}{P _ {c}} \\tag {criticalcondition}\n$$\n\n$$\nT _ {c} = \\frac {8 a}{2 7 b R} P _ {c} = \\frac {a}{2 7 b ^ {2}} V _ {c} = 3 b\n$$\n\nthree degenerate solutions.", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0006", "text": "$a > 0$ attraction  $b > 0$ hand\n\n$$\n\\frac {d G}{d V} = - \\frac {N k _ {B} T}{V - b} + \\frac {a}{V ^ {2}} + P\n$$\n\nLarge T\n\n| V | dG/dV |\n|---|------|\n| 0 | 0    |\n| P | 1    |\n| V | 0    |\n\nsmall T\ndG/dV\nF\nV\n\n* a < 0 replusion b > 0\n\n| V | dG/dV |\n|---|------|\n| 0 | 0    |\n| Peak | ~1.2 |\n| 1 | ~1.1 |\n| 2 | ~1.0 |\n| 3 | ~0.9 |\n| 4 | ~0.8 |\n| 5 | ~0.7 |\n| 6 | ~0.6 |\n| 7 | ~0.5 |\n| 8 | ~0.4 |\n| 9 | ~0.3 |\n| 10 | ~0.2 |\n| 11 | ~0.1 |\n| 12 | ~0.05 |\n| 13 | ~0.03 |\n| 14 | ~0.02 |\n| 15 | ~0.01 |\n| 16 | ~0.005 |\n| 17 | ~0.003 |\n| 18 | ~0.002 |\n| 19 | ~0.001 |\n| 20 | ~0.0005 |\n| 21 | ~0.0003 |\n| 22 | ~0.0002 |\n| 23 | ~0.0001 |\n| 24 | ~0.00005 |\n| 25 | ~0.00003 |\n| 26 | ~0.00002 |\n| 27 | ~0.00001 |\n| 28 | ~0.000005 |\n| 29 | ~0.000003 |\n| 30 | ~0.000002 |\n| 31 | ~0.000001 |\n| 32 | ~0.0000005 |\n| 33 | ~0.0000003 |\n| 34 | ~0.0000002 |\n| 35 | ~0.0000001 |\n| 36 | ~0.00000005 |\n| 37 | ~0.00000003 |\n| 38 | ~0.00000002 |\n| 39 | ~0.00000001 |\n| 40 | ~0.0000   |\n| 41 | ~-0.1 |\n| 42 | ~-0.2 |\n| 43 | ~-0.3 |\n| 44 | ~-0.4 |\n| 45 | ~-0.5 |\n| 46 | ~-0.6 |\n| 47 | ~-0.7 |\n| 48 | ~-0.8 |\n| 49 | ~-0.9 |\n| 50 | ~-1.0 |\n| 51 | ~-1.1 |\n| 52 | ~-1.2 |\n| 53 | ~-1.3 |\n| 54 | ~-1.4 |\n| 55 | ~-1.5 |\n| 56 | ~-1.6 |\n| 57 | ~-1.7 |\n| 58 | ~-1.8 |\n| 59 | ~-1.9 |\n| 60 | ~-2.0 |\n| 61 | ~-2.1 |\n| 62 | ~-2.2 |\n| 63 | ~-2.3 |\n| 64 | ~-2.4 |\n| 65 | ~-2.5 |\n| 66 | ~-2.6 |\n| 67 | ~-2.7 |\n| 68 | ~-2.8 |\n| 69 | ~-2.9 |\n| 70 | ~-3.0 |\n| 71 | ~-3.1 |\n| 72 | ~-3.2 |\n| 73 | ~-3.3 |\n| 74 | ~-3.4 |\n| 75 | ~-3.5 |\n| 76 | ~-3.6 |\n| 77 | ~-3.7 |\n| 78 | ~-3.8 |\n| 79 | ~-3.9 |\n| 80 | ~-4.0 |\n| 81 | ~-4.1 |\n| 82 | ~-4.2 |\n| 83 | ~-4.3 |\n| 84 | ~-4.4 |\n| 85 | ~-4.5 |\n| 86 | ~-4.6 |\n| 87 | ~-4.7 |\n| 88 | ~-4.8 |\n| 89 | ~-4.9 |\n| 90 | ~-5.0 |\n| 91 | ~-5.1 |\n| 92 | ~-5.2 |\n| 93 | ~-5.3 |\n| 94 | ~-5.4 |\n| 95 | ~-5.5 |\n| 96 | ~-5.6 |\n| 97 | ~-5.7 |\n| 98 | ~-5.8 |\n| 99 | ~-5.9 |\n| 100 | ~-6.0 |\n\nV(g)\ninteraction potential\nhardcore\nattraction\nb > 0\na > 0", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0007", "text": "⑥Clapeyton equation:\n\n$$\nG _ {\\mathrm{liquid}} (P, T) \\mathrm{and} G _ {\\mathrm{gas}} (P, T)\n$$\n\nare equal along the phase\n\ntransition line.\n\n$$\nd G _ {e} = V _ {e} d P - S _ {e} d T\n$$\n\n$$\n= d G _ {g} = V _ {g} d P - S _ {g} d T\n$$\n\n$$\n\\Rightarrow \\frac {d P}{d T} = \\frac {\\Delta S}{\\Delta V} = \\frac {L}{T \\Delta V}\n$$\n\nP\ndGg\ndGg\n1.9\nT\n\n$$\nL = T \\Delta S\n$$\n\nlatent heat.\n\nliquid solid transition\n\n$$\n\\Delta V \\simeq 0 \\Rightarrow\n$$\n\nsolid\nliquid\n\nfrom solid → liquid:   $\\Delta S > 0$\n\n$$\ni f \\quad \\Delta V > 0\n$$\n\n$$\n\\Delta V <   0\n$$\n\nwater", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181:page-0008", "text": "20\n\n⑦ scaling and equation of state near critical point\n\n$$\n\\begin{array}{l l l} \\text { Introduce } & \\text { dimensionless } & \\text { quantities } \\\\ \\overline {{P}} = \\frac {P}{P _ {c}} & \\overline {{V}} = \\frac {V}{V _ {c}} & \\overline {{T}} = \\frac {T}{T _ {c}} \\end{array}\n$$\n\nvan den Waals equation of state\n\n$$\n\\begin{array}{l} (\\overline {{V}} - \\frac {b}{V _ {c}}) (\\overline {{P}} + \\frac {a}{\\overline {{V}} ^ {2} P _ {c} V _ {c} ^ {2}}) = \\frac {R T _ {c}}{P _ {c} V _ {c}} \\overline {{T}} \\\\ \\Rightarrow \\boxed {(\\overline {{V}} - \\frac {1}{3}) (\\overline {{P}} + \\frac {3}{\\overline {{V}} ^ {2}}) = \\frac {8}{3} \\overline {{1}}} \\quad \\begin{array}{l} \\text { same   for   all   gases } \\\\ \\text { near   critical   point } \\\\ \\Rightarrow \\text { universality } \\end{array} \\\\ \\end{array}\n$$\n\nVolume jump near the critical point\n\n$$\n\\begin{array}{l} G (t, v) = C _ {1} t (V - V _ {0}) ^ {2} \\\\ + C _ {2} (V - V _ {0}) ^ {4} + c o n s t. \\\\ \\end{array}\n$$\n\nt > 0\n\nt < 0\n\nP\nG\nΔV\nV\nt\nΔV\nT\n\n$$\n\\begin{array}{l} \\frac {d G}{d V} = 2 C _ {1} t (V - V _ {0}) + 4 C _ {2} (V - V _ {0}) ^ {3} = 0 \\\\ \\Rightarrow (V - V _ {0}) = \\pm \\sqrt {- \\frac {c _ {1} t}{2 c _ {2}}} \\\\ \\Rightarrow \\Delta V = \\sqrt {\\frac {- 2 c _ {1} t}{c _ {2}}} \\quad \\propto | t | ^ {1 / 2} \\\\ \\end{array}\n$$\n\nexperiment.\n\n$$\n\\Delta V \\propto | t | ^ {\\beta}\n$$\n\n$$\n\\beta \\simeq \\frac {1}{3}\n$$\n\nUniversality:\n\ndoes not depend on\n\nwhat atom that form the ge", "source": "mit-ocw", "source_doc_id": "0002-ii-interacting-classical-gas-and-van-der-waals-equation-of-state-8675d185-fca24181", "source_title": "Ⅱ Interactions classical gas & van der Waals Eq.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Ⅱ Interactions classical gas & van der Waals Eq.", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-iii-ising-model-3cc4739a-31b3b79d:page-0001", "text": "III. Ising model.\n\nPhase Transition and Ginzburg-Landau theory\n\n① Ising model:\n\n$S_{i} > +1 - 1 + 1 + 1 - 1$\n\n```mermaid\ngraph TD\nA[\"Top Path\"] --> B[\"Left Path\"]\nB --> C[\"Right Path\"]\nC --> D[\"Bottom Path\"]\nD --> E[\"Left Path\"]\nE --> F[\"Right Path\"]\nF --> G[\"Bottom Path\"]\nG --> H[\"Left Path\"]\nH --> I[\"Right Path\"]\nI --> J[\"Bottom Path\"]\nJ --> K[\"Left Path\"]\nK --> L[\"Right Path\"]\nL --> M[\"Bottom Path\"]\n```\n\nSpin configuration  $\\{s; s = \\{s_1, s_2; \\cdots; s = \\{+1, -1, \\cdots\\}\\}$\n\n$E(\\xi s; \\xi) = -J \\sum_{<i,j>} s_i s_j - h \\sum_{i} s_i$\n\nGround slots: (h=0)\n\nJ > 0\n\n↑↑↑↑\n\nFenizomagnetic\n\n$J < 0$\n\n$\\uparrow   \\downarrow   \\uparrow   \\downarrow$\n\nAntiferromagnetic\n\nFinite Temperature:\n\n$$\nz = \\sum_ {i: s _ {i} 3} e ^ {- \\rho E (s _ {i} 3)}\n$$\n\n$$\nA = - k _ {B} T \\ln z\n$$\n\nPhase Transition\n\n$$\n(h = 0) (2 D, N _ {s p i n} \\rightarrow \\infty)\n$$\n\nA\nT\nT_c\n\n2A/2T² has jump\nNspin = ∞\nNspin ≠ ∞", "source": "mit-ocw", "source_doc_id": "0003-iii-ising-model-3cc4739a-31b3b79d", "source_title": "III. Ising model.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["III. Ising model.", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-iii-ising-model-3cc4739a-31b3b79d:page-0002", "text": "② Mean-field theory\n\n$$\n\\begin{array}{r l} & {\\langle (s _ {i}) > s _ {j} + s _ {i} <   s _ {j} > - <   s _ {i} > <   s _ {j};} \\\\ & {= \\langle s _ {i} > <   s _ {j} > \\cong <   s _ {i} > s _ {j} >} \\end{array}\n$$\n\n$$\nE = - J \\sum_ {i j} s _ {i} s _ {j} - h \\geq s _ {j}\n$$\n\n$$\n\\text {   I   sum   over   different   pairs   }\n$$\n\n$$\n\\begin{array}{l} \\simeq - \\sqrt {\\frac {2}{<   i : >}} (<   s _ {i} > s _ {j} + s _ {i} <   s _ {j} > - <   s _ {i} > <   s _ {j} >) - h \\sum_ {i} s _ {i} \\\\ <   s _ {i} > = <   s > (d o n o f d a p e n d o n i) \\\\ \\end{array}\n$$\n\n$$\n= - z J \\sum_ {i} <   s > s _ {i} - h \\overline {{z}} s _ {i} + J <   s > ^ {2} \\frac {1}{2} z N _ {\\text { spin }}\n$$\n\n$$\n= - (\\overbrace {h + z J <   s >}) \\frac {2}{i} s _ {i} + \\frac {1}{2} z N _ {\\text { spin }} J <   s > ^ {2}\n$$\n\n$$\n= E _ {\\text { mean }} \\quad \\text { coordination   number. } \\quad \\text { 44个个 } z = 2\n$$\n\n$$\n\\begin{array}{l} z = \\sum_ {\\{s \\}} e ^ {- \\beta E} \\simeq \\sum_ {\\{s \\}} e ^ {- \\beta E _ {m e o n}} \\# z = 4 \\\\ = (e ^ {- \\beta h _ {\\text { eff }}} + e ^ {\\beta h _ {\\text { eff }}}) ^ {N _ {\\text { spin }}} e ^ {- \\frac {1}{2} \\beta z N _ {\\text { spin }} J <   s > ^ {2}} \\\\ \\end{array}\n$$\n\n$$\nA = \\left[ - k _ {B} T \\ln (e ^ {- \\beta h _ {\\text { eff }}} + e ^ {\\beta h _ {\\text { eff }}}) + \\frac {1}{2} \\geq J <   s > ^ {2} \\right] N _ {\\text { spin }}\n$$\n\n$$\nh _ {a f f} = z J <   s > + h\n$$\n\nBut what is <s>\n\n$$\n\\text { calculate } \\quad <   s > \\text { using } E _ {\\text { mean }} = - h _ {\\text { eff }} \\frac {2}{i} S _ {i} + \\frac {3}{2} N _ {\\text { spin }} J <   s > ^ {2}\n$$\n\nIndependent spin system within meanfield approximation\n\n$$\n<   s > = \\frac {e ^ {+ \\beta h _ {\\text { eff }}} - e ^ {- \\beta h _ {\\text { eff }}}}{e ^ {\\beta h _ {\\text { eff }}} + e ^ {- \\beta h _ {\\text { eff }}}} = \\tanh (\\beta h _ {\\text { eff }})\n$$", "source": "mit-ocw", "source_doc_id": "0003-iii-ising-model-3cc4739a-31b3b79d", "source_title": "III. Ising model.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["III. Ising model.", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-iii-ising-model-3cc4739a-31b3b79d:page-0003", "text": "$$\n\\langle s \\rangle = \\tanh [ (h + z) <   s > \\beta ] \\quad \\text { self   constant } _ {\\text { equation }}\n$$\n\n$$\n\\text { solve   self   consistent   equation   for } \\langle s \\rangle = f (T, h)\n$$\n\n$$\n\\text { then } A (T, h, <   s >) \\rightarrow A (T, h, f (T, h)) \\text {   meaningful   free   energy.   }\n$$\n\n$$\n\\text { Second   way:   minimize   } A (T, h, <   s >) \\text {   resp.   } z _ {0} <   s >\n$$\n\n$$\n\\frac {\\partial A}{\\partial \\langle s \\rangle} = - \\frac {e ^ {\\beta h _ {e f f}} - e ^ {- \\beta h _ {e f f}}}{e ^ {\\beta h _ {e f f}} - e ^ {- \\beta h _ {e f f}}} z J + z J <   s > = 0\n$$\n\n$$\n\\Rightarrow \\boxed {<   s > = \\tanh (\\beta h e f f)} \\quad \\text { same   as   the }\n$$\n\n$$\n\\text { Phase   transition:   for } h = 0\n$$\n\nA(T,0,<s>)\nT>Tc\nA\nT=Tc\nA(T,0,<s>)\nT<Tc\n<s>\n<s>\ntanh(β≥J<s>)\ns>\nsymmetry breaking\n<s>\norder\nparameter\n↑\n<s>≠0\nTc\nT\n\n$$\nF _ {n} \\quad s m a l l <   s) \\quad \\tanh (\\beta z J <   s >) \\simeq \\beta z J <   s >\n$$\n\n$$\na + T _ {c} \\quad \\beta \\in J = 1 \\Rightarrow T _ {c} = \\frac {z J}{k _ {B}}\n$$", "source": "mit-ocw", "source_doc_id": "0003-iii-ising-model-3cc4739a-31b3b79d", "source_title": "III. Ising model.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["III. Ising model.", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-iii-ising-model-3cc4739a-31b3b79d:page-0004", "text": "③Gingburg - Landan Theory\n\nNeon the Transition, order parameter <s> is small\n\nExpand  $A(T, h=0, <s>)$\n\n$A_{0}(T)=-k_{B}T\\ln2N_{sp}$\n\n$$\nA = \\frac {N _ {0} i n}{2} a <   s > ^ {2} + \\frac {N _ {0} i n}{4} b <   s > ^ {4} + A _ {0} ^ {(T)}\n$$\n\n$$\n\\frac {\\partial A}{\\partial \\langle s \\rangle} = 0   \\Rightarrow a <   s > + b <   s s ^ {3} = 0\n$$\n\na, b: function of T.\n\nAt Transition   $a(T_{c})=0$$b(T_{c})>0$\n\nWe may write   $a = a_{0} \\left( \\frac{T}{T_{c}} - 1 \\right)$$a_{0} > 0$\n\n$$\n\\begin{array}{l} {<   s > = \\left\\{ \\begin{array}{l l} {0, T > T _ {c}} \\\\ {\\pm \\sqrt {\\frac {a _ {0}}{b}} \\sqrt {1 - \\frac {T}{T _ {c}}},} & {T <   T _ {c}} \\\\ {\\hline t \\equiv (1 - \\frac {T}{T _ {c}}) ^ {L} \\pm \\sqrt {\\frac {a _ {0}}{b}}} \\end{array} \\right.} \\\\ {\\mathrm{Let}} \\end{array}\n$$\n\n<s>\nTc T\n\n$\\langle s\\rangle\\propto(t)^{\\beta}\\quad\\beta=1/2\\quad\\left(\\begin{array}{c}order\\\\magnetigation\\end{array}\\right)$\n\n$$\nA = \\left\\{ \\begin{array}{l l} A _ {0} (t) & T > T _ {c} \\\\ - \\frac {N _ {\\text { spin }}}{4} \\frac {a _ {0} ^ {2} t ^ {2}}{b} + A _ {0} (t) & T <   T _ {c} \\end{array} \\right.\n$$\n\n| T     | A     |\n|-------|-------|\n| 0     | 0     |\n| 1     | 0.5   |\n| 2     | 1     |\n| 3     | 1.5   |\n| 4     | 2     |\n| 5     | 2.5   |\n| 6     | 3     |\n| 7     | 3.5   |\n| 8     | 4     |\n| 9     | 4.5   |\n| 10    | 5     |\n| 11    | 5.5   |\n| 12    | 6     |\n| 13    | 6.5   |\n| 14    | 7     |\n| 15    | 7.5   |\n| 16    | 8     |\n| 17    | 8.5   |\n| 18    | 9     |\n| 19    | 9.5   |\n| 20    | 10    |\n| 21    | 10.5  |\n| 22    | 11    |\n| 23    | 11.5  |\n| 24    | 12    |\n| 25    | 12.5  |\n| 26    | 13    |\n| 27    | 13.5  |\n| 28    | 14    |\n| 29    | 14.5  |\n| 30    | 15    |\n| 31    | 15.5  |\n| 32    | 16    |\n| 33    | 16.5  |\n| 34    | 17    |\n| 35    | 17.5  |\n| 36    | 18    |\n| 37    | 18.5  |\n| 38    | 19    |\n| 39    | 19.5  |\n| 40    | 20    |\n| 41    | 20.5  |\n| 42    | 21    |\n| 43    | 21.5  |\n| 44    | 22    |\n| 45    | 22.5  |\n| 46    | 23    |\n| 47    | 23.5  |\n| 48    | 24    |\n| 49    | 24.5  |\n| 50    | 25    |\n| 51    | 25.5  |\n| 52    | 26    |\n| 53    | 26.5  |\n| 54    | 27    |\n| 55    | 27.5  |\n| 56    | 28    |\n| 57    | 28.5  |\n| 58    | 29    |\n| 59    | 29.5  |\n| 60    | 30    |\n| 61    | 30.5  |\n| 62    | 31    |\n| 63    | 31.5  |\n| 64    | 32    |\n| 65    | 32.5  |\n| 66    | 33    |\n| 67    | 33.5  |\n| 68    | 34    |\n| 69    | 34.5  |\n| 70    | 35    |\n| 71    | 35.5  |\n| 72    | 36    |\n| 73    | 36.5  |\n| 74    | 37    |\n| 75    | 37.5  |\n| 76    | 38    |\n| 77    | 38.5  |\n| 78    | 39    |\n| 79    | 39.5  |\n| 80    | 40    |\n| 81    | 40.5  |\n| 82    | 41    |\n| 83    | 41.5  |\n| 84    | 42    |\n| 85    | 42.5  |\n| 86    | 43    |\n| 87    | 43.5  |\n| 88    | 44    |\n| 89    | 44.5  |\n| 90    | 45    |\n|      |       |\n\n$$\nC _ {v} = - \\left(\\frac {\\partial^ {2} A}{\\partial T ^ {2}}\\right) T \\propto 1 + 1 ^ {- \\alpha} \\quad \\alpha = 0\n$$\n\n$$\n= T \\frac {2 S}{\\partial T}\n$$\n\nC_V\nT", "source": "mit-ocw", "source_doc_id": "0003-iii-ising-model-3cc4739a-31b3b79d", "source_title": "III. Ising model.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["III. Ising model.", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-iii-ising-model-3cc4739a-31b3b79d:page-0005", "text": "$$\nh \\neq 0 \\cdot \\text {   case   }.\n$$\n\n$$\n- \\frac {2}{i} s _ {i} h\n$$\n\n$$\nA = A _ {0} (T) + \\frac {N _ {\\text { spin }}}{2} a <   s > ^ {2} + \\frac {N _ {\\text { spin }}}{4} b <   s > ^ {4} - N _ {\\text { spin }} <   s > h\n$$\n\n$$\n\\text { Free   energy } \\quad \\text { as } \\quad A (T, N _ {\\text { spin }}, h) = A _ {\\min}\n$$\n\n$$\n\\star T > T _ {c} \\quad (a > 0) \\quad \\& \\quad s m a l l h.\n$$\n\n$$\nA \\simeq A _ {0} + \\frac {N _ {\\text { spin }}}{2} a <   s > ^ {2} - N _ {\\text { spin }} h <   s >\n$$\n\n$$\n\\frac {\\partial A}{\\partial (s)} = \\frac {N _ {\\text { spin }}}{a (s)} - N _ {\\text { spin }} h = 0\n$$\n\n$$\nA _ {\\min} = - \\frac {1}{2} \\frac {h ^ {2}}{a} N _ {\\text { pin }} + O (h ^ {4})\n$$\n\n$$\n\\text { Magnetization }\n$$\n\n$$\nM = - \\frac {\\partial A (T , N _ {s p i n} , h)}{\\partial h} = - \\frac {\\partial A _ {m i n}}{\\partial h} = \\sum_ {i} s _ {i}\n$$\n\n$$\n= + \\frac {N _ {\\text { spin }}}{a} h\n$$\n\n$$\n\\text { magnetic } \\dots \\text { susceptibility }\n$$\n\n$$\nx = \\frac {1}{V} \\frac {\\partial M}{\\partial h} = + \\frac {N _ {\\text { spin }}}{a V}\n$$\n\n$$\n\\text { quick   way   to   get } x:\n$$\n\n$$\nA = \\frac {1}{2} W _ {\\text { spin }} (s) ^ {2} - h W _ {\\text { spin }} (s)\n$$\n\n$$\n\\left(\\frac {N _ {\\text { spin }} ^ {2} k}{N _ {\\text { spin }} a} \\frac {1}{V} = x \\right.\n$$\n\n$$\na s T \\rightarrow T _ {c}\n$$\n\n$$\n\\gamma \\propto t ^ {- \\gamma} \\quad \\gamma = 1 \\quad \\uparrow_ {t _ {1} ^ {- 1}} ^ {\\gamma} \\downarrow_ {t _ {1} ^ {- 1}}\n$$\n\n$$\n\\star T <   \\tau_ {c}\n$$\n\n$$\n<   S > _ {\\min} = \\sqrt {\\frac {a}{b}}\n$$\n\n$$\n\\text { Let } \\quad <   s > = <   s _ {\\min} > + s <   s >\n$$\n\n$$\nA = N _ {\\text { pin }} \\frac {1}{2} (a + 3 b <   s > _ {\\min} ^ {2}) (s <   s >) ^ {2} - N _ {\\text { spin }} h - s <   s >\n$$\n\n$$\nx = \\frac {N _ {\\mathrm{sp} , n}}{- 2 a N _ {\\mathrm{sp} , n}} V \\propto 1 + 1 ^ {- y} \\quad y = 1\n$$", "source": "mit-ocw", "source_doc_id": "0003-iii-ising-model-3cc4739a-31b3b79d", "source_title": "III. Ising model.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["III. Ising model.", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-iii-ising-model-3cc4739a-31b3b79d:page-0006", "text": "④ Phase diagram for   $h \\neq 0$\n\n* if h≠0 & fixed, A $_{min}$ (T)\n\nsmooth function of T.   $N_{0}$\n\nphase transition.\n\nA\nhigh T\n<<>\nlow T\n\nNo singular\n\n* T fixed:\n\nA\nh>0\nh<0\n<s>\nhigh T\nA\nh>0\n<s>\nh<0\nlow T\n\n$M = +\\sqrt{\\frac{-a}{b}} N_{spin}$\n\nif first order\n\nA(h,T) has\nsingularity\n\n$M = -\\sqrt{\\frac{-a}{b}} N_{spin}$\n\nsecond order.(h=0)\n\nA\nh=0\nfirst order\nTransition\npoints\n↓\nA\n<CS>\nsecond order\n\nActions first order transition\n\n$$\n\\Delta M = 2 \\sqrt {\\frac {- a}{b}}\n$$\n\n* Latend heat = ΔST\n\n$$\nA _ {\\min} ^ {1} = - \\frac {N _ {\\min}}{2} a ^ {2} / b = A _ {\\min} ^ {2}\n$$\n\n$$\nS _ {1} = - \\frac {\\partial A}{\\partial T} = \\frac {- 1 6 \\pi}{2} \\frac {\\partial}{\\partial T} (\\frac {a ^ {2}}{b}) = S _ {2}\n$$\n\n$$\n\\Delta S = 0\n$$\n\ndue to  $\\frac{1}{5}\\Rightarrow-\\frac{1}{5};$ symmetry\n\nLatend heat = 0 But ingeneral ∆S ≠ 0\n\nsecond order\nΔM=0\nT\n\nΔm\nh", "source": "mit-ocw", "source_doc_id": "0003-iii-ising-model-3cc4739a-31b3b79d", "source_title": "III. Ising model.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["III. Ising model.", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0003-iii-ising-model-3cc4739a-31b3b79d:page-0007", "text": "27\n\n⑤ Application of G-L theory\n\n* Vapo-Water phase diagram\n\ndensity   $n = n_{e}(P, T) + \\frac{sn}{n}$\n\nChoose  $n_c(P,T)$ to make: (play a role of order parameter\n\nP\nh = 0\nsu\nno\nh = r₀ = 0\ncritical point\nof\nT\n\n$G(T,P;n)=h(T,P)\\sin+\\gamma_{0}(T,P)\\sin^{2}+u_{0}\\sin^{4}$\n\n(make  $\\delta u^{3}$ term disappear).\n\nFirst order transition line is given by\n\nh = 0 near the critical point\n\n$\\left\\{  \\begin{array}{l} h\\left( {{P}_{c},{T}_{c}}\\right)  = 0 \\\\  {r}_{0}\\left( {{P}_{c},{T}_{c}}\\right)  = 0 \\end{array}\\right.$ gives  $\\left( {{P}_{c}{T}_{c}}\\right)$ of the critical point\n\n$N_{em}$  (  $P_{c}, T_{c}$ ):\n\nCritical point for water is the same as\n\nthe critical point for Ising model.\n\ndue to the same G-L theory\n\nWe see again the universality", "source": "mit-ocw", "source_doc_id": "0003-iii-ising-model-3cc4739a-31b3b79d", "source_title": "III. Ising model.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["III. Ising model.", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0001", "text": "IV. Grand canonical ensemble\n\n```mermaid\ngraph LR\nA[\"heat bath T\"] --> B[\"(T,μ) sys\"]\nB <--> C[\"microcanonical ensemble\"]\nC --> D[\"particle reservoir μ\"]\n```\n\ngrand\n\ncanonical\nensemble\n\n① Grand partition function = partition function of combined system\n\n$$\n\\text { Energy   of   combine   system }\n$$\n\n$$\nE _ {n, N} + N _ {\\mathrm{res}} \\mu\n$$\n\n$$\nE _ {\\text { system }} \\quad I _ {\\text { partiil   reservoir }}\n$$\n\n$$\n\\boxed {\\mathrm{sys}} \\longleftrightarrow \\boxed { \\begin{array}{c} \\text { panTilc } \\\\ \\text { reservoir } \\end{array} }\n$$\n\n$$\n\\text { energy } y = N _ {\\text { res }} \\mu\n$$\n\n$$\n= E _ {n, N} - N \\mu + N _ {\\mathrm{tot}} \\mu\n$$\n\n$$\n\\begin{array}{l} {\\mathrm{partitionfunctionofcombinedsystem}} \\\\ {\\mathrm{orgrandpartitionfunctionofthesystem}} \\end{array} = \\sum_ {n, N} e ^ {- \\beta (E _ {n, N} - N \\mu)}\n$$\n\n$$\n= Q _ {G}\n$$\n\n$$\nQ _ {G} (\\mu , V; T) = \\sum e ^ {\\beta \\mu N} Q _ {N} (V, T)\n$$\n\n$$\n\\rightarrow \\text { partition   function   of } N \\text { particles }\n$$\n\n$$\n= \\sum_ {j} s ^ {N} Q _ {N} (V, T) \\quad \\boxed {z = e ^ {\\beta N}} f u g a c i t y\n$$", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0002", "text": "③ thermodynamical quantities:\n\n$$\n* \\frac {\\partial}{\\partial V} \\ln Q _ {G} = \\frac {\\sum_ {n , N} - \\beta \\frac {\\partial E _ {n , N}}{\\partial V} e ^ {- \\beta (E _ {n , N} - N \\mu)}}{\\sum_ {n , N} e ^ {- \\beta (E _ {n , N} - \\mu N)}}\n$$\n\n$$\n= - \\beta <   \\frac {\\partial E _ {n} N}{\\partial V} > = + \\beta P\n$$\n\n$$\n\\Rightarrow \\boxed {P = - \\frac {\\partial \\Omega (V , T , \\mu)}{\\partial V} | _ {\\mu , T}}\n$$\n\n$$\n\\begin{array}{l} N _ {0} T _ {2} \\quad \\Omega = V W (T, \\mu) \\quad \\begin{array}{l l} w & \\text { is   intensive. } \\\\ n o t & \\text { depend   on   extensive } \\end{array} \\\\ \\Rightarrow W (T, \\mu) = - P (T, \\mu) \\\\ \\end{array}\n$$\n\n$$\nn _ {1} \\boxed {\\Omega = - P V}\n$$\n\n$$\n* \\frac {\\partial \\Omega}{\\partial T} \\bigg | _ {v, \\mu} = \\frac {\\partial [ A (v , T , N) - \\mu N ]}{\\partial T \\cap N (\\mu , v , T)} \\bigg | _ {v, \\mu}\n$$\n\n$$\n= \\left. \\frac {\\partial A (v , T , N)}{\\partial T} \\right| _ {V, N} + \\left. \\frac {\\partial A (v , T , N)}{\\partial N} \\right| _ {V, T} \\left. \\frac {\\partial N}{\\partial T} \\right| _ {V, M}\n$$\n\n$$\n= \\frac {\\partial A}{\\partial T} | _ {V, N} = - S \\quad \\text { Dependent   of }\n$$\n\n$$\n\\left| \\frac {\\partial S}{\\partial T} \\right| _ {V, \\mu} = - S\n$$\n\n$$\n\\frac {\\partial \\Omega}{\\partial \\mu} = \\frac {\\partial (A (V , T , N (V , T , \\mu)) - \\mu N (V , T ; \\mu))}{\\partial \\mu}\n$$\n\n$$\n= \\frac {\\partial (A - \\mu N)}{\\partial N} \\frac {\\partial N}{\\partial \\mu} - N = - \\mu \\Rightarrow N = - \\frac {\\partial \\Omega}{\\partial \\mu}\n$$\n\n$$\n\\text { Dependent   of } T\n$$\n\n$$\nx h _ {2 0 0 g} (N (\\mu , V, T))\n$$\n\n$$\n\\text { The   two   lems }\n$$\n\n$$\n\\text { cancel. }\n$$", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0003", "text": "② Average number and number fluctuations.\n\n$$\nQ _ {G} = \\sum_ {n, N} e ^ {- \\beta (E _ {n, N} - N \\mu)}\n$$\n\n→∞ prob. for the system 20 have N particles and in nth state.\n\n$$\n= \\sum_ {N} e ^ {\\beta \\mu N} Q _ {N}\n$$\n\n$\\rightarrow \\infty \\text{ prob. for the system to have N particles}$\n\n$$\n\\langle N \\rangle = \\frac {\\sum_ {N} N e ^ {\\beta \\mu N} Q _ {N}}{\\sum_ {N} e ^ {\\beta \\mu N} Q _ {N}}\n$$\n\nsystem want to\n\nmaximize  $e^{PHN} Q_N = e^{-\\beta t}$\n\nor minimize Ω\n\n$$\n\\langle N \\rangle = - \\frac {\\partial}{\\partial \\mu} \\Omega (\\mu , v, T)\n$$\n\nThe map potential  $\\Omega \\equiv -k_{B}T\\ln Q_{G}$\n\nLet  $\\overline{N}$ maximize  $e^{\\beta_{\\mu}N} Q_N = e^{\\beta_{\\mu}N} e^{-\\beta A_N}$\n\n$$\nQ _ {G} \\approx e ^ {\\beta r \\overline {{N}}} Q _ {\\overline {{N}}} = e ^ {- \\beta (A \\overline {{N}} - r \\overline {{N}})} = e ^ {- \\beta \\Omega}\n$$\n\n$$\n\\Rightarrow \\boxed {\\Omega = A - \\mu N} \\quad \\& \\boxed {\\overline {{N}} = <   N >}\n$$\n\nFluctuation of N:   $e^{\\beta N}e^{-\\beta A_{N}} \\approx e^{-\\frac{1}{2}\\beta\\left.\\frac{\\partial^{2}A_{N}}{\\partial N^{2}}\\right|_{N=\\overline{N}}(N-\\overline{N})^{2}}$\n\nSecond way:   $-\\frac{\\partial^{2}}{\\partial\\mu^{2}}\\Omega = +\\frac{\\beta\\Sigma N^{2}e^{\\beta\\rho N}\\mathcal{Q}_{N}}{\\Sigma e^{\\beta\\rho N}\\mathcal{Q}_{N}} - \\frac{\\beta(\\Sigma N e^{\\beta\\rho N}\\mathcal{Q}_{N})^{2}}{(\\Sigma e^{\\beta\\rho N}\\mathcal{Q}_{N})^{2}}$\n\n$$\n= \\beta \\langle N ^ {2} \\rangle - (\\langle N \\rangle) ^ {2}\n$$\n\n$$\n\\langle (N - \\overline {{N}}) ^ {2} \\rangle = - k _ {B} T \\frac {\\partial^ {2}}{\\partial \\mu^ {2}} \\Omega (\\mu , V, T)\n$$", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0004", "text": "4 Applications\n\n① Two-classical gases\n\nhow  $n_{1}$ &  $n_{2}$ are related:\n\nSimple hand-drawn diagram of a rectangle divided into two sections with diagonal hatching (no text or symbols)\n\nV\nV₀\n\ndensity:   $n_{1}$  ...   $n_{2}$\n\n$$\n\\Omega_ {1} = A _ {1} (V _ {1}, T, N _ {1}) = \\mu N _ {1}\n$$\n\n$$\n= N _ {1} k _ {B} T (\\ln n _ {1} \\lambda_ {1} ^ {3} - 1) - \\mu N _ {1}\n$$\n\n$$\n\\lambda_ {1} = \\sqrt {2 \\pi k ^ {2} / m k T}\n$$\n\n$P_{i,b}: P(N_{i}) \\propto e^{-\\frac{\\Omega}{k_{B}T}}$\n\n$N_{1}$  minimize   $\\Omega$\n\nparticle\n\n```mermaid\ngraph TD\nA[\"Input 1\"] --> C[\"M\"]\nB[\"Input 2\"] --> C[\"M\"]\nC[\"M\"] --> D[\"Output\"]\n```\n\nparticle\nreservoi\n\n$\\Rightarrow  \\frac{\\partial A}{\\partial N}\\bigg|_{V,T} = \\mu  = {k}_{B}T\\left\\lbrack  {{l}_{n}\\left( {n,{\\lambda }_{1}^{3}}\\right)  - 1}\\right\\rbrack   + {k}_{B}T$\n\n$\\mu = k_{B}T \\ln(n, \\lambda_{i}^{3})$ K chemical potential of ideal gas\n\n$$\n\\begin{array}{l} n _ {1} = \\frac {1}{\\lambda_ {1} ^ {3}} e ^ {+ \\frac {\\mu}{k _ {B} T _ {1}}} \\quad \\Omega = N k _ {B} T (\\frac {\\mu}{k _ {B} T} - 1) - \\mu N \\\\ = - N k _ {B} T = - V \\frac {1}{\\lambda^ {2}} e ^ {+ \\frac {\\mu}{k _ {B} T}} \\\\ \\end{array}\n$$\n\n$$\n\\Omega_ {2} = A _ {2} (V _ {2}, T, N _ {2}) - \\mu N _ {2} \\quad \\boxed {\\Omega = - V \\frac {1}{x ^ {3}} e ^ {\\mu / k _ {B} T}}\n$$\n\n$$\n\\begin{array}{l} = N _ {2} k _ {B} T \\left[ \\ln (n _ {2} \\lambda_ {2} ^ {3}) - 1 \\right] - \\mu N _ {2} \\\\ + N _ {2} V _ {0} \\quad \\text {   i.e.   } A = E - S T \\quad E \\text {   is   shifted   by   } N _ {2} V _ {0} \\\\ \\end{array}\n$$\n\n$$\n\\mu = \\frac {\\partial A _ {2}}{\\partial N _ {2}} = k _ {B} T _ {2} \\ln (n _ {2} \\lambda_ {2} ^ {3}) + V _ {0}\n$$\n\n$$\nn _ {2} = \\frac {1}{\\lambda_ {2} ^ {3}} e ^ {+ \\frac {\\mu - V _ {0}}{R _ {B} T _ {2}}}\n$$", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0005", "text": "$n_{1}$ &  $n_{2}$ is related such that\n\nfor O₂\ng mO₂: /kBT 300K\n=1.3×10⁻⁴/m\n=0.13/km\nnO₂(10km)/nO₂(0) = e⁻¹·³ = 0.27\n△ Adiso2p. sion on surface\nλ₁=λ₂⇒\nnZ/n₁ = e⁻V₀/kBT\nnO₂(0)/nN₂(0) ≈ ¼\nnO₂(10km)/nN₂(10km)=?\nT,P\nchemical potentials\nare equal in the\nequilibrium state\n\nsystem 1: classical gas\n\nsystem 2: ☐ in □\n\nenergy cost E0\n\nsystem 1 & system 2 has the same μ.\n\nchemical potential of classical gas\n\n$$\n\\mu = k _ {B} T \\ln (n, \\lambda^ {3}) \\quad P = k _ {B} T n\n$$\n\n$$\n\\begin{array}{r l} {P _ {n} \\propto e ^ {- \\frac {E _ {n} - \\mu N}{k _ {B} T}}} & {= k _ {B} T \\ln (\\frac {P}{k _ {B} T} \\lambda^ {3}) \\quad \\lambda = \\sqrt {2 \\pi k ^ {2} / m k T}} \\\\ {P (T) \\propto e ^ {- \\frac {\\varepsilon_ {0} - \\mu n}{k _ {B} T}} =} & {e ^ {- \\frac {\\varepsilon_ {0} - \\mu}{k _ {B} T}}} \\end{array}\n$$\n\n$$\nP (\\square) \\propto e ^ {\\frac {- 0 - M \\cdot D}{k _ {B} T}} = 1\n$$\n\naverage # of particles per site\n\n$$\nP = \\frac {e ^ {- \\frac {\\xi_ {0} - k}{k _ {B} T}}}{1 + e ^ {- \\frac {\\xi_ {0} - k}{k _ {B} T}}} = \\frac {1}{1 + e ^ {\\frac {\\xi_ {0} - k}{k _ {B} T}}} = \\frac {1}{1 + \\frac {k _ {B} T}{\\lambda^ {3} P} e ^ {\\varepsilon_ {0} / k _ {B} T}}\n$$", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0006", "text": "$$\np = \\frac {1}{1 + e ^ {\\frac {\\epsilon_ {0}}{k _ {B} T}} - \\ln \\frac {P}{k _ {B} T} \\lambda^ {3}} = \\boxed {\\frac {1}{1 + \\frac {k _ {B} T}{\\lambda^ {3} P}} e ^ {\\frac {\\epsilon_ {0}}{k _ {B} T}}}\n$$\n\n$$\np \\rightarrow 1 \\quad \\text { as } \\quad p \\rightarrow \\infty\n$$\n\n$$\np \\rightarrow 0 \\quad \\text {   as   } p \\rightarrow 0\n$$\n\n$$\n\\text { when } \\rho <   1 \\quad \\rho = \\frac {P \\lambda^ {3}}{k _ {B} T} e ^ {- \\varepsilon_ {0} / k _ {B} T}\n$$\n\n$$\n= n \\lambda^ {3} e ^ {- \\epsilon_ {0} / k _ {B} T}\n$$\n\n⑤ Chemical reactions\n\n$$\n2 H \\rightleftharpoons H _ {2} + \\epsilon_ {0}\n$$\n\n$$\n\\mathrm{or} 2 H - H _ {2} = \\varepsilon_ {0}\n$$\n\n$$\nn \\boxed {\\sum_ {i} v _ {i} x _ {i} = \\epsilon_ {0}}\n$$\n\n$$\nv _ {1} = 2 \\quad v _ {2} = - 1\n$$\n\n$$\nX _ {1} = H - X _ {2} = H _ {2}\n$$\n\nAt T, P what is   $n_{H}/n_{H_{2}}$\n\nThe change of  $N_i$ due to the reaction\n\n$$\n\\frac {s N _ {1}}{v _ {1}} = \\frac {s N _ {2}}{v _ {2}} = \\frac {s N _ {i}}{v _ {i}}\n$$\n\nTotal Free energy   $\\sum A_{i}(V,T,N_{i})=A_{tot}$\n\nChemical reaction minimize After and\n\nreach equilibrium.", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0007", "text": "A+ equilibzum :\n\n$$\ns A _ {t o r} = \\sum_ {i} s N _ {i} \\frac {\\partial A _ {i}}{\\partial N _ {i}} = 0\n$$\n\n$$\n\\Rightarrow \\boxed {\\sum_ {i} v _ {i} \\mu_ {i} = 0}\n$$\n\nequilibraim condition\n\nfor one chemical reaction.\n\none condition for each reaction\n\n$$\n2 H \\rightleftharpoons H _ {2} + C _ {0}\n$$\n\ninternal energy\n\n$$\n\\epsilon_ {H}\n$$\n\n$$\n\\epsilon_ {H _ {2}}\n$$\n\n$$\n2 \\epsilon_ {H} - \\epsilon_ {W _ {2}} = \\epsilon_ {0}\n$$\n\n$$\nA _ {H} = N _ {H} k _ {B} T (\\ln n _ {H} \\lambda_ {H} ^ {3} - 1) + N _ {H} \\varepsilon_ {H}\n$$\n\n$$\n\\mu_ {H} = \\frac {\\partial A _ {H}}{\\partial N} = k _ {B} T \\ln (n _ {H} \\lambda_ {H} ^ {3}) + \\epsilon_ {H}\n$$\n\n$$\nM _ {H _ {2}} = k _ {B} T \\ln (n _ {H _ {2}} \\lambda_ {H _ {2}} ^ {3}) + \\epsilon_ {H _ {2}}\n$$\n\n$$\n\\sum_ {i} v _ {i} \\mu_ {i} = 0 \\Rightarrow 2 ^ {\\mu_ {H}} - \\mu_ {H _ {2}} = 0\n$$\n\n$$\n2 k _ {B} T \\ln (n _ {H} \\lambda_ {H} ^ {3}) - k _ {B} T \\ln (m _ {H _ {2}} \\lambda_ {H _ {2}} ^ {5}) = - (2 \\epsilon_ {H} - \\epsilon_ {H _ {2}})\n$$\n\n$$\n\\frac {(n _ {H} \\lambda_ {H} ^ {3}) ^ {2}}{n _ {H _ {2}} \\lambda_ {H _ {2}} ^ {3}} = e ^ {- \\frac {2 \\epsilon_ {H} - \\epsilon_ {H _ {2}}}{k _ {B} T}}\n$$\n\n$$\n\\frac {n _ {H} ^ {2}}{n _ {H _ {2}}} = \\frac {1}{(2 \\pi k ^ {2} / m k T) ^ {3 / 2}} \\cdot \\frac {1}{2 ^ {3 / 2}} \\cdot e ^ {- \\frac {\\varepsilon_ {0}}{k _ {B} T}}\n$$\n\n$$\n\\text { in   general } \\quad \\prod_ {i} (\\lambda^ {3} n _ {i}) ^ {\\nu_ {i}} = e ^ {- \\epsilon_ {0} / k _ {B} T}\n$$\n\nfor:\n\n$$\n\\overline {{2}} \\nu_ {i} X _ {i} = \\epsilon_ {0}\n$$", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0004-iv-grand-canonical-ensemble-c7093fe4-78830874:page-0008", "text": "$$\n2 H = H _ {2} + 4 3 6 k J / m o l\n$$\n\nat 1 atm & 300K  $n_{H}$=?\n\n$$\n\\begin{array}{l} n _ {H _ {2}} = ? \\quad P V = k _ {B} N T \\\\ \\Rightarrow \\begin{array}{c} n _ {H _ {2}} = \\frac {P}{k _ {B} T} \\\\ = 2. 4 \\times 1 0 ^ {1 9} / c m ^ {3} \\end{array} \\\\ \\end{array}\n$$\n\n$$\nC _ {0} = \\frac {(3 6 k J)}{m o l} = 7. 2 4 \\times 1 0 ^ {- 1 2} e g = 4. 5 e V\n$$\n\n$$\n6 0 / k _ {B} \\cdot 3 0 0 = 1 7 5\n$$\n\n$$\n\\begin{array}{l} \\mathrm{cgs} \\quad \\mathrm{unit} \\\\ 1 a T m = 1 0. 1 N / c m ^ {2} \\\\ = 1 0 ^ {6} d y n e / c m ^ {2} \\\\ k _ {B} = 1. 3 8 \\times (0 - 1 6 e g / k \\\\ T = 3 0 0 k \\\\ m _ {p} = 1. 6 8 \\times 1 0 ^ {- 2 4} g \\\\ \\end{array}\n$$\n\n$$\n\\frac {n _ {H} ^ {2}}{n _ {H _ {2}}} = 3. 5 \\times 1 0 ^ {- 2 3} / c m ^ {3} e ^ {- 1 7 5} = 3. 5 \\times 1 0 ^ {- 5 3} / c m ^ {3}\n$$\n\n$$\nn _ {M} = \\sqrt {2 . 4 \\times 1 0 ^ {1 9} \\times 3 . 5 \\times 1 0 ^ {- 5 3}} = 2. 9 \\times 1 0 ^ {- 1 7} / c m ^ {3}\n$$\n\nd) No spin order in ID Ising model   $E = -\\sqrt{2}$$s_{i}$$s_{ii}$\n\n↑↑↑↓↓↓↓↓↑↑↑↑↑↑↓↓↓↓↓\ndomain A B A\nwall   $\\epsilon_{A}=2J$$G_{B}=2J$\n\n$$\n\\begin{array}{l} A + B = \\epsilon_ {A} + \\epsilon_ {B} = \\epsilon_ {0} = 4 J \\\\ \\mathrm{not} \\lambda_ {A} ^ {3} \\mathrm{since} 1 D \\\\ \\Rightarrow (\\lambda_ {A} n _ {A}) (\\lambda_ {B} n _ {B}) = e ^ {\\frac {1}{k _ {B}}} \\\\ \\end{array}\n$$\n\n$$\n\\lambda_ {A} = ? \\quad \\lambda_ {B} = ?\n$$\n\n$$\n\\text { meaning   of } \\lambda^ {3} n = \\frac {\\lambda^ {3}}{v} = \\frac {1}{\\text { of   state   per   particles }}\n$$\n\n$$\ns _ {0} \\lambda_ {A} n _ {A} = n _ {A} (o _ {2} \\lambda_ {A} = \\lambda_ {B} = 1)\n$$\n\n$$\n\\Rightarrow n _ {A} = n _ {B} \\approx e ^ {- 2 J / k _ {B} T} \\quad \\uparrow \\uparrow \\uparrow \\uparrow \\downarrow \\downarrow \\downarrow \\downarrow \\downarrow \\downarrow \\cdot \\uparrow \\uparrow \\uparrow \\uparrow\n$$\n\n$$\n\\mathrm{个} _ {\\text { per   link }}\n$$\n\n$$\nk - e = 1 / k _ {8} T \\rightarrow 1\n$$", "source": "mit-ocw", "source_doc_id": "0004-iv-grand-canonical-ensemble-c7093fe4-78830874", "source_title": "IV. Grand canonical ensemble", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["IV. Grand canonical ensemble", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-v-random-variable-15a531e0-d13b76bb:page-0001", "text": "V. Random variable\n\n① One random variable. s\n\nProb. distribution   $P(s)$ ,   $\\sum_{s} P(s) = 1$\n\n$$\n\\begin{array}{l} \\text { Average } \\quad <   s > = \\sum_ {s} s ^ {-} P (s) \\\\ \\text { moments } \\quad <   s ^ {n} > = \\sum_ {s} s ^ {n} P (s) \\\\ \\end{array}\n$$\n\n$$\nF l u c T u a T i o n:\n$$\n\n$$\n\\Delta S = \\sqrt {\\langle (s - \\langle s \\rangle) ^ {2} \\rangle} = \\sqrt {\\langle s ^ {2} \\rangle - \\langle s \\rangle^ {2}}\n$$\n\n② Two random variables   $s_{1}, s_{2} = 0, 1$\n\n$$\nP (s _ {1} = 1) = P _ {1} (1) \\quad P (s _ {1} = \\Delta) = T - P _ {1} (0) = P _ {1} (0)\n$$\n\n$$\nP (S _ {2} = 1) = P _ {2} (1) \\dots P (S _ {2} = 0) = 1 - P _ {2} (1) = P _ {2} (0)\n$$\n\nJoint prob.\n\n$$\nP (s _ {1} = 1 \\Delta s _ {2} = 1) = P _ {1} (1) P _ {2} (1)?\n$$\n\n$$\n\\boxed {P _ {1 2} (s _ {1}, s _ {2})} = P _ {1 2} (1, 1) \\mathrm{Jointprob.}\n$$\n\n$$\nP _ {1 2} (1 1) + P _ {1 2} (1 0) + P _ {1 2} (0 1) + P _ {1 2} (0 0) = 1\n$$\n\n$$\n\\begin{array}{c} P _ {1} (1) = P _ {1 2} (1, 1) + P _ {1 2} (1, 0) \\\\ I _ {S _ {1} = 1} \\quad \\uparrow \\quad \\uparrow \\quad \\text { regardless } S _ {2} \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0005-v-random-variable-15a531e0-d13b76bb", "source_title": "V. Random variable", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["V. Random variable", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-v-random-variable-15a531e0-d13b76bb:page-0002", "text": "Condition) randomness\n\n$$\n\\underset {i f s _ {2} = 1} {P _ {1} (s _ {1})} \\propto P _ {1 2} (s _ {1}, 1)\n$$\n\n$$\nP _ {i f s _ {2} = 1} (s _ {1}) = \\frac {P _ {1 2} (s _ {1} , 1)}{\\sum_ {s _ {1} = 0 , 1} P _ {1 2} (s _ {1} , 1)}\n$$\n\n$$\nP _ {1 2} (s _ {2} = 0) (s _ {1}) = \\frac {P _ {1 2} (s _ {1} , 0)}{\\sum_ {s _ {1} = 0 , 1} P _ {2} (s _ {1} , 0)}\n$$\n\nexample. I roll a dice n = 1, ..., 6\n\n${S}_{1} = 1\\;{if}\\;n$ is multiple of  $2\\;{S}_{1} = 0$ otherwise\n\n$S_{2}=1$  if n is multiple of 3   $S_{2}=0$  otherwise\n\n$$\nP _ {1} (1) = \\frac {1}{2} \\quad P _ {1} (0) = \\frac {1}{2}\n$$\n\n$$\nP _ {2} (1) = \\frac {1}{3}. \\quad P _ {2} (0) = \\frac {2}{3}\n$$\n\n$$\nP _ {1 2} (1 1) = \\frac {1}{6} \\Big | _ {n = 6} \\quad P _ {1 2} (1 0) = \\frac {1}{3} \\Big | _ {n = 2. 4}\n$$\n\n$$\nP _ {1 2} (0, 1) = \\frac {1}{6} | _ {n = 3} P (0, 0) = \\frac {1}{3} | _ {n = 1. 5}\n$$\n\n$$\nP _ {i f s _ {2} = 1} (s _ {1}) = \\frac {1 / 6}{y _ {6} + y _ {6}}, \\frac {1 / 6}{y _ {6} + y _ {6}} = \\frac {1}{2}, \\frac {1}{2}\n$$\n\n$$\nP _ {1} f s _ {2} = \\textcircled {4} (s _ {1}) = \\frac {1}{2}, \\frac {1}{2}\n$$\n\n$$\n\\begin{array}{r l} {P _ {i f} s _ {1} = 1 (s _ {2})} & {= \\frac {1}{3}, \\frac {2}{3}} \\\\ {P _ {i f} s _ {1} = 0 (s _ {2})} & {= \\frac {1}{3}, \\frac {2}{3}} \\end{array} \\quad \\text {   equal   } \\left[ \\begin{array}{l} {{s _ {1} \\geq s _ {2}}} \\\\ {{\\mathrm{areindependent}}} \\\\ {{\\mathrm{randomvariables.}}} \\end{array} \\right]\n$$", "source": "mit-ocw", "source_doc_id": "0005-v-random-variable-15a531e0-d13b76bb", "source_title": "V. Random variable", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["V. Random variable", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-v-random-variable-15a531e0-d13b76bb:page-0003", "text": "Example Ⅱ.\n\n$$\ns _ {1} = 1 \\quad i f \\quad n = 7, 2, 3 \\quad s _ {1} = 0 \\quad i f \\quad n = 4, 5, 6\n$$\n\n$$\ns _ {2} = 1 \\quad \\text { if } \\quad n \\% 2 = 0 \\quad s _ {2} = 1 \\quad \\text { if } \\quad n \\% 2 \\neq 0\n$$\n\n$$\nP _ {1} (S _ {1}) = \\frac {1}{2}, \\frac {1}{2}\n$$\n\n$$\nP _ {2} (S _ {2}) = \\frac {1}{2}, \\frac {1}{2}\n$$\n\n$$\n\\begin{array}{l} P _ {1 2} (s _ {1}, s _ {2}) = \\begin{array}{c c c} s _ {2} = 1 & s _ {2} = 0 \\\\ \\frac {1}{6} & \\frac {1}{3} & s _ {1} = 1 \\end{array} \\\\ \\frac {1}{3} \\quad \\frac {1}{6} \\quad s _ {1} = 0 \\\\ \\end{array}\n$$\n\n$$\nP _ {i f s _ {2} = 1} (s _ {1}) = \\frac {1}{3}, \\frac {2}{3}\n$$\n\n$$\nP: f _ {S _ {2} = 0} (S _ {1}) = \\frac {2}{3}, \\frac {1}{3} \\quad \\text {   s.   } \\Delta S _ {2} \\text {   are   not   }\n$$\n\nIf  $s_1$ &  $s_2$ are independent\n\nWe can obtain   $P_{12}(s_1, s_2)$  from   $P_1(s_1)$$\\otimes P_2(s_2)$\n\n$$\nP _ {1 2} (s _ {1}, s _ {2}) = P _ {1} (s _ {1}) P _ {2} (s _ {2})\n$$\n\n$$\n\\langle S _ {1} S _ {2} \\rangle = \\langle S _ {1} \\rangle \\langle S _ {2} \\rangle\n$$\n\n$$\n\\begin{array}{l} \\langle s _ {1} s _ {2} \\rangle = \\sum_ {s _ {1} s _ {2}} s _ {1} s _ {2} P _ {1 2} (s, s _ {2}) \\\\ = \\frac {2}{s _ {1} s _ {2}} s _ {1} P _ {1} (s _ {1}) s _ {2} P _ {2} (s _ {2}) \\\\ = \\angle F (s) > \\angle F (0) > = 0 \\\\ i b s \\rightarrow \\infty \\\\ \\end{array}\n$$\n\n$$\n= \\left(\\sum_ {s _ {1}} s _ {1} P _ {1} (s _ {2})\\right) \\left(\\sum_ {s _ {2}} s _ {2} ^ {-} P _ {2} (s _ {2})\\right) = \\langle s _ {1} \\rangle \\langle s _ {2} \\rangle\n$$\n\nfor correlated random variable   $\\langle s_{1}s_{2}\\rangle\\neq\\langle s_{1}\\rangle\\langle s_{2}\\rangle$$\\langle s_{1}s_{2}\\rangle-\\langle s_{1}\\rangle\\langle s_{2}\\rangle\\equiv correlation between s_{1}\\&s_{2}$\n\n$$\n\\begin{array}{l} \\text { Random   force } \\\\ \\angle F (s) F (0) > = \\\\ = \\angle F (s) > \\angle F (0) > = 0 \\\\ i b s \\rightarrow \\infty \\\\ \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0005-v-random-variable-15a531e0-d13b76bb", "source_title": "V. Random variable", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["V. Random variable", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-v-random-variable-15a531e0-d13b76bb:page-0004", "text": "③\n\nBinomial distribution\n\nconsider n independent random variable\n\n$$\ns _ {1}, \\dots , s _ {n} \\quad , \\quad s _ {i} = 0, 1\n$$\n\n$$\nP (s _ {i} = t) = p \\quad , \\quad P (s _ {i} = 6) = 1 - p\n$$\n\nProof the set to take a particular values\n\n$$\nP (s _ {1}, s _ {2} \\dots s _ {n}) = P (s _ {1}) P (s _ {2}) \\dots P (s _ {n})\n$$\n\nProb. for 2s:=k\n\n$$\nB (k; n, p) = \\frac {n !}{k ! (n - k) !} p ^ {k} (1 - p) ^ {n - k} = \\binom {n} {k} p ^ {k} (1 - p) ^ {n - k}\n$$\n\n$$\n\\left\\{ \\begin{array}{l l} {\\langle k \\rangle = \\sum_ {k} k B (k; n, p) = p n} \\\\ {\\langle k ^ {2} \\rangle = n ^ {2} p ^ {2} + n p (1 - p)} \\end{array} \\right.\n$$\n\n$$\n\\text {   How   } x \\text {   calculate   } <   k > \\text {   let   } x = \\frac {p}{1 - p}\n$$\n\n$$\n\\& z = (1 + x) ^ {n}\n$$\n\n$$\n= 1 + n x + \\frac {n (n - 1)}{2} x ^ {2} \\dots + \\binom {n} {k} x ^ {k} \\dots\n$$\n\n$$\nx \\cdot \\frac {d}{d x} \\ln z = \\frac {1}{z} \\overline {{2}} k (n) x ^ {k}\n$$\n\n$$\n\\frac {1}{2} (k) ^ {n} x ^ {k} = (k) \\frac {(p / 1 - p) ^ {k}}{(1 + \\frac {p}{1 - p}) ^ {n}} = (k) \\frac {p ^ {k} (1 - p) ^ {n - k}}{(1 - p + p) ^ {n}}\n$$\n\n$$\n\\begin{array}{l} \\angle k > = x \\frac {d}{d x} \\ln z \\\\ = x n \\frac {1}{1 + x} = n p \\sqrt {B (k , n , p)} \\\\ \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0005-v-random-variable-15a531e0-d13b76bb", "source_title": "V. Random variable", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["V. Random variable", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0005-v-random-variable-15a531e0-d13b76bb:page-0005", "text": "④ Central limit theorem\n\nGiven n independent random variables\n\n$x_{1}, \\cdots, x_{n}$  with distribution P(x)\n\nLet  $X = \\frac{1}{n}\\sum_{i}x_{i}$ .\n\n$I_n \\quad n \\to \\infty \\quad \\text{limit}, \\quad \\text{the distribution of } X$\n\nis given be Gaussian distribution\n\n$$\nP _ {G} (X) = \\frac {1}{\\sqrt {2 \\pi \\sigma}} e ^ {- \\frac {(X - <   X >) ^ {2}}{2 \\sigma}}\n$$\n\n$$\n\\int d X P _ {G} (X) = 1\n$$\n\n$$\n\\int d X \\times P _ {G} (X) = <   X > = \\int d x \\times P (x) = <   x >\n$$\n\n$$\n\\int d X (X - \\langle X \\rangle) ^ {2} P _ {G} (X) = \\sigma\n$$\n\n$$\n= \\frac {1}{n ^ {2}} \\left\\langle \\left(\\frac {\\pi}{i} (x _ {i} - <   x >)\\right) ^ {2} \\right\\rangle\n$$\n\n$$\n= \\frac {1}{n ^ {2}} \\sum_ {i} <   (x _ {i} - <   x >) ^ {2} >\n$$\n\n$$\n+ \\frac {1}{n ^ {2}} \\sum_ {i \\neq j} <   (x _ {i} - <   x >) (x _ {j} - <   x >) >\n$$\n\n$$\n\\sigma = \\frac {1}{n} <   (x - <   x >) ^ {2} >\n$$", "source": "mit-ocw", "source_doc_id": "0005-v-random-variable-15a531e0-d13b76bb", "source_title": "V. Random variable", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["V. Random variable", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-vi-thermal-fluctuations-47d7f829-66bb3151:page-0001", "text": "VI. Thermal fluctuations\n\n① Thermal fluctuations\n\nNumber of particle\nin V: N =   $\\sum_{i=1}^{N_{0}} x_{i}$\n\nnumber of particle\n\nNo particle\nV\nV₀\nparticl reservoir\n\n$$\n\\begin{array}{l} x _ {i} = 1 \\quad i f \\quad i + h \\quad p a n t i l e \\quad i \\quad i n \\quad V \\\\ = 0 \\quad \\text { otherwise } \\\\ \\end{array}\n$$\n\n$$\nP (1) = \\text { probability } \\quad \\text { for } x = 1 \\quad P (1) = \\frac {V}{V _ {0}}\n$$\n\n$$\nP (0) = \\dots x = 0\n$$\n\n$$\n\\begin{array}{l} \\langle N \\rangle = \\sum_ {i} \\langle x _ {i} \\rangle = \\sum_ {i} P (1) = N _ {0} P (1) \\\\ \\langle N ^ {2} \\rangle = \\sum_ {i} \\langle x _ {i} \\rangle^ {2} + \\sum_ {i \\neq j} \\langle x _ {i} - x _ {j} \\rangle \\\\ = N _ {0} P (1) + \\left(N _ {0} ^ {2} - N _ {0}\\right) \\cdot P (1) = \\langle N \\rangle + \\langle N \\rangle^ {2} - 4 N P _ {0}; \\\\ \\end{array}\n$$\n\n$$\n\\sqrt {<   (N - <   N >) ^ {2} >} \\equiv \\overline {{\\Delta N}} = \\sqrt {<   N ^ {2} > - <   N > ^ {2}}\n$$\n\n$$\n= \\sqrt {\\langle N \\rangle (1 - P (1))} \\rightarrow \\sqrt {\\langle N \\rangle} \\quad \\text { as } \\frac {V}{V _ {4}} \\rightarrow 0\n$$\n\n$$\n\\frac {\\frac {\\overline {{\\Delta N}}}{\\langle N \\rangle}}{= \\frac {1}{\\sqrt {N}}}\n$$", "source": "mit-ocw", "source_doc_id": "0006-vi-thermal-fluctuations-47d7f829-66bb3151", "source_title": "VI. Thermal fluctuations", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VI. Thermal fluctuations", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-vi-thermal-fluctuations-47d7f829-66bb3151:page-0002", "text": "* How to calculate the fluctuation of N\n\nfor a generic system?\n\nGeneral ideas:\n\nThermal dynamical function S, A, Ω, G\n\n$\\Rightarrow  {P}_{\\text{robability }} :$\n\n$P \\propto e^{\\frac{s}{k_{b}}} < \\# f_{stat}$$P \\propto e^{-\\beta A}$  = total prob.\n\n$P \\propto e^{-\\beta \\Omega} \\quad P \\propto e^{-\\beta G}$\n\nFor ideal gas\n\n$$\nA (T, V, N) = N k _ {B} T [ \\ell_ {n} (n \\lambda^ {3}) - 1 ]\n$$\n\n$$\n= N k _ {B} T \\ln N + c o n s t\n$$\n\n$$\nP (N) = e ^ {- \\beta A} \\quad \\text { is } \\quad \\text { peaked } \\quad a + N \\sim \\frac {1}{2}\n$$\n\nwith particle reservoir:   $P_{w1}=e^{-\\beta A_{tot}}$\n\n$$\nA _ {+ \\mathrm{tot}} (T, V, N) = A - \\mu N\n$$\n\n$$\n= \\mathrm{const} + \\frac {1}{2} \\left. \\frac {\\partial^ {2} A}{\\partial N ^ {2}} \\right| _ {\\overline {{N}}} (N - \\overline {{N}}) ^ {2}\n$$\n\n$$\n\\overline {{N}} \\text { minimize } A _ {\\text { tot }}\n$$\n\n$$\n\\frac {\\partial A}{\\partial N} = k _ {B} T \\ln N + k _ {B} T, \\frac {\\partial^ {2} A}{\\partial N ^ {2}} = \\frac {k _ {B} T}{N}\n$$\n\n$$\nP = e ^ {- \\frac {\\beta}{2} \\frac {\\partial^ {2} A}{\\partial N ^ {2}} (N - \\overline {{N}}) ^ {2}} \\Rightarrow \\left\\langle (N - \\overline {{N}}) ^ {2} \\right\\rangle = \\frac {k _ {B} T}{\\frac {\\partial^ {2} A}{\\partial N ^ {2}}} = \\overline {{N}}\n$$\n\nA\nI\n\nA_{to+}\nN\nN̅", "source": "mit-ocw", "source_doc_id": "0006-vi-thermal-fluctuations-47d7f829-66bb3151", "source_title": "VI. Thermal fluctuations", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VI. Thermal fluctuations", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-vi-thermal-fluctuations-47d7f829-66bb3151:page-0003", "text": "$$\n\\text {   What   is   } \\left. \\frac {\\partial^ {2} A}{\\partial N ^ {2}} \\right| _ {V, T} = \\left. \\frac {\\partial^ {M}}{\\partial N} \\right| _ {V, T}? \\Rightarrow \\text {   compressibility   }\n$$\n\n$$\nA (v, T, N) = N a (T, \\frac {V}{N}) = \\frac {V}{\\sigma} a (T, \\sigma)\n$$\n\n$$\nv = \\frac {V}{N}\n$$\n\n$$\n\\text {   change   variable   } (V, T, N) \\Rightarrow (V, T, \\sigma)\n$$\n\n$$\n\\left. \\frac {\\partial}{\\partial N} \\right| _ {V, T} = \\left. \\frac {\\partial U}{\\partial N} \\right| _ {V, T} \\left. \\frac {\\partial}{\\partial U} \\right| _ {V, T}\n$$\n\n$$\n= - \\frac {V}{N ^ {2}} \\left. \\frac {\\partial}{\\partial v} \\right| _ {v, T} = - \\frac {v ^ {2}}{V} \\left. \\frac {\\partial}{\\partial v} \\right| _ {v, T}\n$$\n\n$$\n\\frac {\\partial A}{\\partial N} = - \\frac {v ^ {2}}{V} \\frac {\\partial}{\\partial v} (\\frac {V}{v} a (v, T)) | _ {v, T} \\quad \\boxed {\\frac {\\partial a}{\\partial v} | _ {T} = \\frac {\\partial A}{\\partial V} | _ {T, N} = P}\n$$\n\n$$\n= a (0, T) - v \\frac {\\partial a}{\\partial v} | _ {T} = a (v, T) - v P (v, T)\n$$\n\n$$\n\\frac {\\partial^ {2} A}{\\partial N ^ {2}} = - \\frac {v ^ {2}}{V} \\frac {\\partial}{\\partial v} [ a (0, T) - o P (0, T) ] | _ {V, T}\n$$\n\n$$\n= - \\frac {v ^ {2}}{V} \\left[ \\frac {\\partial a}{\\partial v} \\right| _ {T} - p - o \\frac {\\partial p}{\\partial v} | _ {T} ] \\underbrace {\\text {   iso   thermal   }} _ {\\text {   compressibility   }}\n$$\n\n$$\n= \\frac {v ^ {3}}{V} \\frac {\\partial P}{\\partial v} = - \\frac {v ^ {2}}{V} \\frac {1}{k _ {T}}\n$$\n\n$$\n\\Rightarrow \\left\\langle (N - \\overline {{N}}) ^ {2} \\right\\rangle = k _ {B} T \\frac {V K _ {T}}{v ^ {2}}\n$$\n\n$$\n\\begin{array}{r l} {K _ {T}} & {= - \\frac {1}{U} \\left(\\frac {\\partial U}{\\partial P}\\right) | _ {T}} \\\\ & {= - \\frac {1}{V} \\left. \\frac {\\partial V}{\\partial P} \\right| _ {T}} \\end{array}\n$$\n\n$$\n<   (N - \\bar {N}) ^ {2} > \\Rightarrow f l u c T a x T r o n\n$$\n\n$$\nK _ {T} = - \\frac {1}{V} \\frac {\\partial V}{\\partial p} \\Rightarrow \\text { response }\n$$\n\n$$\n\\Leftrightarrow \\frac {\\partial^ {2} A}{\\partial N ^ {2}}\n$$\n\n$$\n\\text { closely   related }\n$$", "source": "mit-ocw", "source_doc_id": "0006-vi-thermal-fluctuations-47d7f829-66bb3151", "source_title": "VI. Thermal fluctuations", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VI. Thermal fluctuations", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-vi-thermal-fluctuations-47d7f829-66bb3151:page-0004", "text": "sN has large fluctuations near the critical point\n\n$$\n\\begin{array}{l} \\frac {\\delta n}{n} = \\frac {\\sqrt {<   (N - \\overline {{N}}) ^ {2})}}{N} \\\\ = \\sqrt {k _ {B} T \\frac {k _ {T}}{v}} \\frac {1}{\\sqrt {N}} \\\\ = \\sqrt {k _ {0} T k _ {T}} \\frac {1}{\\sqrt {V}} \\\\ = \\sqrt {\\frac {V _ {c}}{V}} \\\\ \\end{array}\n$$\n\n$$\nV _ {c} = k _ {B} T K _ {T}\n$$\n\n| T     | P (liquid) | T (gas) | V (critical point) |\n|-------|------------|---------|-------------------|\n| T3    | -          | -       | -                 |\n| T2    | -          | -       | -                 |\n| T1    | -          | -       | -                 |\n| T1    | -          | -       | -                 |\n| T2    | -          | -       | -                 |\n| T3    | -          | -       | -                 |\n| T3    | -          | -       | -                 |\n| T3    | -          | -       | -                 |\n| T3    | -          | -       | -                 |\n| T3    | -          | -       | -                 |\n| T3    | -          | -       | -                 |\n| T3    | -          | -       | -                 |\n| T3    = ∞  | -          | -       | -                 |\n| T3    = ∞  | -          | -       | -                 |\n| T3    = ∞  | -          | -       | -                 |\n| T3    = ∞  | -          | -       | -                 |\n| T3    = ∞  | -          | -       | -                 |\n| T3    = ∞  | -          | -       | -                 |\n\n| Point | Temperature (T) |\n|-------|-----------------|\n| T1    | -               |\n| T2    | -               |\n| T3    | -               |\n\n⇒ Water density in a smaller volume has a stronger fluctuation\nThe fluctuations become of order 1 when V = Vc\n\n$$\nU _ {s u a l l y} V _ {c} \\sim (1 A ^ {0}) ^ {3}\n$$\n\n$$\nV _ {c} \\rightarrow a o \\quad \\text { at } \\quad \\text { the   critical   point }\n$$\n\nWhen  $V_c \\sim (500\\text{Å})^3$, the water turn milky", "source": "mit-ocw", "source_doc_id": "0006-vi-thermal-fluctuations-47d7f829-66bb3151", "source_title": "VI. Thermal fluctuations", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VI. Thermal fluctuations", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-vi-thermal-fluctuations-47d7f829-66bb3151:page-0005", "text": "② Fluctuations of V:\n\n$$\n\\begin{array}{l} - \\beta (A (V, T) + V P) - \\frac {1}{2} \\beta \\frac {\\partial^ {2} A}{\\partial V ^ {2}} (V - \\overline {{V}}) ^ {2} \\\\ * e = \\# e \\\\ \\end{array}\n$$\n\n$$\n\\left. \\frac {\\partial^ {2} A}{\\partial v ^ {2}} \\right| _ {T} = - \\left. \\frac {\\partial P}{\\partial V} \\right| _ {T} = - \\frac {1}{\\frac {\\partial V}{\\partial p} | _ {T}} = \\frac {1}{V K _ {T}} \\sqrt {K _ {T} : \\text { compressibility }} = \\frac {- 1}{V} \\left. \\frac {\\partial V}{\\partial p} \\right| _ {T}\n$$\n\n$$\nF _ {o r} \\text { ideal   gas: }\n$$\n\n$$\nV = \\frac {N k _ {B} T}{P}, V K _ {T} = + \\frac {N k _ {B} T}{P ^ {2}} = \\frac {V}{P} = \\frac {V ^ {2}}{N k _ {B} T}\n$$\n\n$$\n\\begin{array}{r l} & {\\left\\langle (V - \\overline {{{V}}}) \\right\\rangle^ {2} = \\frac {i d e a l}{g a s}} \\\\ & {= \\frac {k _ {B} T}{\\partial^ {2} A / g v ^ {2}} = k _ {B} T k _ {T} V = \\frac {V ^ {2}}{N}} \\end{array} \\Rightarrow \\boxed { \\begin{array}{l} {\\delta V = \\sqrt {\\langle (V - \\overline {{{V}}}) ^ {2} \\rangle} = \\frac {V}{\\sqrt {N}}} \\\\ {\\mathrm{idealgas}} \\end{array} }\n$$\n\n* Second way:\n\n$$\n\\frac {\\partial}{\\partial p} G = - k _ {B} T \\frac {\\int d V (- \\beta) V e ^ {- \\beta [ A (V , T) + V P ]}}{\\int d V e ^ {- \\beta [ A (V , T) + V P ]}}\n$$\n\n$$\n= + <   V >\n$$\n\n$$\n\\begin{array}{l} \\frac {\\partial^ {2} G}{\\partial p ^ {2}} = - \\beta \\frac {\\int d V v ^ {2} e ^ {- \\beta (A + V P)}}{\\int d V e ^ {- \\beta (A + V P)}} \\\\ - \\frac {- \\beta (\\int d V V e ^ {- \\beta (A + V P)}) ^ {2}}{(\\int d V e ^ {- \\beta (A + V P)}) ^ {2}} \\\\ \\end{array}\n$$\n\n$$\n= [ \\langle V ^ {2} \\rangle - (\\langle V \\rangle) ^ {2} ] (- p)\n$$\n\n$$\n\\left\\langle (V - \\overline {{V}}) ^ {2} \\right\\rangle = - k _ {B} T \\frac {\\partial^ {2} G}{\\partial p ^ {2}} = k _ {B} T \\cdot V \\cdot K _ {T}\n$$\n\n$$\n\\begin{array}{l} \\frac {\\partial G (P , T)}{\\partial P} = V \\\\ P = - \\frac {2 A}{\\partial V} \\\\ V = \\frac {\\partial G}{\\partial P} \\\\ \\sqrt {<   (V - \\overline {{V}}) ^ {2} >} \\\\ V \\leftrightarrow P \\\\ \\text { conjugate } \\\\ = \\frac {k _ {B} T}{\\partial^ {2} A / \\partial V ^ {2}} \\\\ A (V, T) \\\\ = - k _ {B} T \\frac {\\partial^ {2} G}{\\partial P ^ {2}} \\\\ G (P, V) \\\\ \\end{array}\n$$\n\n$$\n\\begin{array}{l} \\sqrt {<   (V - \\overline {{V}}) ^ {2} >} \\\\ = \\frac {k _ {B} T}{\\partial^ {2} A / \\partial V ^ {2}} \\\\ = - k _ {B} T \\frac {\\partial^ {2} G}{\\partial P ^ {2}} \\\\ \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0006-vi-thermal-fluctuations-47d7f829-66bb3151", "source_title": "VI. Thermal fluctuations", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VI. Thermal fluctuations", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-vi-thermal-fluctuations-47d7f829-66bb3151:page-0006", "text": "$$\n\\mathrm{Totalentropy} \\left\\{\\frac {S _ {\\mathrm{tot}}}{T} = S (E) - \\frac {E ^ {2}}{T} \\right. \\quad \\begin{array}{l} \\text { entropy   of   heat   bath } \\\\ \\boxed {\\mathrm{sys}} \\leftrightarrow \\boxed {\\mathrm{heat}} \\\\ \\boxed {\\mathrm{bath}} \\end{array}\n$$\n\n③ Fluctuation of energies\n\n$$\n\\text { Prob. } \\propto e ^ {\\frac {1}{2} \\left[ \\frac {d ^ {2}}{d E ^ {2}} \\frac {1}{k _ {B}} [ S (E) - \\frac {E}{T} ] \\right] (E - \\overline {{E}}) ^ {2}}\n$$\n\n$$\n= e ^ {+ \\frac {1}{2} k _ {B} ^ {- 1} \\frac {d S (E)}{d E ^ {2}} (E - \\overline {{E}}) ^ {2}}\n$$\n\n$$\n\\Rightarrow <   (E - \\overline {{E}}) ^ {2} > = - \\frac {k _ {B}}{d ^ {3} S (E) / d E ^ {2}}\n$$\n\n$$\n\\begin{array}{r l} {- \\frac {\\partial^ {2} S}{\\partial E ^ {2}}} & {= \\frac {1}{T ^ {2}} \\frac {\\partial T}{\\partial E}} \\\\ & {= \\frac {1}{T ^ {2} C _ {V}}} \\end{array}\n$$\n\n$$\n= - N \\frac {k _ {B}}{d ^ {2} s / d \\epsilon^ {2}} = k _ {B} T _ {-} ^ {2} c _ {v} \\quad \\begin{array}{l l l} {{s = S / N}} & {{n}} & {{S / V}} \\\\ {{\\epsilon = E / N}} & {{n}} & {{E / V}} \\end{array}\n$$\n\n$$\n\\Rightarrow \\delta E = \\sqrt {<   (E - \\overline {{E}}) ^ {2} >} \\simeq \\sqrt {k _ {B} T ^ {2} c _ {V}} \\sim \\sqrt {N}\n$$\n\n$$\n\\overline {{E}} \\sim N \\Rightarrow \\overline {{E}} > \\sqrt {<   (E - \\overline {{E}}) ^ {2} >}\n$$\n\n$$\n\\text { small   fluctuation }\n$$\n\n$$\n\\boxed {\\frac {S E}{E} \\sim \\frac {1}{\\sqrt {N}}}\n$$\n\n$$\n\\text {   For   ideal   gas   } S = k _ {B} \\ln \\frac {\\sigma (2 m \\epsilon) ^ {3 / 2}}{h ^ {3}}\n$$\n\n$$\n\\frac {d ^ {2} s}{d \\epsilon^ {2}} = k _ {B} \\frac {3}{2} \\frac {1}{\\epsilon^ {2}}\n$$\n\n$$\n\\delta E = \\sqrt {N} \\sqrt {\\frac {2}{3}} \\epsilon = \\sqrt {\\frac {2}{3}} \\frac {E}{\\sqrt {N}}\n$$", "source": "mit-ocw", "source_doc_id": "0006-vi-thermal-fluctuations-47d7f829-66bb3151", "source_title": "VI. Thermal fluctuations", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VI. Thermal fluctuations", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0006-vi-thermal-fluctuations-47d7f829-66bb3151:page-0007", "text": "④ Fluctuation and response\n\nIn general\n\nA(x)\n\nI some parameter\n\nwhich can fluctuate\n\n$$\np \\propto e ^ {- \\beta A (x)}\n$$\n\n$$\nA = \\mathrm{const} + \\frac {1}{2} \\frac {\\partial A}{\\partial x ^ {2}} (x - \\overline {{x}}) ^ {2}\n$$\n\n$$\n\\Rightarrow \\quad <   (x - \\overline {{x}}) ^ {2} = \\frac {k _ {B} T}{\\frac {\\partial^ {2} A}{\\partial x ^ {2}} | _ {\\overline {{x}}}}\n$$\n\nP₀\nA\nAₜₒₜ\n\n$$\nA _ {t o t} = A + P _ {0} V\n$$\n\n$$\n<   (V - \\overline {{V}}) ^ {2} > = \\frac {k _ {B} T}{\\frac {\\partial^ {2} A _ {T B +}}{\\partial V ^ {2}}}\n$$\n\n$$\n= \\frac {k _ {B} T}{\\frac {\\partial^ {2} A}{\\partial V ^ {2}}} = \\frac {k _ {B} T}{\\left(- \\frac {\\partial P}{\\partial r}\\right) _ {T}}\n$$\n\n$$\n\\text { Ideal   gas } \\quad P V = k _ {\\mathrm{B}} T N\n$$\n\n$$\n- \\frac {\\partial P}{\\partial V} = \\frac {k _ {3} T N}{V ^ {2}}\n$$\n\n$$\n\\Rightarrow <   (V - \\overline {{V}}) ^ {2} > = \\frac {\\overline {{V}} ^ {2}}{N} = \\Delta V ^ {2} \\Rightarrow\n$$\n\n$$\n\\Delta V = \\frac {V}{\\sqrt {N}}\n$$\n\n$$\n\\langle (V - \\overline {{V}}) ^ {2} \\rangle = \\frac {k _ {B} T}{\\left(- \\frac {\\partial P}{\\partial V}\\right) _ {T}} = k _ {B} T \\left(- \\frac {\\partial V}{\\partial P}\\right)\n$$\n\nflucTunFion\n\n2 response.\n\n$$\n\\boxed {\\frac {\\Delta V}{V} = \\frac {1}{\\sqrt {N}}}\n$$\n\n$\\frac{\\partial V}{\\partial p}$$p \\leftrightarrow f_{onue}$\n\nThermal fluctuations - response\n\nA\n↓kBT\nx\nΔx\n\nfluctuation\n\nA\nx = K\n\n$$\n\\langle \\Delta x ^ {2} \\rangle = k _ {B} T K\n$$\n\n$$\nx = K F\n$$\n\nresponse\n\n$$\n= F _ {\\mathrm{once}}\n$$\n\n$$\nA - x F\n$$\n\n$$\nk = \\frac {1}{\\frac {2 x ^ {2}}{2 x ^ {2}}}\n$$", "source": "mit-ocw", "source_doc_id": "0006-vi-thermal-fluctuations-47d7f829-66bb3151", "source_title": "VI. Thermal fluctuations", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VI. Thermal fluctuations", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0001", "text": "# VII The Bose Gas.\n\n① Photons :\n\n④ Photons are bosons\n\n⑤ No particle number conservation → :Caunonical Ensemble\n\nHow to label photon state: no Grand cannonical ensemble\n\n$$\n\\begin{array}{l l} {\\mathrm{one}} & {\\mathrm{bosen:}} \\\\ {\\mathrm{two}} & {\\mathrm{bosens:}} \\end{array} \\quad \\begin{array}{l l} {| \\vec {k} _ {1} >,} & {\\vec {k} _ {1} = \\frac {2 \\pi}{L} (n _ {x}, n _ {y}, n _ {z})} \\\\ {| \\vec {k} _ {1}, \\vec {k} _ {2} >} & \\end{array}\n$$\n\n$$\n= 1 \\overrightarrow {k _ {2}}, \\overrightarrow {k _ {1}} > \\quad \\overrightarrow {k _ {1}} = \\overrightarrow {k _ {2}} \\quad O K\n$$\n\n$$\n\\hat {I} \\text { identical   particles }\n$$\n\n$$\nN = \\text { bosons } \\quad | \\vec {k} _ {1}, \\vec {k} _ {2}, \\dots , \\vec {k} _ {N} >\n$$\n\n$$\n\\begin{array}{l l} {3} & {\\vec {k} _ {5} = \\vec {k} _ {6} = \\vec {k} _ {7}} \\\\ {0} & \\\\ {1} & {\\vec {k} _ {8}} \\\\ {2} & {\\vec {k} _ {2} = \\vec {k} _ {3}} \\\\ {0} & \\\\ {n _ {\\vec {k} _ {2}} = 1} & {\\vec {k} _ {1}} \\\\ {n _ {\\vec {k} _ {1}} = 0} & \\end{array}\n$$\n\n$$\n\\text { Another   label } \\quad \\ln_ {\\vec {k} _ {1}} n _ {\\vec {k} _ {2}} \\dots >\n$$\n\n$$\nn _ {k} = 0, 1, 2, \\dots , \\infty \\quad \\text { independently }\n$$\n\n$$\n[ n \\frac {1}{n} = 0, 1 (\\text { fermion }) ]\n$$\n\n$$\nQ = \\sum_ {n _ {k _ {1}} = 0, 1, \\dots} \\sum_ {n _ {k _ {2}} = 0, 1, \\dots} e ^ {- \\beta \\sum_ {k} n _ {k} t _ {k}}\n$$\n\n$$\n= \\pi \\sum e ^ {- \\beta n _ {k} \\epsilon_ {k}}\n$$\n\n$$\n= \\prod_ {k} \\frac {1}{1 - e ^ {- \\beta \\epsilon_ {k}}} \\quad \\text {   Prob.   for   } \\vec {k} \\text {-state   to   have   } n _ {k}\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0002", "text": "Free energy:\n\n$$\n\\begin{array}{l} A = - k _ {B} T \\ln Q \\\\ = + k _ {3} T \\frac {2}{k} \\ln (1 - e ^ {- \\beta E \\frac {1}{k}}) \\times 2 \\\\ \\end{array}\n$$\n\nThe Total system is a sum of independent\n\nsub systems: each k-level = one subsystem\n\nThe state of each subsystem\n\nin labeled by   $n_{k}=0,1,\\ldots$  energy=  $n_{k}\\epsilon_{k}$\n\n$$\nQ _ {k} = \\sum_ {n _ {k}} e ^ {- \\beta n _ {k} \\epsilon_ {k}} = \\frac {1}{1 - e ^ {- \\beta \\epsilon_ {k}}}\n$$\n\n$$\n\\Delta_ {k} = k _ {B} T \\ln (1 - e ^ {- p \\varepsilon \\vec {x}})\n$$\n\ntotal Free energy\n\n$$\nA = \\sum_ {k} A _ {k} = k _ {B} T \\sum_ {k} \\ln (1 - e ^ {- \\beta \\epsilon_ {k}})\n$$\n\nEach subsystem = a harmonic oscillator\n\n$$\nw i t h \\quad k w = \\varepsilon_ {k}\n$$\n\n$$\n\\text { energy } E = n \\hbar w + \\frac {1}{2} \\hbar w\n$$\n\nA photon system = A collection of oscillators labeled by   $\\overline{k}$  with   $\\tau w_{e} = G_{e}$\n\nProb. for level-k to have  $n_k$ photons\n\n$$\nP (n _ {k}) = \\frac {e ^ {- \\beta n _ {k} \\epsilon_ {k}}}{\\sum_ {n _ {k}} e ^ {- \\beta n _ {k} \\sigma_ {k}}} = e ^ {- \\beta \\epsilon_ {k} n _ {k}} (1 - e ^ {- \\beta \\epsilon_ {k}})\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0003", "text": "$$\n\\text { Average } * \\text { of   photon   on   level-} k\n$$\n\n$$\n<   n _ {k} > = \\sum_ {n _ {k}} P (n _ {k}) n _ {k}\n$$\n\n$$\n= (1 - e ^ {- \\beta \\epsilon_ {k}}) \\sum_ {n _ {k}} n _ {k} e ^ {- \\beta n _ {k} \\epsilon_ {k}}\n$$\n\n$$\n= (1 - e ^ {- \\beta \\epsilon_ {k}}) (-) \\frac {\\partial}{\\partial (\\beta \\epsilon_ {k})} \\dots \\sum_ {n _ {k}} e ^ {- (\\beta \\epsilon_ {k}) n _ {k}}\n$$\n\n$$\n= (1 - e ^ {- \\beta \\varepsilon_ {n}}) (-) \\frac {\\partial}{\\partial (\\beta \\varepsilon_ {n})} \\left(\\frac {1}{1 - e ^ {- \\beta \\varepsilon_ {n}}}\\right)\n$$\n\n$$\n= (1 - e ^ {- \\beta \\epsilon_ {k}}) \\frac {e ^ {- \\beta \\epsilon_ {k}}}{(1 - e ^ {- \\beta \\epsilon_ {k}}) ^ {2}}\n$$\n\n$$\n\\langle n _ {k} ^ {+} \\rangle = \\frac {1}{e ^ {\\beta E _ {k}} - 1}\n$$\n\n$$\n\\left\\langle n _ {k} \\right\\rangle = \\frac {1}{e ^ {\\beta \\epsilon_ {k} + 1}} \\text { fermion }\n$$\n\n$$\n\\text {   of   photons   on   level - k   }\n$$\n\n$$\n\\text { with } \\quad \\text { a given } \\quad \\text { polarization }\n$$\n\n$$\n\\text { Total } \\# \\text { of photons }\n$$\n\n$$\n\\begin{array}{l} \\text { Each   } k \\text {-level has a volume} (\\frac {2 \\pi}{L}) ^ {3} \\text { in } k \\text {-space} \\\\ \\sum_ {k} \\Longleftrightarrow V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\end{array}\n$$\n\n$$\nN = 2 \\sum_ {k} \\Pi_ {k}\n$$\n\n$$\n= 2 V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {1}{e ^ {\\beta \\epsilon_ {0} - 1}}\n$$\n\n$$\n\\begin{array}{r l} {\\leftarrow \\vec {k}} & {= k \\omega_ {\\vec {k}}} \\\\ & {= k | \\vec {k} | c} \\end{array}\n$$\n\n$$\n\\omega = c k\n$$\n\n$$\n\\begin{array}{r l} {n} & {= \\frac {2}{(2 \\pi) ^ {3}} \\int_ {0} ^ {\\infty} d k \\frac {4 \\pi k ^ {2}}{e ^ {\\beta k c k} - 1}} \\\\ {\\hat {I} _ {\\mathrm{photon}}} & {= \\frac {1}{\\pi^ {2} c ^ {3}} \\int_ {0} ^ {\\infty} d w \\frac {w ^ {2}}{e ^ {\\beta w h} - 1}} \\end{array}\n$$\n\n$$\n\\text { density } = \\kappa \\left(\\frac {k _ {B} T}{k _ {C}}\\right) ^ {3}\n$$\n\n$$\nK = \\frac {1}{\\pi^ {2}} \\int_ {0} ^ {\\infty} d x \\frac {x ^ {2}}{e ^ {x} - 1} = 0. 2 3 \\dots\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0004", "text": "Physical picture:\n\ntypical energy of a photon  $\\sim k_{B}T$\n\ntypical wave length:  $k \\sim \\frac{k_B T}{k_C}$\n\none photon per  $\\lambda^{3}$$\\lambda \\sim \\frac{hc}{kBT} <$ thermal wave length\n\n$\\Rightarrow  {density}n \\approx  \\frac{1}{{\\lambda }^{3}} \\sim  {\\left( \\frac{{k}_{B}T}{hC}\\right) }^{3}$\n\n* Energy density U ~ n·kBT ~ (kBT)/(kC)³ kBT\n\n$$\n\\propto T ^ {4}\n$$\n\n$$\n\\begin{array}{l} U = 2 \\sum_ {k} \\epsilon_ {k} u _ {k} \\\\ = \\frac {8 \\pi c V k}{(2 \\pi) ^ {3}} \\int_ {0} ^ {\\infty} d k \\frac {k ^ {3}}{e ^ {\\beta c \\pi k} - 1} \\\\ = \\frac {V h}{\\pi^ {2} c ^ {3}} \\int_ {0} ^ {\\infty} d w \\frac {w ^ {3}}{e ^ {\\beta w k _ {1}}} \\\\ \\end{array}\n$$\n\n$$\n\\frac {U}{V} = \\sigma T ^ {4}\n$$\n\n$$\n\\sigma = \\frac {\\pi^ {2} k _ {3} ^ {4}}{1 5 (k <  ) ^ {3}}\n$$\n\nStefan's law\n\n$$\n\\begin{array}{l} \\text { Classical }: \\\\ U = 2 \\sum_ {k} k _ {0} T \\\\ = 2 V k _ {B} T \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\\\ = \\frac {2 V k _ {B} T}{(2 \\pi) ^ {3}} \\int 4 \\pi k ^ {2} d k = 0 0 \\\\ \\end{array}\n$$\n\nStefan's const.\n\n$$\n\\frac {U}{V} = \\int_ {0} ^ {\\infty} d \\omega U (\\omega , T)\n$$\n\n$$\nu (w) T = \\frac {k}{\\pi^ {2} c ^ {3}} \\frac {\\omega^ {3}}{e ^ {i k \\omega} - 1}\n$$\n\nPbanck distribution\n\n$$\nd w u (w, T)\n$$\n\n$$\n= \\text { energy   density   with }\n$$\n\nfrequency between W & W+di\n\n$u(w,T)$ spectral density\n\nT = 2.73 K", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0005", "text": "Black body radiation\n\nTotal power\n\narea\nA\n\n$$\n\\frac {W}{A} = \\frac {U}{V} \\frac {1}{2} C \\cdot \\int_ {0} ^ {\\frac {\\pi}{2}} d \\theta \\sin \\theta \\cdot \\cos \\theta = \\frac {1}{2}\n$$\n\n$$\n\\left( \\begin{array}{c} \\frac {1}{\\int_ {0} ^ {\\frac {\\pi}{2}} d \\theta \\sin \\theta} = 1 \\\\ <   \\cos \\theta) \\\\ \\int_ {0} ^ {1} d (\\cos \\theta) \\cos \\theta = \\frac {1}{2} \\end{array} \\right)\n$$\n\n$$\n= \\frac {1}{4} \\frac {U}{V} c\n$$\n\n$$\n\\frac {W}{A} = \\frac {c}{4} \\sigma T ^ {4}\n$$\n\nTotal power per area\n\n$$\n\\text { Power   spectrum }\n$$\n\n$$\n\\frac {\\omega}{A} = \\frac {c}{4} U (\\omega , T)\n$$\n\n$$\n= \\frac {c}{4} \\frac {\\kappa}{\\pi^ {2} c ^ {3}} \\frac {\\omega^ {3}}{e ^ {\\beta \\kappa \\omega} - 1}\n$$\n\n* Pressure :\n\n$$\n\\begin{array}{l} P = - \\frac {\\partial A}{\\partial V} = - 2 \\sum_ {k} \\frac {\\frac {\\partial E _ {k}}{\\partial V} e ^ {- \\beta E _ {k}}}{1 - e ^ {- \\beta E _ {k}}} \\\\ = \\frac {1}{3 V} 2 \\sum_ {k} \\frac {\\varepsilon_ {k} e ^ {- \\beta \\varepsilon_ {k}}}{1 - e ^ {- \\beta \\varepsilon_ {k}}} = \\frac {1}{3 V} U \\\\ \\end{array}\n$$\n\n$$\n\\Rightarrow P = \\frac {1}{3} \\frac {U}{V}\n$$\n\nradiation pressure.", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0006", "text": "② Phonons: vibration of lattice\n\nPhonons are just like photons\n\nThree differences:\n\n- ① Three polarizations (3D)\n- ② velocity 0 << c\n- ③ Upper bond of k\n\n$$\n\\# \\text {   of   } \\vec {k} - \\text {   levels   } = \\# \\text {   of   lattice   site   }\n$$\n\nSimple line drawing of a curved path with marked points (no text or symbols)\n\nSimple line drawing of a wavy pattern with no text or symbols\n\nSimple line drawing of a periodic wave with no text or symbols\n\n$$\n\\begin{array}{l} \\text {   of   } \\overline {{k}} = \\text {   levels   } = \\frac {\\overline {{\\Sigma}}}{\\overline {{k}}} 1 \\\\ = V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\\\ = V \\frac {4 \\pi}{3} k _ {\\max} ^ {3} = N _ {\\text { site }} \\\\ \\Rightarrow k _ {\\max} = \\left(\\frac {3 N _ {s i T _ {4}}}{V}\\right) ^ {1 / 3} \\\\ \\end{array}\n$$\n\n$$\nD e l y _ {e} - m o d e\n$$\n\n- ①   $\\vec{k}=\\frac{2\\pi}{L}(n_{x},n_{y},n_{z}),|\\vec{k}|<k_{max}$\n- ⑬   $\\epsilon_{k} = \\hbar v|\\vec{k}|$\n\n$$\nE i n s t e i n m o d e l\n$$\n\n- ① allowed k-levels =   $N_{site}$  (or   $|\\vec{k}| < k_{max}$ )\n- ③   $\\epsilon_{k} = \\epsilon_{0}$\n\nkz\nallowed\nk\nkmax\nks\nkx", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0007", "text": "Density of states in Debye model\n\n$$\n\\begin{array}{l} D (e) = 3 V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} S (E - h \\sigma k) \\\\ = 3 \\frac {V}{2 \\pi^ {2}} \\int d k k ^ {2} S (c - t o k) \\\\ = \\frac {3}{2 \\pi^ {2}} \\frac {V}{(x , 0) ^ {3}} \\int_ {0} ^ {\\epsilon_ {\\max}} d t ^ {\\prime} \\delta (t - t ^ {\\prime}) (t ^ {\\prime}) ^ {2} \\\\ = \\left\\{ \\begin{array}{l l} \\frac {3}{2 \\pi} \\frac {V}{(k U) ^ {3}} \\epsilon^ {2} & \\epsilon <   \\epsilon_ {\\max} = k o k _ {\\max} \\\\ 0 & \\epsilon > \\epsilon_ {\\max} \\end{array} \\right. \\\\ \\end{array}\n$$\n\nreal material\n\n- Ⓐ allowed k - levels\n- ⑧   $E_{k}=0\\sqrt{\\sin^{2}k_{x}a+\\sin^{2}k_{y}a+\\sin^{2}k_{z}a}$\n\nSimple line drawing of a cube with internal lines and shading (no text or symbols)\n\n| ε    | D(ε) |\n| ---- | ---- |\n| 0    | 0    |\n| Peak | High |\n\n* Dodger specific heat\n\ninternal energy\n\n$$\n\\frac {U}{N} = \\int d \\varepsilon D (\\varepsilon) \\frac {\\varepsilon}{e ^ {\\beta \\varepsilon} - 1} / \\frac {1}{3} \\int d \\varepsilon D (\\varepsilon)\n$$\n\n$$\n= 3 \\int_ {0} ^ {\\varepsilon_ {\\max}} d t \\frac {t ^ {3}}{e ^ {\\beta t} - 1} / \\int_ {0} ^ {\\varepsilon_ {\\max}} d t\n$$\n\n$$\n= 9 \\in_ {\\max} ^ {- 3} \\int_ {0} ^ {\\varepsilon_ {\\max}} d \\varepsilon \\frac {\\varepsilon^ {3}}{a b \\varepsilon - 1}\n$$\n\n$$\n= 3 k _ {B} T D (u) \\boxed {u = \\frac {T D}{T}} e ^ {p c - 1}\n$$\n\n$$\nD (u) = \\frac {3}{u ^ {3}} \\int_ {0} ^ {u} d t + \\frac {t ^ {3}}{e ^ {t} - 1} = \\left\\{ \\begin{array}{l l} 1 - \\frac {\\pi}{8} u & (u <   t) \\\\ \\frac {\\pi^ {4}}{5 u ^ {3}} & (u > 1) \\\\ & T <   T _ {0} \\end{array} \\right.\n$$\n\n| T_D | C_V = (dU/dT) / 3kTN |\n|-----|----------------------|\n| 0   | 0                    |\n| T_D | ~1.5                 |\n| T   | ~2.0                 |\n\n$$\nt = \\beta \\epsilon\n$$\n\n$$\nT _ {D} = \\frac {\\epsilon_ {m a r}}{k _ {B}}\n$$\n\n$$\nD e l i g e t e m p e r n T o n e\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0008", "text": "③ Boson Condensation\n\nBoson system with conserved boson number.\n\nGrand partition function\n\n$$\nQ _ {G} = \\sum_ {\\{n _ {k} \\}} e ^ {- \\beta (\\sum_ {k} n _ {k} \\epsilon_ {k} - \\mu N)} L _ {\\sum_ {k} n _ {k}}\n$$\n\n$$\n= \\prod_ {k} \\underbrace {\\sum_ {n _ {k}} e ^ {- \\beta n _ {k} (\\varepsilon_ {k} - \\mu)}} _ {} = \\prod_ {k} \\frac {1}{1 - e ^ {\\beta (\\varepsilon_ {k} - \\mu)}}\n$$\n\n$$\n\\text { partition   function }\n$$\n\n$$\n\\begin{array}{l} \\text {   f   one   oscillata   } \\\\ \\text {   with   } \\boxed {\\hbar w = \\epsilon_ {k} - \\mu} \\end{array}\n$$\n\n$$\n\\epsilon_ {k _ {4}} - \\text { energy }\n$$\n\n$$\n\\begin{array}{c} \\varepsilon_ {k _ {3}} \\\\ \\varepsilon_ {k _ {2}} \\\\ \\varepsilon_ {k _ {1}} \\\\ \\varepsilon_ {k _ {0}} \\end{array} = \\frac {\\mathrm{co,r}}{\\mu - \\mu}\n$$\n\n$$\ne ^ {- \\beta n _ {k} (t _ {k} - \\mu)} \\propto p r o t g. f o r \\text { the   oscillator }\n$$\n\n$$\n\\pi_ {0} \\text { in } - t h e n _ {u} ^ {t h} s t a t e\n$$\n\n$$\n= p _ {2} b _ {0} f _ {n} x _ {0} l e n e l - k\n$$\n\n$$\n\\text { have } n _ {k} \\text { bosers. }\n$$\n\n$$\n= e ^ {- \\beta n _ {k} (\\epsilon_ {k} - \\mu)} (1 - e ^ {- \\beta (\\epsilon_ {k} - \\mu)})\n$$\n\n$$\n\\langle n _ {k} \\rangle = \\frac {1}{e ^ {- \\beta (\\varepsilon_ {k} - \\mu)} - 1}\n$$\n\nbose distribution", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0009", "text": "56\n\nOccupation numbers of N-hoseon system\n\n$$\nN = \\sum_ {k} n _ {k}\n$$\n\n$$\n\\leftarrow \\mu \\text { is   such   that }\n$$\n\n$$\n\\text { there   are } N \\text { be   long }\n$$\n\n$$\n= V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {1}{e ^ {\\beta (\\epsilon_ {k} - \\mu)} - 1}\n$$\n\n$$\nT \\rightarrow \\infty\n$$\n\n$$\n\\mu \\rightarrow - 1 0\n$$\n\n$$\nn _ {k} \\rightarrow 0\n$$\n\n$$\nN = V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- \\beta (\\epsilon_ {k} - \\mu)}\n$$\n\n$$\n\\lambda = \\sqrt {2 \\pi k ^ {2} / m k _ {B} T}\n$$\n\n$$\n= V e \\beta \\mu \\lambda^ {3}\n$$\n\n$$\nn \\lambda^ {3} = e ^ {\\beta r} = 8\n$$\n\nIn general\n\n$$\n\\text {   beson - density   } = n = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\sum_ {k = 1} ^ {\\infty} e ^ {- m \\beta (\\epsilon_ {k} - \\mu)}\n$$\n\n$$\n= \\sum_ {m = 1} ^ {\\infty} e ^ {m \\beta \\mu} \\int \\frac {d ^ {3} k}{(2 - t) ^ {3}} e ^ {- m \\beta \\epsilon_ {k}}\n$$\n\n$$\n\\frac {1}{m ^ {3 / 2}} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- \\beta \\epsilon_ {k}}\n$$\n\n$$\n\\Rightarrow \\left| \\begin{array}{c} n \\lambda^ {3} = \\sum_ {m = 1} ^ {\\infty} \\frac {z ^ {m}}{m ^ {3 / 2}} \\\\ = g _ {3 / 2} (z) \\end{array} \\right|\n$$\n\n$$\nz = e ^ {\\beta \\mu} \\frac {a}{\\sum_ {m = 1} ^ {n} m ^ {- 3 / 2}} = 2. 6 1 2 \\dots\n$$\n\nT₂ nλ₃²\nT₁ > T₂\nT₁ nλ₁³\n↑\nfixed\n0 1 3", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0010", "text": "But  $\\mu (n,3)$ has no solution if\n\n$$\nn \\lambda^ {3} > g _ {3 / 2} (1) \\quad n k _ {B} T <   k _ {B} T _ {c} = \\frac {2 \\pi h ^ {2}}{m} \\left(\\frac {n}{g _ {3 / 2} (1)}\\right) ^ {2 / 3}\n$$\n\n* What happens: n > nc = g(1)/2 (m k₀T)^(1/2)\n\nwhen  $\\mu=0$\n\nthe excited levels\n\n$$\n\\left. \\begin{array}{c} \\hline \\\\ \\hline \\\\ \\hline \\end{array} \\right\\} \\sim \\uparrow\n$$\n\ncontain most bosons\n\n↑μ T\n\nAs we add more bosons\n\nwe cannot increase μ further\n\nBut when  $\\mu=0$, the  $k=0$ level\n\ncan have any numbers of boson.\n\n( adding lawcon to k = 0 level cost\n\nenergy μ for each boson)\n\n→ energy level of particle reservoir\n\ncannonical\nensemble.\nFermion\nμ=0 → ← → system\nμ>0 → ↑\n\nFermion\nClassical\ngas\nBozon\nsingularity\nphase transition\n\nreduce\n\nfree energy\n\nun stalle", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 9, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 10"], "page_start": 10, "page_end": 10, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0011", "text": "In the condensed state   $\\left\\{\\begin{array}{l}T<T_{c}\\\\ a_{n}>n_{c}\\end{array}\\right.$\n\n$$\nN = N _ {0} + \\frac {V}{\\lambda^ {3}} g _ {3 / 2} (1)\n$$\n\n$$\n\\begin{array}{r l} {\\Rightarrow \\frac {N _ {0}}{N}} & {= 1 - \\frac {g _ {3 / 2} (1)}{n \\lambda^ {3}}} \\\\ & {= 1 - \\left(\\frac {T}{T _ {c}}\\right) ^ {3 / 2}} \\end{array}\n$$\n\nN/N\nTc\nT\n\n* Order parameter\n\ncondensed phase   $\\frac{N_{0}}{N} \\neq 0$\n\ngas phase   $\\frac{N_{0}}{N} = 0$\n\nSo   $\\frac{N_{0}}{N}$  is an order parameter\n\nT>Tc\nD(ε)\n1\nβ(ε-μ)\nT<Tc\nNo\npenticles in\nthe excited levels\n\nNo phase transition in finite system.\n\n$$\nN = \\frac {2}{k} \\frac {1}{e ^ {\\beta (\\epsilon_ {k} - \\mu)} - 1} = \\frac {2}{k} \\frac {1}{3 e ^ {\\beta \\epsilon_ {k} - 1}}\n$$\n\nV\nV<∞\nV=∞\ny/N\ng=e^βμ\n\n$$\nN _ {0} = \\frac {1}{3 - 1}\n$$\n\n$1 - 3 : \\frac{2}{N} \\rightarrow \\frac{1}{N}\\;N_{0} = \\frac{N}{2} \\rightarrow N$\n\nfixed T\n\nsingularity = phase\nTrans\nv=00\nv<00 sharp turn\nbut no phase\ntrans.", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 10, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 11"], "page_start": 11, "page_end": 11, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0012", "text": "* Equation of state.\n\nThermal potential,\n\n$$\n\\Omega = - k _ {B} T \\ln Q _ {G} = + k _ {B} T \\sum_ {k} \\ln [ 1 - e ^ {- \\beta (\\varepsilon_ {u} - \\mu)} ]\n$$\n\n$$\np _ {\\mathrm{pressure}}\n$$\n\n$$\nP = - \\frac {\\partial \\Omega}{\\partial V} \\Big | _ {\\mu , T} = - k _ {B} T \\sum_ {k} \\frac {e ^ {- \\beta (k _ {n} - \\mu)}}{1 - e ^ {- \\beta (k _ {n} - \\mu)}} \\quad \\beta \\frac {\\partial \\epsilon_ {k}}{\\partial V}\n$$\n\n$$\n= \\sum_ {k} n _ {k} \\left(- \\frac {\\partial \\varepsilon_ {k}}{\\partial V}\\right) \\quad \\varepsilon_ {k} = \\frac {k k ^ {2}}{2 m}\n$$\n\n$$\n= \\frac {2}{3 V} \\sum_ {k} n _ {k} \\epsilon_ {k} \\Rightarrow \\boxed {P V = \\frac {2}{3} U} \\quad \\frac {\\partial \\epsilon_ {k}}{\\partial V} = - \\frac {2}{3} \\frac {\\epsilon_ {k}}{V}\n$$\n\n$$\nU = \\sum_ {k} n _ {k} \\varepsilon_ {k} \\frac {1}{\\bar {z} ^ {- 1} e ^ {\\beta \\varepsilon_ {k}} - 1}\n$$\n\n$$\n= V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\in_ {k} \\sum_ {m = 1} ^ {\\infty} \\delta^ {m} e ^ {- m \\beta \\epsilon_ {k}}\n$$\n\n$$\n= - V \\left. \\frac {\\partial}{\\partial \\beta} \\right| _ {z} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\sum_ {m = 1} ^ {\\infty} \\frac {1}{m} z ^ {m} e ^ {- m \\beta \\epsilon_ {k}}\n$$\n\n$$\n= - V \\frac {\\partial}{\\partial \\beta} | _ {z} \\sum_ {m = 1} ^ {\\infty} \\frac {1}{m} z ^ {m} m ^ {- 3 / 2} \\lambda^ {- 3}\n$$\n\n$$\n= - V \\frac {\\partial}{\\partial \\beta} | _ {3} \\lambda^ {- 3} \\sum_ {m = 1} ^ {\\infty} \\frac {1}{m ^ {5 / 2}} 3 ^ {m}\n$$\n\n$$\n= - \\sqrt {g _ {5 h} (3)} \\left. \\frac {\\partial}{\\partial \\beta} \\right| _ {0} \\lambda^ {- 3}\n$$\n\n$$\n= \\frac {3}{2} V \\lambda^ {- 3} g _ {s / 2} (3) k _ {B} T\n$$\n\n$$\n\\frac {g _ {k} ^ {(2)} = \\sum_ {m = 1} ^ {\\infty} m ^ {- k} g _ {m}}{\\frac {1}{3} \\int_ {0} ^ {3} d \\tilde {z} g _ {k} (z) = g _ {k + 1} (z)}\n$$\n\n$$\n\\left\\{ \\begin{array}{l l} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- \\beta \\epsilon_ {k}} = \\lambda^ {- 3} \\\\ \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- m \\beta \\epsilon_ {k}} = m ^ {- 3 / 2} \\lambda^ {- 3} \\end{array} \\right.\n$$\n\n$$\n\\lambda = \\sqrt {2 \\pi h ^ {2} / m k _ {B} T}\n$$\n\n$$\n\\begin{array}{l} \\lambda^ {- 3} \\propto T ^ {3 / 2} \\propto F ^ {- 3 / 2} \\\\ \\frac {\\partial}{\\partial \\beta} \\lambda^ {- 3} = - \\frac {2}{2} \\lambda^ {- 3} / \\beta \\\\ \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 11, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 12"], "page_start": 12, "page_end": 12, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0013", "text": "$$\n\\frac {U}{V} = \\frac {3}{2} k _ {8} T \\frac {g _ {5 / 2} (3)}{\\lambda^ {3}}\n$$\n\n$$\nP = \\frac {2}{3} \\frac {U}{V} = k _ {B} T \\frac {g _ {5 / 2} (3)}{\\lambda^ {3}}\n$$\n\n$$\nn \\lambda^ {3} = g _ {3 / 2} (z) = \\frac {N}{V} \\lambda^ {3} \\quad a \\quad z = 1\n$$\n\n→ determine   $3(n,T)$  as\na function of   $\\frac{N}{V}$  & T\n\n| V    | T1   | T2   | T3   |\n|------|------|------|------|\n| 0    | 1.0  | 1.0  | 1.0  |\n| 1    | 0.95 | 0.98 | 0.97 |\n| 2    | 0.9  | 0.95 | 0.94 |\n| 3    | 0.85 | 0.92 | 0.91 |\n| 4    | 0.8  | 0.88 | 0.87 |\n| 5    | 0.75 | 0.85 | 0.84 |\n| 6    | 0.7  | 0.82 | 0.81 |\n| 7    | 0.65 | 0.79 | 0.78 |\n| 8    | 0.6  | 0.76 | 0.75 |\n| 9    | 0.55 | 0.73 | 0.72 |\n| 10   | 0.5  | 0.70 | 0.69 |\n| 11   | 0.45 | 0.67 | 0.66 |\n| 12   | 0.4  | 0.64 | 0.63 |\n| 13   | 0.35 | 0.61 | 0.60 |\n| 14   | 0.3  | 0.58 | 0.57 |\n| 15   | 0.25 | 0.55 | 0.54 |\n| 16   | 0.2  | 0.52 | 0.51 |\n| 17   | 0.15 | 0.49 | 0.48 |\n| 18   | 0.1  | 0.46 | 0.45 |\n| 19   | 0.05 | 0.43 | 0.42 |\n| 20   | 0.0  | 0.40 | 0.39 |\n| 21   | -0.05| 0.37 | 0.36 |\n| 22   | -0.1 | -0.34| -0.33 |\n| 23   | -0.15| -0.31| -0.30 |\n| 24   | -0.2 | -0.28| -0.27 |\n| 25   | -0.25| -0.25| -0.24 |\n| 26   | -0.3 | -0.22| -0.21 |\n| 27   | -0.35| -0.19| -0.18 |\n| 28   | -0.4 | -0.16| -0.15 |\n| 29   | -0.45| -0.13| -0.12 |\n| 30   | -0.5 | -0.10| -0.09 |\n| 31   | -0.55| -0.07| -0.06 |\n| 32   | -0.6 | -0.04| -0.03 |\n| 33   | -0.65| -0.01| 0.00 |\n| 34   | -0.7 | 0.02 | 0.03 |\n| 35   | -0.75| 0.05 | 0.06 |\n| 36   | -0.8 | 0.08 | 0.09 |\n| 37   | -0.85| 0.11 | 0.12 |\n| 38   | -0.9 | 0.14 | 0.15 |\n| 39   | -0.95| 0.17 | 0.18 |\n| 40   | -1.0 | 0.2     | 0.21 |\n| 41   | -1.05| 0.23 | 0.24 |\n| 42   | -1.1 | 0.26 | 0.27 |\n| 43   | -1.15| 0.29 | 0.3   |\n| 44   | -1.2 | 0.32 | 0.33 |\n| 45   | -1.25| 0.35 | 0.36 |\n| 46   | -1.3 | 0.38 | 0.39 |\n| 47   | -1.35| 0.41 | 0.42 |\n| 48   | -1.4 | 0.44 | 0.45 |\n| 49   | -1.45| 0.47 | 0.48 |\n| 50   | -1.5 | 0.5   | 0.51 |\n| 51   | -1.55| 0.53 | 0.54 |\n| 52   | -1.6 | 0.56 | 0.57 |\n| 53   | -1.65| 0.59 | 0.6   |\n| 54   | -1.7 | 0.62 | 0.63 |\n| 55   | -1.75| 0.65 | 0.66 |\n| 56   | -1.8 | 0.68 | 0.69 |\n| 57   | -1.85| 0.71 | 0.72 |\n| 58   | -1.9 | 0.74 | 0.75 |\n| 59   | -1.95| 0.77 | 0.78 |\n| 60   | -2.0 | 0.8   | 0.81 |\n| 61   | -2.05| 0.83 | 0.84 |\n| 62   | -2.1 | 0.86 | 0.87 |\n| 63   | -2.15| 0.89 | 0.9   |\n| 64   | -2.2 | 1.      |        |\n|    |      |      |      |\nP = N fixed\n\ndensity\nof excited\nbosons is\nind. of V\n\nSpecific heat:\n\nCondensed phase 3=1\n\n$$\nU = \\frac {2}{2} k _ {B} T \\frac {g _ {5 / 2} (1)}{\\lambda^ {3}} \\propto T ^ {5 / 2}\n$$\n\n$$\nC _ {v} = \\left. \\frac {\\partial U}{\\partial T} \\right| _ {V, N} = \\frac {5}{2} \\cdot \\frac {3}{2} k _ {B} \\frac {g _ {7 / 2} (1)}{\\lambda^ {3}} V = \\boxed {\\frac {1 5}{4} k _ {B} \\frac {g _ {5 / 2} (1)}{\\lambda^ {3}} V = C _ {v}}\n$$\n\n| T     | Value |\n|-------|-------|\n| 2.17  | Cᵥ    |\n\n$T^{3/2}$ conserved boson with  $\\epsilon_{k} \\propto k^{2}$\n\nT conserved formation with  $\\epsilon_{k}$ = anything\n\n$T^{3}$ non-conserved bosons with  $E_{h} \\propto k$\n\nT $^{3}$ conserved interacting bosons with  $\\epsilon_{k}=anything$\n\n$$\nc _ {v} = k _ {B} V \\left[ \\frac {1 5}{4} \\frac {g _ {5 / 2} (3)}{\\lambda^ {3}} - \\frac {9}{4} \\frac {g _ {3 / 2} (3)}{g _ {1 / 2} (3)} \\right]\n$$\n\n$$\ng _ {1 / 2} (1) = \\infty\n$$\n\nCv\nFree boson\ngas\nT³/₂\nTc\nT", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 12, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 13"], "page_start": 13, "page_end": 13, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0014", "text": "Boson condensation in ultracold atom system\n\nTransition temperature\n\n$$\nn \\lambda^ {3} = g _ {3 / 2} (1) \\Rightarrow k _ {B} T _ {c} = \\frac {2 \\pi h ^ {2}}{m} \\left(\\frac {n}{g _ {3 / 2} (1)}\\right) ^ {2 / 3}\n$$\n\n$$\n\\hat {L} _ {\\lambda} = \\sqrt {2 \\pi k ^ {2} / m k _ {B} T}\n$$\n\n$$\ng _ {3 / 2} (1) = 2. 6 1 2 \\dots\n$$\n\n$$\n\\begin{array}{l} \\mathrm{for} N a \\quad \\mathrm{atoms} \\quad \\mathrm{if} n = 1 0 ^ {1 4} \\mathrm{cm} ^ {- 3} \\\\ z = 1 1, A = 2 3 \\quad T _ {c} = 1. 5 \\mu K \\\\ f _ {n} H e p = 0. 1 2 g c m ^ {- 3} n = 1. 8 \\times 1 0 ^ {2 2} c m ^ {- 3} \\\\ z = 2, A = 4 \\quad T _ {c} = 1. 8 K \\quad a c t u r a l T _ {c} = 2. 1 7 \\\\ \\end{array}\n$$\n\nMomentum distribution\n\n$N(k) dk = \\# of \\text{ bosons with wave vector}$\n\nbetween k & k+dk\n\n$$\n= \\sqrt {4 \\pi k ^ {2}} \\frac {1}{e ^ {\\beta (k - \\mu)} - 1}\n$$\n\n| T     | N(k) |\n|-------|------|\n| < Tc  | Peak |\n| > Tc  | Decreasing |\n\nCondensation   $N_{0}=N$$\\frac{V}{x^{3}}g_{3/2}(1)$\n\nDetecting condensation\n\nTime fly:\n\nT>Tc\nT<Tc\n→\n←condensation\n\n| Point | Value |\n|---|---|\n| N₀ | 1 |\n| Tc | 0 |\n\nInterference :", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 13, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 14"], "page_start": 14, "page_end": 14, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0015", "text": "④ Interacting bosons and superfluidity\n\n$$\n\\begin{array}{l} T = 0 \\quad \\text { All } \\quad \\text { bosons } \\quad \\text { condense. } \\\\ \\text { All   bosons   are   in   the   same   state. } \\\\ \\end{array}\n$$\n\nField Theory for free condensed bosons:\n\n$$\n\\text { One } \\quad \\text { bosen } \\quad S - q g\n$$\n\n$$\ni k \\frac {\\partial N}{\\partial t} = \\underbrace {\\left(- \\frac {h ^ {2}}{2 m} \\nabla^ {2} + U\\right) \\psi} _ {H}\n$$\n\n$$\nx _ {0} + 1 \\text {   and   particle   for   state   } y\n$$\n\n$$\n\\int d ^ {3} x \\quad 4 ^ {x} + = 1\n$$\n\n$$\n\\text { total   energy   for   state } +\n$$\n\n$$\nE _ {1} = \\int d ^ {3} x - 4 * H \\psi\n$$\n\n$$\n= \\int d ^ {3} x. \\left(\\frac {\\xi^ {2}}{2 m} | \\nabla + 1 | ^ {2} + U | 4 | ^ {2}\\right)\n$$\n\n$$\n\\vert \\partial_ {x} + 1 ^ {2} + \\vert \\partial_ {y} + 1 ^ {2} + \\vert \\partial_ {z} + 1 ^ {2}\n$$\n\n* N. boson describe by the same wave function.\n\n$$\n\\int d ^ {3} x \\vert + 1 ^ {2} = N, \\quad \\vert + 1 ^ {2} = \\rho (\\text { boson   densit } _ {j})\n$$\n\n$$\n\\int d ^ {3} x \\left(\\frac {c _ {0} ^ {2}}{2 m} | \\nabla + 1 ^ {2} + U | + 1 ^ {2}\\right) = E = N E,\n$$\n\nNot quite right N-bosen wave function 互  $(\\vec{x},\\cdots,\\vec{x}_{d})$\n\n$$\n= \\pi + (\\vec {x} _ {i})\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 14, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 15"], "page_start": 15, "page_end": 15, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0016", "text": "4\n\nGround state is given by 40 that\n\nminimize E and satisfy   $\\int d^{3}x|N_{a}|^{2}=N$\n\nexample: U = 0   $\\psi_{0} = \\sqrt{n} e^{i\\theta}$  I any phase.\n\n$x^{2}+y^{2}=r^{2}$\n\n\"Grand canonical\" ensemble. (N is not fixed)\n\n$$\n\\Omega = E - \\mu N = \\int d ^ {3} x \\left[ \\frac {\\hbar^ {2}}{2 m} | \\nabla + 1 ^ {2} + (U - \\mu) 1 + 1 ^ {2} \\right]\n$$\n\nGround state is given by  $t_0$ that minimize\n\nΩ ( no other constant )\n\nEq. of motion: (EOM)\n\n$$\ni \\hbar \\frac {\\partial}{\\partial t} 4 = (- \\frac {\\hbar^ {2}}{2 m} \\nabla^ {2} + U - \\mu) 4\n$$\n\nHow to find   $\\psi_{0}$  (and relation between   $\\Omega$  and Eom)\n\n$$\n\\begin{array}{l} \\zeta \\Omega = \\int d ^ {3} x \\frac {\\hbar^ {2}}{2 m} \\nabla \\delta \\psi^ {*} \\nabla \\psi + (U - \\mu) \\delta \\psi^ {*} \\psi \\\\ + \\frac {x ^ {2}}{2 n} \\nabla 4 ^ {*} \\nabla s 4 + (U - \\mu) 4 ^ {*} s 4 + O (s t ^ {2}) \\\\ = \\int d ^ {3} x \\delta \\psi \\left(- \\frac {\\chi^ {2}}{2 m} \\nabla^ {2} + U - \\mu\\right) \\psi + \\delta \\psi \\left(- \\frac {\\chi^ {2}}{2 m} \\nabla^ {2} + U - \\mu\\right) \\psi^ {4} \\\\ \\end{array}\n$$\n\n$$\n\\Rightarrow \\sqrt {\\left[ - \\frac {k ^ {2}}{2 m} \\nabla^ {2} + (U - \\mu) \\right] - k _ {0} = 0} \\tag {*}\n$$\n\nRight hand side of equation of motion", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 15, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 16"], "page_start": 16, "page_end": 16, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0017", "text": "How to use:\n\n⑨ Pick a μ. ⑥ Find a 40 xh.t satisfies (*)\n\n② check if   $\\int dx^{3}|t_{0}|^{2}=N$  ① if not find another\n\n40 and/or pick a new μ\n\nExample: U = const.\n\n①   $\\mu < U: \\left(-\\frac{k^{2}}{2m}\\nabla^{2} + U - \\mu\\right) t_{0} = 0$\n\nhas only one solution   $4_{0}=0$\n\n$\\Rightarrow  {N}_{0}$ bosons\n\n②   $\\mu = U$$(- \\frac{x^{2}}{2m} \\sigma^{2}) + f_{0} = 0$\n\nhas many solutions   $\\gamma_{0}=c e^{i\\theta}$\n\nC, θ = √ numbers\n\n⇒ any number bosons\n\n③   $\\mu > U$$(- \\frac{c^{2}}{2m} \\nabla^{2} + (U - \\mu)) \\times_{0} = 0$\n\nhas many solutions   $y_{0}=ce^{i\\frac{1}{k}\\cdot\\frac{1}{x}}$\n\n$$\n\\frac {b ^ {2} k ^ {2}}{2 m} = \\mu - U\n$$\n\n$B_{n}+$$\\Omega=\\int d^{3}x\\quad\\psi_{0}^{*}\\left(-\\frac{\\pi^{2}}{2n}\\sigma^{2}+U-\\mu\\right)\\psi_{0}=0$\n\nif  ${40} = \\sqrt{n} \\Rightarrow  \\Omega  = \\int {d}^{3}x\\;n\\left( {U - \\mu }\\right)  < 0$\n\nThus  $\\psi_{0}=c e^{i\\vec{k}\\cdot\\vec{x}}$ is a maximum NOT minimum", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 16, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 17"], "page_start": 17, "page_end": 17, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0018", "text": "To get a N-bosou slate we must\n\n$$\n\\mathrm{set} \\quad \\mu = U \\quad \\mathrm{and} \\quad \\underline {{\\underline {{\\mathrm{choose}}}}} \\quad \\psi_ {0} = \\sqrt {n} e ^ {i \\theta} n = \\frac {N}{V}\n$$\n\n(In general there is only one solution  $\\mathcal{L}_0$\n\nfor each choice of μ. We need to\n\n$\\text{tune } \\mu \\text{ so make } \\int d^{3} x |y|^{2} = N)$\n\nCollective excitations\n\n$$\n4 = 4 0 + 8 - 4\n$$\n\nL excitations\n\n$$\n\\alpha_ {0} (x, t), \\quad \\text { satisfies } \\quad i _ {k} \\frac {\\partial \\psi_ {0}}{\\partial t} = (- \\frac {x ^ {2}}{2 m} \\nabla^ {2} + U - \\mu) x _ {0}\n$$\n\n$$\n= 0\n$$\n\n$$\n\\Rightarrow \\psi_ {0} = \\psi_ {0} (\\vec {x})\n$$\n\n$\\text{I no } t \\text{ dependence}$\n\n$$\ni \\frac {\\partial f}{2 t} = (- \\frac {k ^ {2}}{2 m} \\nabla^ {2} + U - \\mu) +\n$$\n\n$$\n\\Rightarrow \\boxed {i \\hbar \\frac {\\partial f}{\\partial t} = (- \\frac {\\hbar^ {2}}{i m} \\nabla^ {2} + U - \\mu) \\delta - 4}\n$$\n\nEq. of motion for excitations\n\n$$\n\\text {   example   } U = \\text {   const.   } \\quad \\mu = U\n$$\n\n$$\n\\Rightarrow s - 4 = * e ^ {i \\vec {k} \\cdot \\vec {x} - i \\omega_ {k} t}\n$$\n\nground state\n\n$$\n\\hbar W _ {k} = \\frac {\\hbar^ {2} k ^ {2}}{2 m} \\leftarrow e n e g y \\quad f e x c i t a t r o n\n$$\n\n$$\n\\vec {k} = \\underset {\\mathrm{ofexcitation}} {\\mathrm{momentum}} (\\mathrm{oneexcitedbozon})\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 17, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 18"], "page_start": 18, "page_end": 18, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0019", "text": "k₂\nk₁\nk = 0\n\nMore general\n\n$$\n\\begin{array}{l} u (x, t) = \\psi_ {0} + c _ {1} e ^ {i k _ {1} x - i \\omega_ {1} t} + c _ {2} e ^ {i k _ {2} \\vec {x} - i \\omega_ {2} t} \\\\ \\mathrm{企} c _ {0} e ^ {i \\theta} \\\\ \\end{array}\n$$\n\n$$\n\\vert c _ {0} \\vert^ {2} \\quad \\text { density   of   boous   in   the } \\vec {k} = 0 \\quad \\text { level }\n$$\n\n$$\n\\vert c _ {1} \\vert^ {2} \\dots \\dots \\quad k = k _ {1} \\quad \\text { low }\n$$\n\n$$\n\\begin{array}{l l} {\\vec {k _ {1}}} & {\\mathrm{momentum}} \\\\ {\\vec {k _ {w _ {1}}}} & {\\mathrm{energy}} \\end{array} \\geq \\mathrm{ofoneboson}\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 18, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 19"], "page_start": 19, "page_end": 19, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0020", "text": "Interacting bosons.\n\n$$\n\\begin{array}{l} \\Omega = \\int d ^ {3} x \\left[ \\frac {\\hbar^ {2}}{2 m} | \\nabla + 1 | ^ {2} + U | + 1 | ^ {2} \\right] \\\\ + \\int d ^ {3} x d ^ {3} x ^ {\\prime} \\frac {1}{2} V (\\vec {x}, \\vec {x} ^ {\\prime}) (4 (\\vec {x}) ^ {2} | - 4 (\\vec {x} ^ {\\prime}) | ^ {2} = \\int d ^ {3} x m | + | ^ {2} \\\\ \\end{array}\n$$\n\n$(x)$$(x')$\n\nEq. of motion\n\n$$\n\\therefore \\frac {0}{\\sigma t} \\psi = (- \\frac {\\zeta^ {2}}{2 m} \\nabla^ {2} + V _ {\\text { eff }} - \\mu) 4\n$$\n\n$V(\\vec{x},\\vec{x}^{\\prime}) \\text{ potential } 1$\n\nbetween a boson at  $\\frac{1}{x}$\n\nand a boson at   $\\vec{x}$\n\nnon-linear S-g\n\n$$\nU _ {2 8 f} (\\vec {x}, \\vec {y}) = U (\\vec {x}) + \\int d ^ {3} \\vec {x} ^ {\\prime} V (\\vec {x}, \\vec {x} ^ {\\prime}) 1 - 4 (\\vec {x} ^ {\\prime}) ^ {2}\n$$\n\nGround state  $V_{0}$ minimizes  $\\Omega : \\left[-\\frac{h^{2}}{2m} \\nabla^{2} + U_{eff}(x, V_{0}) - \\mu\\right] V_{0} = 0$\n\n* Short range interaction  $V(\\vec{x},\\vec{x}^{\\prime})=U_{0}\\delta^{3}(\\vec{x}-\\vec{x}^{\\prime})$\n\n$$\n\\Omega = \\int d ^ {3} x [ \\frac {x ^ {2}}{2 m} | 0 + 1 | ^ {2} + (U - \\mu) | 4 1 | ^ {2} + \\frac {V _ {0}}{2} | 4 1 | ^ {4} ]\n$$\n\n$$\n\\text {   i.e.   } \\frac {\\partial}{\\partial t} + = (- \\frac {\\hbar^ {2}}{2 m} \\nabla^ {2} + U - \\mu + \\sigma_ {0} | + 1 ^ {2}) + G _ {\\mathrm{loss}} - P _ {\\mathrm{if~or~s}}\n$$\n\nE8.\n\nEq. motion for collective excitations   $S_{4}=4-\\frac{1}{\\hat{I}_{no}}$\n\n$$\n\\text {   i.e.   } \\frac {\\partial}{\\partial t} \\delta \\psi = (- \\frac {k ^ {2}}{2 m} \\nabla^ {2} + U - \\mu + 2 U _ {0} | - k _ {0} | ^ {2}) \\delta \\psi\n$$\n\n$$\n+ v _ {0} (4) ^ {2} \\delta 4 ^ {*} + O (\\delta 4 ^ {2})\n$$\n\n$\\hat{L}_{nc} + t$\ndep.", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 19, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 20"], "page_start": 20, "page_end": 20, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0021", "text": "$$\n\\star \\text {   Example:   } U = 0 \\quad \\Omega = \\int d ^ {3} x \\left(\\frac {\\pi^ {2}}{2 n} | 0 + 1 | ^ {2} - \\mu | 4 | ^ {2} + \\frac {1}{2} \\sigma_ {0} | 4 | ^ {4}\\right)\n$$\n\n$$\nⒶ G r o u n d s t a t e: \\quad \\left(- \\frac {x ^ {2}}{2 m} \\nabla^ {2} - \\mu + v _ {0} | t _ {0} | ^ {2}\\right) t _ {0} = 0\n$$\n\n$$\ny _ {0} = \\frac {\\sqrt {n} e ^ {i 0}}{c o n s t} \\quad \\text { minimize } \\quad \\Omega \\quad V (x) = - \\mu | x | ^ {2} + \\frac {1}{2} \\sigma_ {0} | x | ^ {4}\n$$\n\n$$\n\\Rightarrow (- \\mu + v _ {0} n) \\sqrt {n} e ^ {i \\theta} = 0\n$$\n\nμ<0\nμ>0\nφ\n\n$$\n\\Rightarrow n = \\frac {\\mu}{v _ {0}} \\quad \\text { or } \\quad 0\n$$\n\n$$\nⓛ \\mu <   0 \\quad n _ {0} = 0, n = \\frac {\\mu}{v _ {0}} <   0 | _ {n o t a l l o w e d}\n$$\n\n$$\nⓏ \\mu > 0 \\quad y _ {o} = \\sqrt {\\frac {\\mu}{\\nu_ {o}}} e ^ {i \\theta}, n = 0 \\mid_ {\\max i m a m}\n$$\n\n$$\nF _ {n} = N - b o s o n \\quad \\text {   system   } \\quad \\frac {N}{V} = \\frac {\\mu}{v _ {0}} \\quad \\text { or } \\quad \\mu = v _ {0} n\n$$\n\n$$\n\\Omega = \\left\\{ \\begin{array}{l l} 0 & \\mu <   0 \\\\ - \\frac {1}{2} \\frac {\\mu^ {2}}{\\sigma_ {0}} & \\mu > 0 \\end{array} \\right. \\quad \\psi_ {0} = 0 \\quad \\psi_ {0} \\neq 0\n$$\n\nsingularity\nphase Trans.\nno boson\nμ\nboson condensed\n\n$$\nⒷ \\text {   Collectiie   excitations   }\n$$\n\n$$\nT = 0 \\quad T _ {u n s i t i o n}\n$$\n\n$$\n\\text { quantum   phase   trans. }\n$$\n\n$$\n\\mathrm{法} \\frac {\\partial}{\\partial t}   \\delta t = (- \\frac {k ^ {2}}{2 m} \\partial^ {2} + \\mu)   \\delta t + \\mu   \\delta 4 ^ {*}\n$$\n\n$$\n\\delta x = \\delta x _ {1} + i \\delta x _ {2}\n$$\n\n$$\n\\left\\{ \\begin{array}{l l} - \\hbar \\frac {\\partial}{\\partial t} \\delta \\psi_ {2} = (- \\frac {\\hbar^ {2}}{2 m} \\nabla^ {2} + 2 \\mu) \\delta \\psi_ {1} \\\\ \\hbar \\frac {\\partial}{\\partial t} \\delta \\psi_ {1} = - \\frac {\\hbar^ {2}}{2 m} \\nabla^ {2} \\delta \\psi_ {2} \\end{array} \\right.\n$$\n\n$$\n- \\hbar^ {2} \\frac {\\partial^ {2}}{\\partial t ^ {2}} \\delta \\psi_ {2} = (- \\frac {\\hbar^ {2}}{2 m} \\nabla^ {2} + 2 \\mu) (- \\frac {\\hbar^ {2}}{2 m} \\nabla^ {2}) \\delta \\psi_ {2}\n$$", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 20, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 21"], "page_start": 21, "page_end": 21, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd:page-0022", "text": "$$\n\\left\\{ \\begin{array}{l l} {\\delta \\psi_ {2} = c \\sin (\\vec {k} \\cdot \\vec {x} - \\omega_ {k} t + \\theta)} \\\\ {\\delta \\psi_ {1} = \\frac {\\hbar k ^ {2}}{2 m \\omega_ {k}} c \\cos (\\vec {k} \\cdot \\vec {x} - \\omega_ {k} t + \\theta)} \\end{array} \\right.\n$$\n\n$$\n\\omega_ {k} ^ {2} = \\left(\\frac {\\hbar^ {2}}{2 m} k ^ {2} + 2 \\mu\\right) \\frac {k ^ {2}}{2 m}\n$$\n\n$$\n\\omega_ {k} = \\sqrt {\\frac {k ^ {2}}{2 m} \\left(\\frac {k ^ {2}}{2 m} k ^ {2} + 2 \\mu\\right)} = \\sqrt {\\frac {k ^ {2}}{2 m} \\left(\\frac {k ^ {2}}{2 m} k ^ {2} + 2 v _ {0} a\\right)}\n$$\n\n$$\n\\begin{array}{r l} & {= \\frac {\\psi_ {s} | \\vec {k} |}{\\text { small }} \\frac {1}{k}} \\\\ & {= \\frac {k \\vec {k} ^ {2}}{\\frac {2 m}{\\text { large }} \\vec {k}}} \\end{array}\n$$\n\n$$\n\\boxed {\\begin{array}{l}v _ {s} = \\sqrt {\\frac {\\mu}{m}} = \\sqrt {\\frac {v _ {0} n}{m}}\\\\\\rightarrow \\text { sound   velocity }\\end{array}}\n$$\n\nk\\omega_k\nenergy of\ncollective mode\nint. bosons\nfree bosons\nk\\omega_k = \\frac{k^2}{2m}\n\n$$\n\\text { Low   energy   excitations } \\hbar \\omega_ {k} = 0. 1 \\vec {k}\n$$\n\n$$\n\\text { not } \\quad E _ {k} = \\frac {k ^ {2} k ^ {2}}{2 m} | \\text { free   boson }\n$$\n\n$$\nc _ {v} \\propto T ^ {3} \\quad \\text { at   low } T \\quad \\text { mot } c _ {v} \\propto T ^ {3 / 2} | _ {\\mathrm{free}}\n$$\n\n```mermaid\ngraph TD\nA[\"→\"] --> B[\"↑\"]\nB --> C[\"←\"]\nC --> D[\"↓\"]\nD --> E[\"→\"]\n```\n\nsymmetry breaking & gapless mode.", "source": "mit-ocw", "source_doc_id": "0007-vii-the-bose-gas-1-2-6-mb-2-b191ba44-cac5bcdd", "source_title": "VII The Bose Gas.", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 21, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VII The Bose Gas.", "Page 22"], "page_start": 22, "page_end": 22, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32:page-0001", "text": "Superfluidity\n\n$$\n\\begin{array}{l} \\Omega = \\int d x ^ {5} \\frac {\\hbar^ {2}}{2 m} | \\nabla + 1 | ^ {2} \\\\ - \\mu | \\psi | ^ {2} + \\frac {1}{2} v _ {0} | \\psi | ^ {4} \\\\ \\end{array}\n$$\n\nHand-drawn diagram with labeled arrows and regions, including 'x' and 'R'\n\nV(t)\n4\n4₀ ≠ 0\n\ndensity current\n\n$$\nJ _ {x} = \\sigma_ {x} \\rho = \\frac {1}{m} R e (\\psi^ {*} \\frac {\\kappa}{i} \\partial_ {x} \\psi)\n$$\n\nif   $t = t_{0} = const.$$J_{x} = 0$  no flow\n\nif  $y = {40}{e}^{i\\frac{n}{R}x} = \\frac{n}{k{R}m} \\mid  {40}{1}^{2}\\;{flow} \\neq  0$\n\nKey: 4.0e $^{i\\frac{n}{R}x}$ minimizes Ω and\n\nsatisfies the eg. of motion.\n\nIn order for the flow to decay to zero\n\nn must decay to zero. But n is\n\nquantized as integer and cannot decay\n\nThus the\n\nfluid keep\n\nflowing   $\\Rightarrow$  super fluid\n\ntraped\na\n-2 -1 0 1 2 n\nd", "source": "mit-ocw", "source_doc_id": "0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32", "source_title": "72", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["72", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32:page-0002", "text": "$$\nI _ {n} \\cdot \\alpha d e _ {2} \\quad f o r \\quad n = 1 \\Rightarrow n = 0\n$$\n\n$$\n\\therefore 4 = e ^ {i \\frac {x}{2}} 4 _ {0} \\Rightarrow 4 _ {0}\n$$\n\n$$\n\\text { must   pass } \\quad \\nu = 0\n$$\n\n$$\n\\text { potential   barrier, } n \\text { cannot   change }\n$$\n\n$$\n\\text { Only } \\quad \\text { vortex.   tunneling   can   change   n }\n$$\n\n$$\n\\text { and } \\quad \\text { reduce } \\quad \\text { the } \\quad \\text { superfluid } \\quad \\text { flow }\n$$\n\nn=1\nbarrie2\nn=0\n\n* What is votex\n\n$$\n\\psi = f (r) \\psi_ {0} e ^ {i \\varphi}\n$$\n\n$$\n(x = r \\cos \\varphi , y = r \\sin \\varphi)\n$$\n\nf(Y)\nsize f vortex cove\nφ = π/2\nφ = π\nφ = 0, 2π\nφ = (3/2)π\n\n$$\n\\text { velocity } \\quad \\vec {v} _ {s} = \\frac {k}{m} \\nabla \\varphi (x, y)\n$$\n\n$$\n\\text { density   current } \\quad \\frac {\\vec {J}}{J} = n \\frac {\\vec {S}}{v _ {s}}\n$$\n\n$$\n\\text { quantization   of   vozticity }\n$$\n\n$$\n\\oint d \\vec {s} \\cdot \\vec {v _ {s}} = \\frac {h}{2 m} \\cdot 2 \\pi \\cdot i n t = \\frac {h}{m} \\times i n t e g e s\n$$\n\nNo uniform rotation\n→U(→Y) = →ω × Y\nin superfluid!\nx integer\nG ∪ G\nG G G\nG G\n\nTunneling:\n\n2π(n+1)\n2π\n2πh\n\nTop view\n\nvorlex\nΔphase = 2π\nΔphase = 2πn\nΔphase = 2π (n+1)", "source": "mit-ocw", "source_doc_id": "0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32", "source_title": "72", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["72", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32:page-0003", "text": "# 72\n\n* Excitation on a flowing superfluid\n\n$$\n\\frac {t = t _ {0} , v _ {s} = 0}{\\underline {{\\underline {{}}}}}\n$$\n\n$$\nw _ {k} = v | k |\n$$\n\nw_k\nk\n\n$$\n\\frac {4 - 4 _ {0} e ^ {i k x} , v _ {s} = \\frac {\\hbar K}{m}}{\\Rightarrow \\Rightarrow \\Rightarrow}\n$$\n\n$$\nw _ {k} = \\left\\{ \\begin{array}{l l} (U + U _ {s}) | k | & k > 0 \\\\ (U - U _ {s}) | k | & k <   0 \\end{array} \\right.\n$$\n\n$$\n= v | k | + v _ {s} k\n$$\n\nstable flow no friction\n\n$v_{s} < v$\n\nW_k\nV_s > V\n← un st al k\n\nReal system\n\nw_k\nv_c k\nv_s = 0\n\nW_k\nU_s = U_c\n\n${v}_{c}$ critical velocity of superfluid flow.\n\n* No superfluid flows for free boson condensed states\n\nw_k\nv_c = 0\nk\n\n4 = e^im\\frac{x}{R}\n\nεm\nN\nm\nεm\nN-1\nm", "source": "mit-ocw", "source_doc_id": "0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32", "source_title": "72", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["72", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32:page-0004", "text": "Remark\n\nBoson condensation: single-particle state  $\\psi_{0}(\\lambda)$\n\nBoson condensed state = all bosons are in No state\n\n$\\Rightarrow N - \\text{bosch wave function } f(x_1, \\ldots, x_N) = f_0(x_1) \\cdots f_0(x_N)$\n\ndensity : single particle state, N-boson state\n\nn₀\n← →\nL\n\nNn₀\nL\n\nCollective excitations\n\nchange  $a_{0}(x) \\rightarrow a_{1}(x)$ ( $a_{1} \\& a_{0}$ do not have to\n\nNew N-boson state\n\neach other)\n\n$\\psi_{1}(x_{1}\\cdots x_{N})=\\psi_{1}(x_{1})\\cdots\\psi_{1}(x_{N})$\n\nTwo types of collective modes: (ground state   $\\chi_{0}=const.$ )\n\na) density wave   $\\psi_{1} = \\psi_{0} + x e^{i k \\cdot x}$\n\nb)  $v_{n} + a_{n}$$+1 = f(r)e^{i\\theta} + 0$\n\n(minimum) (r, θ, θ) poln coordinate\n\nn-vintex   $\\psi_{1}=f(r)e^{in\\theta}\\psi.$", "source": "mit-ocw", "source_doc_id": "0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32", "source_title": "72", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["72", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32:page-0005", "text": "74\n\n* G-L theory of boson condensation\n\nBoson condensed state: All bosons are in\n\nthe same single-particle state   $\\psi(x)$ .\n\n( ie N- Boson wave function   $f(x_{1},\\cdots,x_{n})=\\psi(x_{1})\\cdots\\psi(x_{n})$ )\n\n$\\text{order parameter} = \\text{amplitude of condensed bosons}$\n\nA+ T=0 the free energy (on energy) of interacting\n\nboson\n\n$$\nA = \\int d ^ {3} x \\left[ \\frac {\\hbar^ {2}}{2 m} | \\nabla \\psi | ^ {2} + (U - \\mu) | \\psi | ^ {2} + \\frac {U _ {0}}{2} | \\psi | ^ {4} \\right]\n$$\n\nFor finite T. 4 amplitude of condensed bosons\n\n$$\nA = \\int d ^ {3} x \\left[ \\frac {\\hbar^ {2}}{2 m} | \\nabla \\psi | ^ {2} + a (T) | \\psi | ^ {2} + \\frac {v _ {0}}{2} | \\psi | ^ {4} \\right] + A _ {0} (T)\n$$\n\npictau:\n\ncondensed\n↔\nnon-condensed\nμ(T)\n\n$a(T)>0$$\\eta=0$  no condensation\n\n$a(T) < 0$$4 \\neq 0$ finite condensation\n\nA   $U_{(1)}$  symmetry   $\\rightarrow$$e^{i\\theta}$  A   $\\rightarrow$  A\n\nSimple line drawing of a curved shape with a small arrow and a cross mark (no text or symbols)", "source": "mit-ocw", "source_doc_id": "0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32", "source_title": "72", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["72", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32:page-0006", "text": "Minimize  $a(T) |t|^{2} + \\frac{v_{0}}{2} |t|^{4}$\n\n$$\n\\Rightarrow a (r) + v _ {0} (r + 1) ^ {2} = 0\n$$\n\nl\nTc\n\n$$\n\\Rightarrow \\frac {A}{V} = \\frac {A _ {0}}{V} + \\left\\{ \\begin{array}{l l} 0 & a (T) > 0 \\\\ - \\frac {1}{2} \\frac {a ^ {2}}{U _ {0}} & a (T) <   0 \\end{array} \\right.\n$$\n\n$$\nS = - \\frac {\\partial A}{\\partial T}\n$$\n\n$$\nC = T \\frac {\\partial S}{\\partial T}\n$$\n\n| Tc | S     |\n|----|-------|\n| 0  | 0     |\n| >1.5 | >2A₀  |\n\n| T    | e_r  |\n| ---- | ---- |\n| 0    | 0    |\n| 3    | 1    |\n\n$$\n\\mathrm{na} a \\left(H _ {e} ^ {4}\\right)\n$$\n\n$$\n\\text { free   bosons }\n$$\n\n| T | C |\n|---|---|\n| 0 | 0 |\n| 2.17 K | Peak (labeled as '2.17 K') |\n\n| Temperature (T) | Concentration (C) |\n| :--- | :--- |\n| 0 | 0 |\n| T_c | 3½ |\n| T | 1 |", "source": "mit-ocw", "source_doc_id": "0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32", "source_title": "72", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["72", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32:page-0007", "text": "Solid as \"boson condensation\" (CDW)\n\nliquid\n\nsolid\n\nwhat is order parameters?\n\n$$\nn (x) = n _ {0} + \\alpha \\cos (k x + \\phi) = n _ {0} + R e (\\phi e ^ {i k x})\n$$\n\nL complex num bin.\n\n$\\phi$ is order parameter for a solid (charge-density-wave) (CDW)\n\n$$\n\\sim = 0 \\Rightarrow n o C D W\n$$\n\nphase of   $\\sim\\Rightarrow$  position of CDW\n\nTranslation symmetry  $\\Rightarrow$ energy does not depend\n\non xu position of CDW\n\n$=) \\text{free energy} \\quad \\text{does not depend}$\n\non the phase of   $\\psi(U(1)\\text{ symmetry})$$b(T)$\n\n$\\Rightarrow  G - L$ theory  $A = \\int d^{D} x a(T) |t|^{2} + |t|^{4} + \\cdots$\n\nCDW = Boson condensation   $\\left\\{\\begin{array}{l} no +4+4, +4 + +4 - 4. \\end{array}\\right.$  terms\n\n$\\psi(x) = f(x)e^{i\\theta} \\psi_0 \\Rightarrow \\text{vortex in BC} \\left\\{ \\begin{array}{l} \\text{since they} \\\\ \\text{symmetry} \\end{array} \\right. \\text{break the } U(1)$\n\nwhat   $n(x)=n_{0}+Re(\\psi(\\vec{x})e^{i\\vec{k}-\\vec{x}})$  looks like?\n\nX", "source": "mit-ocw", "source_doc_id": "0008-vii-the-bose-gas-1-2-6-mb-2-f5dd4bcd-5b1a4e32", "source_title": "72", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["72", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0001", "text": "VIII The Fermi gas:\n\n① Free Fermions =\n\n* Single particle states  $k > \\vec{k} = \\frac{2\\pi}{L}\\left( {{n}_{x},{n}_{y},{n}_{z}}\\right)$\n\nMulti-particle states\n\n$$\n\\vert n _ {k _ {1}} n _ {k _ {2}} \\dots > \\equiv \\vert \\{n _ {\\frac {1}{k}} \\}\n$$\n\n$$\nn _ {\\vec {k}} \\text { is   of   particles   on } | \\vec {k} \\rangle\n$$\n\n$$\nF _ {e r} \\quad f e r m i o n s \\quad n \\frac {1}{k} = 0, 1.\n$$\n\n$$\n\\text { Energy   of   state } 1 \\text {   enk   } >\n$$\n\n$$\nE (\\varepsilon n _ {k} 3) = \\sum_ {k} \\epsilon_ {k} n _ {k} \\quad \\epsilon_ {k} = \\frac {k ^ {2} k ^ {2}}{2 m}\n$$\n\nGrand partition function\n\n$$\nQ _ {G} = \\sum_ {\\xi n _ {k} \\in \\mathbb {Z}} e ^ {- \\beta (\\overline {{2}} \\epsilon_ {k} n _ {k} - \\mu N)} = \\overline {{2}} n _ {k}\n$$\n\n$$\n= \\frac {2}{\\varepsilon n _ {k} 3} e ^ {- \\beta \\frac {2}{2} (\\epsilon_ {k} - \\mu) n _ {k}}\n$$\n\n$$\n= \\prod_ {k} \\frac {2}{n _ {k}} e ^ {- \\beta - (\\epsilon_ {k} - \\mu) n _ {k}}\n$$\n\n$$\n= \\pi_ {k} [ 1 + e ^ {- \\beta (\\epsilon_ {k} - \\mu)} ]\n$$\n\nThermopotential:\n\n$$\n\\Omega = - k _ {B} T \\ln Q _ {G} = - k _ {B} T \\sum_ {k} \\ln (1 + e ^ {- \\beta (\\epsilon_ {k} - \\mu)})\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0002", "text": "* Total * of particle\n\n$$\n\\begin{array}{l} N = - \\frac {\\partial \\Omega}{\\partial \\mu} \\bigg | _ {y, T} = \\frac {2}{k} \\quad \\frac {e ^ {- \\beta (\\epsilon_ {k} - \\mu)}}{1 + e ^ {- \\beta (\\epsilon_ {k} - \\mu)}} \\\\ = \\frac {2}{k} \\frac {1}{1 + e ^ {\\beta (\\epsilon_ {k} - \\mu)}} \\\\ \\end{array}\n$$\n\n$x$ of particle in state  $(k) >$\n\n$$\nn _ {u} = \\frac {1}{1 + e ^ {\\beta (\\epsilon_ {u} - \\mu)}}\n$$\n\n$$\nN = \\sum_ {k} n _ {k} = V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {1}{1 + e ^ {\\beta (\\epsilon_ {k}) ^ {n}}}\n$$\n\nFermi-Diac distribution\n\n* Equation of state.\n\n$$\nP = - \\frac {\\partial \\Omega}{\\partial V} \\Big | _ {\\mu , T} = - \\frac {2}{k} \\frac {\\frac {\\partial \\varepsilon_ {k}}{\\partial V} e ^ {- \\beta (\\varepsilon_ {k} - \\mu)}}{1 + e ^ {- \\beta (\\varepsilon_ {k} - \\mu)}}\n$$\n\n$$\n= - \\sum_ {k} \\frac {\\partial E _ {k}}{\\partial V} n _ {k}\n$$\n\n$$\n= \\frac {2}{3} \\frac {1}{V} \\sum_ {k} \\epsilon_ {k} n _ {k}\n$$\n\n$$\n= \\frac {z}{3} \\frac {1}{V} L ^ {3} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {k ^ {2} k ^ {2}}{2 m} \\frac {1}{1 + e ^ {\\beta (t _ {4} - \\mu)}}\n$$\n\n$$\n\\epsilon_ {k} = \\frac {k ^ {2} k ^ {2}}{2 m}\n$$\n\n$$\n= \\frac {k ^ {2} (n _ {x} ^ {2} + n _ {y} ^ {2} + n _ {z} ^ {2}) \\frac {(2 \\pi) ^ {2}}{\\sqrt {3 5}}}{2 m}\n$$\n\n$$\n\\frac {\\partial \\varepsilon_ {k}}{\\partial V} = - \\frac {2}{3} \\frac {1}{V} \\varepsilon_ {k}\n$$\n\n$$\nP = \\frac {2}{3} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {k ^ {2} k ^ {2}}{2 m} \\frac {1}{1 + e ^ {\\beta (\\varepsilon_ {k} - \\mu)}}\n$$\n\n$$\n\\begin{array}{l} {n = \\frac {N}{V} = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {1}{1 + e ^ {\\beta (\\varepsilon_ {k} - \\mu)}}} \\\\ {\\Rightarrow \\mu (N, T, V) \\Rightarrow P V = \\frac {2}{3} U (T, V, \\mu (T, V _ {N}))} \\end{array} \\Rightarrow \\left[ \\begin{array}{l} {P V = \\frac {2}{3} U (\\tau , V, \\mu)} \\\\ {U = V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\varepsilon_ {k} n _ {k}} \\end{array} \\right]\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0003", "text": "High temperature (classical) limit\n\n$$\nn \\quad f i x e d \\quad T \\rightarrow \\infty \\quad \\mu \\rightarrow ?\n$$\n\n$$\n\\text { to   have   fixed } n \\quad \\mu \\rightarrow - \\infty\n$$\n\n$$\n\\text { and } \\quad n _ {k} \\propto \\frac {1}{T} \\rightarrow 0\n$$\n\n$$\nP = \\frac {2}{3} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {k ^ {2} k ^ {2}}{2 m} e ^ {- \\beta \\epsilon_ {k}} e ^ {+ \\beta \\mu}\n$$\n\n$$\nh = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- \\beta \\epsilon_ {k}} e ^ {+ \\beta \\mu}\n$$\n\n$$\n\\frac {p}{n} = \\frac {\\frac {2}{3} \\int d ^ {3} k \\epsilon_ {k} e ^ {- \\beta \\epsilon_ {k}}}{\\int d ^ {3} k e ^ {- \\beta \\epsilon}} = \\frac {2}{3} \\langle \\epsilon_ {k} \\rangle \\propto \\beta^ {- 3 / 2}\n$$\n\n$$\n= \\frac {2}{3} \\frac {- \\frac {\\partial}{\\partial \\beta} I (\\beta)}{J (\\beta)} = \\frac {1}{\\beta} = k _ {B} T \\quad \\boxed {P V = \\frac {2}{3} U}\n$$\n\n$$\nP \\forall = N k _ {B} T \\quad \\text { classical   gas   eq.   of   state }\n$$\n\nAt high temperature\nFermi gas = classical gas\n\n* Zero - temperature (quantum) limit\n\n$$\nT = 0 \\quad n _ {k} = 1 \\quad \\text { if } \\quad \\varepsilon_ {k} <   \\mu = \\varepsilon_ {F} \\text { or } k <   k _ {F} = \\frac {\\sqrt {2 m \\mu}}{\\pi}\n$$\n\n$$\nn _ {k} = 0 \\quad \\text { if } \\quad \\varepsilon_ {k} > \\mu = \\varepsilon_ {F}\n$$\n\n$$\nP = \\frac {2}{3} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {\\hbar^ {2} k ^ {2}}{2 m} = \\frac {2}{3} \\frac {\\hbar^ {2}}{(2 \\pi) ^ {3} 2 m} \\int_ {0} ^ {k _ {F}} 4 \\pi k ^ {4} d k\n$$\n\n$$\n= \\frac {2}{3} \\frac {k ^ {2}}{(5 \\pi) ^ {3} 2 m} \\frac {4 \\pi}{\\sqrt {3}} k _ {p} ^ {5}\n$$\n\n$$\n= \\frac {k ^ {2}}{3 ① \\pi^ {2} m} k _ {F} ^ {5}\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0004", "text": "$$\n\\begin{array}{l} n = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} = \\frac {1}{(2 \\pi) ^ {3}} \\int_ {0} ^ {k _ {F}} 4 \\pi k ^ {2} d k \\\\ = \\frac {1}{(2 \\pi) ^ {3}} \\cdot \\frac {4 \\pi}{3} k _ {F} ^ {3} = \\frac {1}{6 \\pi^ {2}} k _ {F} ^ {3} \\\\ \\end{array}\n$$\n\n$$\nk _ {F} = (6 \\pi^ {2} n) ^ {1 / 3} \\quad \\epsilon_ {F} = \\frac {\\hbar^ {2} k _ {F} ^ {2}}{2 m}\n$$\n\nP\nP×(1/V)\nP×(1/V)^(1/3)\nT→∞\nT=0\n\n$$\n\\begin{array}{l} P = \\frac {k ^ {2}}{3 0 \\pi^ {2} m} (6 \\pi^ {2} n) ^ {5 / 3} \\\\ = \\frac {h ^ {2} (6 \\pi^ {2}) ^ {5 / 3}}{3 0 \\pi^ {2} m} n ^ {5 / 3} \\\\ \\end{array}\n$$\n\n$$\nP = \\frac {(6 \\pi^ {2}) ^ {\\frac {5}{3}}}{1 5 \\pi^ {2}} \\frac {n ^ {2} n ^ {2 / 3}}{2 m} n\n$$\n\n$$\nV ^ {\\frac {5}{3}} P = \\frac {(6 \\pi^ {2}) ^ {5 / 2} x _ {1} ^ {2}}{3 0 \\pi^ {2} m} N ^ {\\frac {5}{3}} = c o n s t.\n$$\n\n* Quantum limit and classical limit\n\n$$\nT = 0 \\quad P \\sim \\text { energy   per   volume } \\quad (E \\sim P V)\n$$\n\n$$\n= \\text { energy   per   particle } \\times n\n$$\n\n$$\n\\sim \\frac {h ^ {2} n ^ {3 / 3}}{2 m} \\times n\n$$\n\n2 Fermi: pressure\n\nhigh temperature (classical)\n\n$$\n\\text { classical } \\quad \\text { limit } \\quad \\left[ \\begin{array}{c c} \\frac {\\sqrt {2 m k _ {B} T}}{h} & \\ll n ^ {1 / 3} \\\\ n k _ {B} T & > \\frac {h ^ {2} k _ {F} ^ {2}}{2 m} \\equiv \\epsilon_ {F} \\end{array} \\right]\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0005", "text": "* High Temperature expansion - correction to ideal gas.\n\nFree energy at high temperatures\n\n$$\n\\begin{array}{l} \\Omega = - k _ {B} T \\sum_ {k} l n (1 + e ^ {- \\beta (\\epsilon_ {k} - \\mu)}) \\\\ = - k _ {B} T V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} 2 n (1 + e ^ {- \\beta (\\epsilon_ {k} - \\mu)}) \\\\ \\end{array}\n$$\n\n$$\nA = \\Omega + \\mu N \\Bigg | _ {\\mu = \\mu (V, N, T)}\n$$\n\n$$\n\\frac {3 e ^ {- \\beta \\epsilon_ {k}}}{1 + 3 e ^ {- \\beta \\epsilon_ {k}}}\n$$\n\n$$\nN = - \\frac {\\partial \\Omega}{\\partial \\mu} = V \\int \\frac {d ^ {3} k}{(2 \\xi) ^ {3}} \\frac {1}{i + e ^ {\\beta (\\varepsilon_ {k} - \\mu)}}\n$$\n\n$$\ns d o u e f a \\mu (V, N, T)\n$$\n\n$$\nI _ {n} \\quad h i g h \\quad T e m p a r a t e r e \\quad e ^ {- \\beta (\\epsilon_ {k} - \\mu)} = 3: e ^ {- \\beta \\epsilon_ {k}} <   <   1\n$$\n\n$$\n\\text {   expand   } x _ {0} \\text {   second   order   in   } z ^ {\\prime \\prime}\n$$\n\n$$\nN = V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} (8 e ^ {- \\beta \\epsilon_ {k}} - 8 ^ {2} e ^ {- 2 \\beta \\epsilon_ {k}})\n$$\n\n$$\n= V (z \\lambda^ {3} - z ^ {2} 2 ^ {- 3 / 2} \\lambda^ {- 3})\n$$\n\n$$\n\\Rightarrow n \\lambda^ {3} = 3 - 3 ^ {2} / 2 ^ {3 / 2}\n$$\n\n$$\n\\simeq 3 - \\frac {(n \\lambda^ {3}) ^ {2}}{2 ^ {3 / 2}}\n$$\n\n$$\nz = n \\lambda^ {3} + \\frac {(n \\lambda^ {3}) ^ {2}}{2 ^ {3 / 2}} \\quad \\mu = k _ {B} T \\ln z\n$$\n\n$$\n\\begin{array}{l} \\int \\frac {\\partial^ {3} k}{(2 \\pi) ^ {3}} e ^ {- \\beta \\epsilon_ {k}} \\\\ = \\left(\\sqrt {2 \\pi k ^ {2} / m k _ {B} T}\\right) ^ {- 3} \\\\ = \\lambda^ {- 3} \\\\ \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0006", "text": "$$\nA = - k _ {B} T V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} (3 e ^ {- \\beta \\varepsilon_ {k}} - \\frac {1}{2} z ^ {2} e ^ {- 2 \\beta \\varepsilon_ {k}})\n$$\n\n$$\n+ k _ {B} T \\left[ \\ln n \\lambda^ {3} + \\ln (1 + \\frac {n \\lambda^ {3}}{2 ^ {3 / 2}}) \\right] N\n$$\n\n$$\n= - k _ {8} T V \\frac {1}{\\lambda^ {3}} (n \\lambda^ {3} + \\frac {(n \\lambda^ {3}) ^ {2}}{2 ^ {3 / 2}}) z ^ {2}\n$$\n\n$$\n+ \\frac {1}{2} k _ {B} T V \\frac {1}{2 ^ {3 / 2} \\lambda^ {3}} \\overbrace {(n \\lambda^ {3} + \\frac {(n \\lambda^ {3}) ^ {2}}{2 ^ {3 / 2}})} ^ {2}\n$$\n\n$$\n+ k _ {B} T \\left(\\ln n \\lambda^ {3} + \\frac {n x ^ {3}}{2 ^ {3 / 2}}\\right) N\n$$\n\n$$\n= - k _ {B} T N (1 + \\frac {n \\lambda^ {3}}{2 ^ {3 / 2}})\n$$\n\n$$\n+ \\frac {1}{2} k _ {B} T N \\frac {n \\lambda^ {3}}{2 ^ {3 / 2}} + k _ {B} T (\\ln n \\lambda^ {3} + \\frac {n \\lambda^ {3}}{2 ^ {3 / 2}}) \\sqrt {}\n$$\n\n$$\nA = k _ {B} T N \\left(\\ln n \\lambda^ {3} - 1 + \\frac {1}{2} \\frac {n \\lambda^ {3}}{2 ^ {3 / 2}}\\right)\n$$\n\n$$\n\\text { quantum   correction }\n$$\n\n$$\n\\text {   small   if   } n \\lambda^ {3} <   <   1\n$$\n\n$$\nE q n. f \\quad s t a t A = k _ {3} T N (- \\ln V + \\frac {1}{2} \\frac {N \\lambda^ {3}}{2 ^ {3 / 2} V})\n$$\n\n$$\nP = - \\frac {\\partial A}{\\partial V} = k _ {B} T / V + k _ {B} T \\frac {N ^ {2} \\lambda^ {3}}{2 ^ {5 / 2} V ^ {2}}\n$$\n\n$$\n\\text { extra   pressure }\n$$\n\n$$\n\\text { Viziml   expansion }\n$$\n\n$$\n\\frac {P V}{k _ {3} T} = 1 + \\frac {n \\lambda^ {3}}{2 ^ {5 / 2}} = 1 + \\frac {\\sqrt {\\lambda^ {3} / 2 ^ {5 / 2}}}{V} = 1 + \\frac {c _ {3}}{V} + \\frac {c _ {3}}{V}.\n$$\n\n$$\nc _ {2} = \\frac {N \\lambda^ {3}}{2 ^ {5 / 2}} > 0\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0007", "text": "82\n\n* Low temperature properties:\n\nDensity of states:\n\n$D(\\epsilon)d\\epsilon = \\#$ of states with energy\n\nbetween   $\\in$  and   $\\epsilon+d\\epsilon$\n\n$D\\left( \\varepsilon \\right)  = V\\int \\frac{{d}^{3}k}{\\left( {2\\pi }\\right) ^{3}}\\;\\delta \\left( {{\\varepsilon }_{k} - \\varepsilon }\\right) \\;\\left( {f_{n},3D}\\right)$\n\n$$\n\\begin{array}{l} N (\\epsilon_ {F}) = \\# \\text {   of   states   below   } \\epsilon_ {F} = \\int_ {0} ^ {\\epsilon_ {F}} D (\\epsilon) d \\epsilon \\\\ \\begin{array}{r l} {= V \\int_ {\\epsilon_ {k} <   \\epsilon_ {F}} \\frac {d ^ {3} k}{(2 \\pi) ^ {3}}} & {= \\frac {V}{(2 \\pi) ^ {3}} \\frac {4 \\pi}{3} k _ {F} ^ {3}} \\\\ {\\mathrm{m} k <   k _ {F}} & {k _ {F} = \\frac {\\sqrt {2 m \\epsilon_ {F}}}{\\hbar}} \\end{array} = \\boxed {\\frac {V}{6 \\pi^ {2}} (\\frac {\\sqrt {2 m}}{\\hbar}) ^ {3} \\epsilon_ {F} ^ {3 / 2}} \\\\ \\end{array}\n$$\n\n$$\nN (G) = \\mathrm{H} f f e r m i o n s = \\frac {V}{6 \\pi^ {2}} k _ {F} ^ {3}\n$$\n\n( spin less, one fermion\npen state\n\n$$\n\\boxed {k _ {F} ^ {3} \\sim \\text { number   density }} = 2 \\times \\frac {\\sqrt {6 \\pi^ {2}} k _ {F} ^ {3}}{1} \\text { from   spin }\n$$\n\n( spin =   $\\frac{1}{2}$ , two fermion\npor state )\n\n$$\nD (\\epsilon) = \\frac {\\partial N (\\epsilon)}{\\partial \\epsilon} = \\frac {\\sqrt {2}}{2 \\pi^ {2}} \\frac {m ^ {3 / 2}}{h ^ {3}} \\epsilon^ {1 / 2}\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0008", "text": "Zew temperature:\n\nGround state energy\n\n$$\nU _ {0} = \\int_ {0} ^ {\\varepsilon_ {p}} d \\varepsilon \\in D (\\varepsilon)\n$$\n\n$$\n= \\frac {\\sqrt {2}}{2 \\pi^ {2}} \\frac {m ^ {3 / 2}}{t _ {n} ^ {3}} V \\int_ {0} ^ {\\epsilon_ {F}} d \\epsilon \\epsilon \\epsilon^ {1 / 2}\n$$\n\n$$\n= \\frac {\\sqrt {2}}{2 \\pi^ {2}} \\frac {m ^ {3 / 2}}{\\pi^ {3}} V \\frac {2}{5} \\epsilon_ {F} ^ {5 / 2}\n$$\n\n$$\nN _ {0} = \\int_ {0} ^ {\\varepsilon_ {1}} d \\varepsilon D (\\varepsilon)\n$$\n\n$$\n= \\frac {\\sqrt {2}}{2 \\pi^ {2}} \\frac {m ^ {3 / 2}}{k _ {0} ^ {3}} V \\int_ {0} ^ {\\epsilon_ {F}} d \\epsilon \\in 1 / 2\n$$\n\n$$\n= \\frac {\\sqrt {2}}{2 \\sqrt {2}} \\frac {m ^ {3 / 2}}{t ^ {3}} \\sqrt {\\frac {2}{3}} \\epsilon_ {F} ^ {3 / 2}\n$$\n\n$$\nU _ {0} = N \\frac {3}{5} \\epsilon_ {F}\n$$\n\nenergy per particle  $\\sim$ 6F\n\n$$\n\\begin{array}{l} V P = - V \\frac {\\partial \\epsilon_ {k}}{\\partial V} = \\frac {2}{3} V \\epsilon_ {k} \\\\ P V = \\frac {2}{3} V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {h ^ {2} k ^ {2}}{2 m} \\frac {1}{1 + e ^ {\\lambda (\\epsilon k - \\mu)}} \\boxed {\\epsilon_ {k} \\propto V ^ {- \\frac {2}{3}}} \\\\ = \\frac {2}{3} V \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\varepsilon_ {k} n _ {k} = \\frac {2}{3} U \\\\ \\end{array}\n$$\n\nWe have shown that\n\n$$\nP _ {0} = \\frac {2}{3} \\frac {U}{V} = \\frac {2}{5} n \\in_ {F} T _ {n \\rightarrow k _ {F} \\rightarrow 6 _ {F}}\n$$\n\nin metal  $n \\sim 10^{22} / \\mathrm{cm}^3$$P_0 \\sim 10^4 \\mathrm{atm}$\n\n$$\n\\epsilon_ {F} \\sim a \\text { few   eV } > k _ {B} T \\sim \\frac {1}{4 0} e V\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0009", "text": "$$\n\\begin{array}{l} \\text {   Compressibility   of   a   metal   } \\\\ F (N _ {1} + N _ {2}) \\\\ \\end{array}\n$$\n\n$$\nU _ {0} = N \\frac {3}{5} \\epsilon_ {F} = C N n _ {e} ^ {2 / 3}\n$$\n\n$$\n= C N ^ {5 / 3} / V ^ {2 / 3}\n$$\n\n$$\n= \\frac {3}{5} \\frac {4 ^ {2} (3 \\pi^ {2}) ^ {2 / 3}}{2 m}\n$$\n\n| U    | V     |\n| ---- | ----- |\n| 84   | 0     |\n| 84   | V₀    |\n\n$$\n\\epsilon_ {F} \\propto n ^ {2 / 3}\n$$\n\n$$\n\\varepsilon_ {F} = \\frac {k ^ {2} k _ {F} ^ {2}}{2 m}\n$$\n\n$$\nn _ {\\text { spin } \\frac {1}{2}} = 2 \\cdot \\frac {k _ {F} ^ {3}}{6 \\pi^ {2}}\n$$\n\n$$\nn _ {\\text { spin - 0 }} = \\frac {k _ {F} ^ {3}}{6 \\pi^ {2}}\n$$\n\n$$\n\\text { Model }\n$$\n\n$$\nk _ {F} = (3 \\pi^ {2} n _ {e}) ^ {1 / 3}\n$$\n\n$$\nU _ {t + t} = \\frac {C N ^ {5 / 3}}{V ^ {2 / 3}} + P _ {0} V\n$$\n\n$$\n\\varepsilon_ {F} = \\frac {h ^ {2} (3 \\pi^ {2}) ^ {2 / 3}}{2 m} n _ {e} ^ {2 / 3}\n$$\n\n$$\ne f u. v o l u m e\n$$\n\n$$\n\\frac {\\partial V _ {t o t}}{\\partial V} = 0 \\Rightarrow P _ {0} = \\frac {2}{3} \\frac {C N ^ {5 / 2}}{V _ {0} ^ {5 / 3}}\n$$\n\n$$\nV _ {0} = \\left(\\frac {3 P _ {0}}{2 C N ^ {5 / 2}}\\right) ^ {3 / 5}\n$$\n\n$$\n\\text { Compressibility }\n$$\n\n$$\nx = - \\frac {1}{V} \\frac {d V}{d P} = \\frac {- 1}{V \\frac {d P}{d V}} = \\frac {1}{V \\frac {\\partial^ {2} U _ {t o t}}{\\partial V ^ {2}}}\n$$\n\n$$\n= \\frac {1}{V \\frac {2}{3} \\frac {5}{3} \\frac {C N ^ {5 / 3}}{V ^ {8 / 3}}} = \\frac {9}{1 0} \\frac {1}{C n ^ {5 / 3}}\n$$\n\n$$\n\\gamma = 9 \\frac {m}{n ^ {2} (3 \\pi^ {2}) ^ {2 / 3}} \\frac {1}{n ^ {5 / 3}} = \\frac {9}{2} \\frac {1}{\\epsilon_ {F} n} \\sim \\frac {1}{1 0 ^ {4} a t _ {m}}\n$$\n\n$$\nN e o d 1 0 ^ {4} \\mathrm{atm} \\text { to   reduce } V \\rightarrow \\frac {1}{2} V\n$$\n\n$$\n1 \\mathrm{atm} \\sim 1 0 \\mathrm{m} \\mathrm{water} \\rightarrow 1 0 ^ {5} \\mathrm{m} \\mathrm{of~water}\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0010", "text": "a Spin susceptibility of malal.\n\nmagnetic moment  $\\uparrow\\quad\\mu_{B}\\quad\\downarrow\\quad-\\mu_{B}$\n\n$$\n\\text { Induced } M = \\mu_ {B} (N _ {T} - N _ {U})\n$$\n\n$$\n\\begin{array}{l} \\Delta N = 2 \\mu_ {B} B D (\\epsilon_ {F}) \\\\ = 2 \\mu_ {B} B \\frac {\\sqrt {3}}{2 \\pi^ {2}} \\frac {m ^ {3 / 2}}{h ^ {3}} E _ {F} ^ {1 / 2} V \\\\ \\end{array}\n$$\n\n$$\n\\begin{array}{l} n = \\frac {1}{6 \\pi^ {2}} \\left(\\frac {\\sqrt {2 m}}{x}\\right) ^ {3} \\epsilon_ {F} ^ {3 / 2} \\times 2 \\\\ \\alpha \\epsilon_ {F} ^ {3 / 2} \\end{array}\n$$\n\n$$\n\\epsilon_ {F} \\propto n ^ {2 / 3}\n$$\n\n$$\n\\Delta N \\propto n ^ {1 / 3}\n$$\n\nμ\n↓\nO …… → μαB\n↑\nμB\nε↓ = k²ħ²/2m + μB B\n\n$$\nG _ {T} = \\frac {k ^ {2} h ^ {3}}{2 m} - \\mu_ {B} B\n$$\n\n2μsB\n\n$$\n\\begin{array}{l} x = \\frac {M}{B} = 2 \\mu_ {B} ^ {2} \\frac {\\sqrt {2}}{(2 \\pi) ^ {2}} \\frac {m ^ {3 / 2}}{\\hbar^ {3}} \\epsilon_ {F} ^ {1 / 2} V \\\\ = \\mu_ {B} ^ {2} \\frac {\\sqrt {2} 3 ^ {1 / 3}}{\\pi} \\frac {m}{k ^ {2}} n ^ {1 / 3} V \\\\ x = \\frac {M}{B} = 2 \\mu_ {B} ^ {2} \\frac {\\sqrt {2}}{(2 \\pi) ^ {2}} \\frac {m ^ {3 / 2}}{\\hbar^ {3}} \\epsilon_ {F} ^ {1 / 2} V \\\\ = \\mu_ {B} ^ {2} \\frac {\\sqrt {2} 3 ^ {1 / 3}}{\\pi} \\frac {m}{k ^ {2}} n ^ {1 / 3} V \\\\ \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 9, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 10"], "page_start": 10, "page_end": 10, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0011", "text": "Low temperature specific heat\n\n$$\nU = \\int d \\xi D (\\xi) \\in n _ {F} (\\xi) n _ {F} (\\xi) = \\frac {1}{1 + e ^ {\\beta (\\xi - \\mu)}}\n$$\n\n$$\nN = \\int d t D (\\epsilon) n _ {F} (\\epsilon) \\Rightarrow f i n d \\mu = \\mu (N, V, T)\n$$\n\n$$\nU (\\mu , v, T) = U (\\mu (N, v, T), v, T ]\n$$\n\n$$\nC = \\left. \\frac {\\partial U}{\\partial T} \\right| _ {N, V}\n$$\n\n$$\n\\left. \\frac {\\partial N}{\\partial T} \\right| _ {N, V} = \\int d \\epsilon D (\\epsilon) \\left. \\frac {\\partial n _ {F} [ \\epsilon , T , \\mu_ {f (N , V , T)} ]}{\\partial T} \\right| _ {N, V} = 0\n$$\n\n$$\n\\Rightarrow \\int d \\epsilon   \\mu   D (\\epsilon) \\left. \\frac {\\partial n _ {P}}{\\partial T} \\right| _ {N, V} = 0\n$$\n\n$$\n\\Rightarrow C = \\int d \\epsilon D (\\epsilon) \\in \\frac {\\partial n _ {F}}{\\partial T} | _ {N, V}\n$$\n\n$$\n= \\int d \\epsilon D (\\epsilon) (\\epsilon - \\mu) \\frac {\\partial n _ {F}}{\\partial T} | _ {N, V}\n$$\n\n$$\n\\left. \\frac {\\partial n _ {F}}{\\partial T} \\right| _ {N, V} = \\frac {\\varepsilon - \\mu}{k _ {B} T ^ {2}} \\quad \\frac {e ^ {\\beta (\\varepsilon - \\mu)}}{[ 1 + e ^ {\\beta (\\varepsilon - \\mu)} ] ^ {2}}\n$$\n\n$$\n+ \\frac {1}{k _ {B} T} \\left. \\frac {\\partial \\mu}{\\partial T} \\right| _ {N, V} \\frac {e ^ {\\beta (\\varepsilon - \\mu)}}{\\{1 + e ^ {\\beta (\\varepsilon - \\mu)} \\} ^ {2}}\n$$\n\n$$\n\\underbrace {b \\sim T} _ {L \\sim O (T ^ {0})} d i o p e d\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 10, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 11"], "page_start": 11, "page_end": 11, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0012", "text": "$$\nC = \\frac {D (\\mu)}{k _ {B} T ^ {2}} \\int d t (\\varepsilon - \\mu) ^ {2} \\frac {e ^ {\\beta (\\varepsilon - \\mu)}}{\\left[ 1 + e ^ {\\beta (\\varepsilon - \\mu)} \\right] ^ {2}}\n$$\n\n$$\nt = \\beta (\\epsilon - \\mu)\n$$\n\n$$\n= \\frac {D (\\mu)}{k _ {B} T ^ {2}} k _ {B} ^ {3} T ^ {3} \\int_ {- \\infty} ^ {+ \\infty} d t t ^ {2} \\frac {e ^ {t}}{(1 + e ^ {t}) ^ {2}}\n$$\n\n$$\n\\begin{array}{r l} {\\boxed {C = \\frac {\\pi^ {2}}{3} k _ {B} ^ {2} T D (\\mu)}} & {\\mathrm{works}} \\\\ & {\\mathrm{for~v~dimensions}} \\\\ & {= \\frac {3 N}{2 \\mu}, \\quad \\mu = \\epsilon_ {F}} \\end{array}\n$$\n\n$$\nI _ {n} = 2 \\int_ {0} ^ {\\infty} d t \\frac {t ^ {n} e ^ {t}}{(1 + e ^ {t}) ^ {2}}\n$$\n\n$$\nI _ {0} = 1 \\quad I _ {2} = \\frac {\\pi^ {2}}{3}\n$$\n\n$$\n\\mu = \\epsilon_ {F}\n$$\n\n$$\n\\mathrm{C} (V) \\rightarrow \\frac {3}{2} k _ {B}\n$$\n\n$$\nC = \\frac {\\pi^ {2}}{2} k _ {B} \\frac {k _ {B} T}{G _ {F}} N \\tag {3D}\n$$\n\n$$\n= \\frac {3}{2} k _ {B} \\frac {\\frac {\\pi^ {2}}{3} k _ {B} T}{G _ {P}} N\n$$\n\nClassical\n\n$$\nV = N \\frac {3}{2} k _ {B} T\n$$\n\n$$\nC = N \\frac {3}{2} k _ {B}\n$$\n\n$$\nw h _ {y} \\frac {\\partial m}{\\partial T} l _ {N, v} \\sim T\n$$\n\n$$\nN = \\int d \\epsilon D (\\epsilon) n _ {F} (\\epsilon)\n$$\n\n$$\n= N _ {0} + \\mathrm{※} T ^ {2}\n$$\n\n$$\n\\Rightarrow \\mu = \\mu_ {0} + x T ^ {2}\n$$\n\nD(ε) M_F(ε)\nT\nk_B T\n\nn_F(t)\nkT", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 11, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 12"], "page_start": 12, "page_end": 12, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0013", "text": "White dwarf & neutron star\n\n$$\nP \\approx \\frac {h ^ {2} n ^ {5 / 3}}{m}\n$$\n\n$$\n\\begin{array}{l} P _ {t o t} = P _ {e} + P _ {p} \\quad P _ {p} <   P _ {e} \\\\ = P _ {e} = \\frac {\\hbar^ {2} n ^ {5 / 3}}{m _ {e}} \\\\ \\end{array}\n$$\n\nBalance :\n\n$$\n\\begin{array}{l} P _ {i, t} \\approx G \\frac {M m}{R ^ {2}} = m \\\\ = G \\frac {M}{R ^ {2}} (\\overbrace {n m _ {p} R}) \\\\ = G M m _ {p} n / R = \\frac {h ^ {2} n ^ {5 / 3}}{m _ {e}} \\\\ P _ {i, t} \\sim G \\frac {M m}{R ^ {2}} = m \\\\ = G \\frac {M}{R ^ {2}} (\\overbrace {n m _ {p} R}) \\\\ = G M m _ {p} n / R = \\frac {h ^ {2} n ^ {5 / 3}}{m _ {e}} \\\\ \\end{array}\n$$\n\n$$\n\\begin{array}{l} P _ {i, t} \\approx G \\frac {M m}{R ^ {2}} = m \\\\ = G \\frac {M}{R ^ {2}} (\\overbrace {n m _ {p} R}) \\\\ = G M m _ {p} n / R = \\frac {h ^ {2} n ^ {5 / 3}}{m _ {e}} \\\\ P _ {i, t} \\sim G \\frac {M m}{R ^ {2}} = m \\\\ = G \\frac {M}{R ^ {2}} (\\overbrace {n m _ {p} R}) \\\\ = G M m _ {p} n / R = \\frac {h ^ {2} n ^ {5 / 3}}{m _ {e}} \\\\ \\end{array}\n$$\n\n$$\n\\begin{array}{l} G M m _ {p} m _ {e} / R = k ^ {2} \\left(\\frac {M}{m _ {p} R ^ {3}}\\right) ^ {2 / 3} \\\\ = \\hbar^ {2} \\frac {M ^ {3 / 3}}{m _ {p} ^ {2 / 3} R ^ {2}} \\\\ \\end{array}\n$$\n\n$$\nR _ {w D} = m ^ {- 1 / 3} \\frac {h ^ {2}}{m _ {e} m _ {p} ^ {3 / 3}} G ^ {- 1}\n$$\n\n$$\n= \\left(\\frac {M _ {\\theta}}{m}\\right) ^ {1 / 3} M _ {\\theta} ^ {- 1 / 3} \\frac {x ^ {2}}{m _ {e m p} ^ {5 7 / 3} G}\n$$\n\n$R_{WD} = \\left( \\frac{M_0}{M} \\right)^{1/3} 6200 km$\n\nFor neutron star we replace me\n\n$$\nR _ {N S} = (\\frac {M _ {\\theta}}{m}) ^ {1 / 3} 3. 4 k m\n$$\n\nm\nR\n\n$$\nm _ {n} = 9 3 9. 5 6 5 \\mathrm{MeV}\n$$\n\n$$\nm _ {P} = 9 3. 8, 2 7 1 M e V\n$$\n\n$$\n\\Delta m = 1. 3   \\text { MeV }\n$$\n\n$$\nm _ {e} = 0. 5 1 m _ {e} V\n$$\n\n$$\nn \\simeq \\frac {M}{m _ {p} R ^ {3}}\n$$\n\n$$\n[ \\frac {h ^ {2}}{m ^ {3} G} ]\n$$\n\n$$\n= [ t ^ {2} ] [ m ^ {- 1} L ^ {- 1} ]\n$$\n\n$$\n[ \\frac {L}{m ^ {2} G} ]\n$$\n\n$$\n= \\left[ \\frac {\\xi^ {2}}{m L E} \\right] \\quad \\checkmark\n$$\n\n$$\nm _ {p}: M _ {\\textcircled {1}} = 1. 9 9 \\times 1 0 ^ {3 3} g\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 12, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 13"], "page_start": 13, "page_end": 13, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0014", "text": "$$\nP _ {2 0} b. 1 2. 6\n$$\n\n$$\nQ _ {1} = \\sum e x p (- f \\cdot \\epsilon_ {n}) = 2 e ^ {\\beta (b x ^ {2} - c x / 2)} + e ^ {- \\beta (b x ^ {2} + c x)}\n$$\n\n$$\nA _ {(x)} = - k _ {B} T N \\ln Q _ {1}\n$$\n\nA\nT > Tc = \\frac{cL}{4b/kB}\nxmin = 0\nx\n\nA\nT < \\frac{c^2}{4b/b_B}\nxmin≠0\nx\n\n| Curve Type | X-axis Label | Y-axis Label |\n|------------|--------------|--------------|\n| Top Curve  | Tc           | A            |\n| Bottom Curve | Tc           | A            |", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 13, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 14"], "page_start": 14, "page_end": 14, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0015", "text": "Semi conductor\n\nBand theory\n\n$$\nf _ {k} (x) = e ^ {i k x}\n$$\n\n$$\n\\epsilon_ {k} = \\frac {k ^ {2} r _ {h} ^ {2}}{2 m}\n$$\n\non lattice\n\n$$\nx = n a\n$$\n\n$$\nn = 0, \\pm 1, \\pm 2 \\dots\n$$\n\n$$\n\\mathcal {H} _ {k} (x) \\rightarrow \\mathcal {H} _ {k} (n) = e ^ {i k n a}\n$$\n\n$$\n\\epsilon_ {k} = \\frac {k ^ {2} k ^ {2}}{2 m}\n$$\n\n$$\nB _ {u t} \\quad \\gamma_ {k + k} (n) = \\gamma_ {k}\n$$\n\n$$\n\\therefore k = \\frac {2 \\pi}{a}\n$$\n\nBrillouin zone\n\n$$\n\\epsilon_ {k} = \\frac {h ^ {2}}{m a ^ {2}} [ 1 - \\cos (k a) ]\n$$\n\n令  ${P}_{n}$ small  $k\\;{\\epsilon }_{k} = \\frac{{k}^{2} \\cdot  {k}^{2}}{2m}$\n\n-π/a\n0\nπ/a\nk\n\nBand structure:\n\nFemmin surface\nMetal\n\nInsulator\n# of levels in each band\n= unit cell.", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 14, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 15"], "page_start": 15, "page_end": 15, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0016", "text": "Method 1.\n\n$$\nn _ {e} = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {\\beta}} \\frac {1}{e ^ {\\beta (\\varepsilon_ {e} + \\frac {\\Delta}{2} - \\mu) + 1}}\n$$\n\nεc = \\frac{1}{2m_e} k^2 t^2\nΔ/2\nμ\nΔ/2\nε_h = \\frac{1}{2m_h} k^2 t^2\n\n$$\nn _ {h} = \\int \\frac {d ^ {2} k}{(2 \\pi) ^ {3}} \\frac {1}{e ^ {\\beta (\\epsilon_ {h} + \\frac {\\Delta}{2} + \\mu)} + 1}\n$$\n\n$$\n\\text { Adjust } \\mu \\text { to   make } n _ {e} = n _ {h}\n$$\n\n$$\nI f \\quad m _ {e} = m _ {h} \\quad \\Rightarrow \\quad \\mu = 0\n$$\n\n$$\nn _ {e} = n _ {h} = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {1}{e ^ {\\beta (\\epsilon + \\frac {\\Delta}{2})} , + 1}\n$$\n\n$$\nF _ {a} \\quad l a n g e T (k _ {B} T > > \\Delta)\n$$\n\n$$\nn e \\simeq \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\frac {1}{e ^ {\\beta E} + 1} = \\frac {1}{2 \\pi^ {2}} \\left(\\frac {2 m k _ {B} T}{h ^ {2}}\\right) ^ {3 / 2} \\int_ {0} ^ {\\infty} d t \\frac {t ^ {2}}{e ^ {t} + 1}\n$$\n\n$$\n\\sim \\left(\\frac {m k _ {B} T}{\\hbar^ {2}}\\right) ^ {3 / 2} \\sim \\frac {1}{\\lambda^ {3}} \\quad (\\text { quantum })\n$$\n\n$$\nF _ {m} \\dots \\text { small } T (k _ {B} T <   <   \\Delta)\n$$\n\n$$\nn _ {e} \\approx \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- \\beta \\epsilon} e ^ {- \\beta \\frac {\\Delta}{2}} = \\lambda^ {- 3} e ^ {- \\beta \\Delta / 2} (\\text { classical })\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 15, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 16"], "page_start": 16, "page_end": 16, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0017", "text": "Method 2\n\n$$\n\\epsilon_ {e} = \\frac {1}{2 m _ {c}} k ^ {2}\n$$\n\n$$\n\\epsilon_ {k} = \\frac {1}{2 m _ {v}} k ^ {2}\n$$\n\n$$\n(e l e) + (h o l e) = \\Delta\n$$\n\nεₖ\n91\nband\nΔ\nk\nband\n\n$$\n\\begin{array}{r l} {\\nu_ {1} X _ {1} + \\nu_ {2} X _ {2} + \\dots} & {= \\epsilon_ {0}} \\\\ {\\Rightarrow (\\lambda_ {1} ^ {3} n _ {1}) ^ {\\nu_ {1}} (\\lambda_ {2} ^ {5} n _ {2}) ^ {\\nu_ {2}}} & {\\dots = e ^ {- \\frac {\\epsilon_ {0}}{k _ {3} T}}} \\end{array}\n$$\n\n$k_{B}T<<\\Delta$$n_{k}<<1$  0   $1-n_{k}<<1$\n\n$$\n\\lambda_ {e} ^ {3} n _ {e} \\quad \\lambda_ {h} ^ {3} n _ {h} = e ^ {- \\Delta / k _ {B T}} \\quad \\begin{array}{l l} & \\text { classical   gas } \\\\ & f \\text { particles } \\\\ & \\text { & holes } \\end{array}\n$$\n\n$$\n\\lambda = \\sqrt {2 \\pi \\xi^ {2} / m k _ {B} T}\n$$\n\n$$\n\\begin{array}{r l} {n _ {e}} & {= n _ {h} = 2 e ^ {- \\Delta / 2 k _ {B} T} \\left(\\frac {\\sqrt {m _ {e} m _ {v}} - k _ {B} T}{2 \\pi k ^ {2}}\\right) ^ {3 / 2}} \\\\ & {\\simeq T ^ {3 / 2} e ^ {- \\frac {\\Delta}{2 k _ {B} T}}} \\end{array}\n$$\n\n$$\n\\begin{array}{c} \\lambda^ {3} n <   <   1 \\\\ \\text { classical } \\\\ \\lambda^ {3} n \\geq 1 \\\\ \\text { Quantum } \\end{array}\n$$\n\nConductivity of semi-coductors (no doping)\n\n$$\n\\sigma \\propto n _ {e} n n _ {h} \\propto T ^ {3 / 2} e ^ {- \\frac {\\Delta}{2 k _ {B} T}}\n$$", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 16, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 17"], "page_start": 17, "page_end": 17, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0009-viii-the-fermi-gas-9f1cf2b2-36771958:page-0018", "text": "Diode (p-n junction)\n\nSimple line drawing of a container with liquid and a handle, no text or symbols present\n\nn-type\n\n(extra electrons)\n\nμ\n1171710\n\np-type\n\n(extra holes)\n\npolenleal\nbarries\nεB\nE\nη\n\npositive bias\n\nsmall\n\nbarriers\n\nμ₁\nε₃\nE\nμ₂\n\nelectron flow\n\nnegative bias\n\nlayer\nBarrier\nE_B\nM\nE\n\nelectron flow\n\n$I \\sim Ve^{-\\epsilon_B / k_0 T}$\n\nHandwritten mathematical diagram showing a curve intersecting a horizontal line with a vertical axis", "source": "mit-ocw", "source_doc_id": "0009-viii-the-fermi-gas-9f1cf2b2-36771958", "source_title": "VIII The Fermi gas:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 17, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["VIII The Fermi gas:", "Page 18"], "page_start": 18, "page_end": 18, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0001", "text": "Transport:\n\nTwo limits Collisionless and hydrodynamic\n\nMean - free path.\n\n$\\sigma$ scattering cross section  $\\rightarrow 0\\sigma$ cross section\n\nλ σ : contain one\n\ncollision center\n\non avenge,\n\nSimple hand-drawn sketch of a wavy line with a small oval mark above it (no text or symbols)\n\nλ\nσ\n\n$$\n\\lambda = \\frac {1}{\\sigma n}\n$$\n\n• ← collision center\n\nn: density of collision center\n\nScattering cross-section of two hand balls\n\n$$\n\\sigma = \\pi (2 r) ^ {2} = 4 \\pi r ^ {2}\n$$\n\n→\n2r\n\nd < \\lambda\n\neffusion\n\nAbstract sketch of a stylized human figure with jagged lines, no text or symbols present\n\nhydrodynamic flow", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0002", "text": "Effusion\n\ndistribution of momentum\n\nd³p\n→p\n\n$$\nf (\\vec {p}) d ^ {3} \\vec {p} = \\begin{array}{l l} \\text { density } & \\text { for   Trile } \\\\ \\text { in } & d ^ {3} p \\end{array} \\text { with   momentum }\n$$\n\n$$\nf (\\vec {p}) = c e ^ {- \\beta \\vec {p} ^ {2} / 2 m}\n$$\n\nBoltzmann distribution\n\n$$\n\\int d ^ {3} p f (\\vec {p}) = n _ {\\text { total   density }} f \\text { particles. }\n$$\n\n$$\n\\Rightarrow C = n (2 \\pi m k _ {B} T) ^ {- 3 / 2}\n$$\n\n$$\n\\text { Flux   of   effusion: } (\\# \\text { of   particle } n \\neq 0 \\quad n = 0\n$$\n\n$$\nI = \\int_ {v _ {x} > 0} v _ {x} f (p) d ^ {3} \\vec {p}\n$$\n\n$$\n= c \\int_ {p _ {x} > 0} \\frac {p _ {x}}{m} e ^ {- \\beta (p _ {x} ^ {2} + p _ {y} ^ {2} + p _ {z} ^ {2}) / 2 m} d p _ {x} d p _ {y} d p _ {z}\n$$\n\n$$\n= n \\sqrt {\\frac {k _ {B} T}{2 \\pi m}} \\quad \\begin{array}{l l} & I \\propto n T ^ {1 / 2} m ^ {- 1 / 2} \\\\ & W h y? \\end{array}\n$$\n\n$$\n\\text { Physical   picture }\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0003", "text": "$$\n\\text { Non - viscous } \\quad \\text { hydrodynamics }\n$$\n\n$$\n= \\text { Three   equations }\n$$\n\n$$\n\\lambda / _ {l} \\rightarrow 0 \\quad\\begin{array}{l l}&\\lambda : \\text { mean - free   path }\\\\&l: \\text { scale   of   interest, } +\\end{array}\n$$\n\n$$\n\\text { Partiles   are   \"caged\"   by   their   neighbors }\n$$\n\npicture\n\n$$\n\\begin{array}{l l} {\\mathrm{mass}} & {\\mathrm{density:} \\rho (\\vec {x}, t) = m n (\\vec {x}, t)} \\\\ {\\mathrm{mass}} & {\\mathrm{(current:} \\rho \\overrightarrow {u}} \\end{array}\n$$\n\n$$\n\\text {   I   average   velocity   }\n$$\n\n$$\n\\text { of   panT1:lec }\n$$\n\n$$\n\\text {   a.   velocity   of   gas / fluid   }\n$$\n\nno excl\n\nno exchange of particle,\n\nheat,\n\n① Mass conservation\n\n$$\n\\boxed {\\frac {\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\vec {u}) = 0} \\quad \\text { continuity   equation }\n$$\n\n$$\n\\Rightarrow \\frac {d}{d t} \\int_ {V} d ^ {3} \\vec {x} \\rho + \\oint_ {\\partial V} \\vec {d s} \\cdot \\rho \\vec {u} = 0\n$$\n\n② Newton's law.\n\n$$\n\\frac {d \\vec {u}}{d t} = \\frac {1}{m} \\vec {f} \\leftarrow \\text { force   on   the   point }\n$$\n\n$$\nI i f w e f o l l o w o n e p a r t i c l e\n$$\n\n$$\n\\frac {d \\vec {u}}{d t} = ? \\quad \\frac {d \\vec {u}}{d t} = \\underbrace {\\left(\\frac {\\partial}{\\partial t} + \\vec {u} \\cdot \\nabla\\right) \\frac {\\vec {u}}{\\vec {u} (\\vec {x} , t)}} _ {= \\frac {d}{d t}.} \\text {   velocity   field   }\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0004", "text": "$$\n\\begin{array}{l} F _ {x} = A (P (x) - P (x + \\Delta x)) ] \\\\ = - \\Delta x A \\partial_ {x} P \\quad \\text { on } N _ {\\text { particle }} \\\\ \\end{array}\n$$\n\n$$\n\\text { on   one   particle } \\quad f _ {x} = \\frac {F _ {x}}{N} = - \\frac {1}{n} \\partial_ {x} p\n$$\n\n$$\nP _ {(x)} \\left[ \\begin{array}{c} \\text { area } A \\\\ \\text { P } (x + \\Delta x) \\\\ \\leftrightarrow \\\\ \\Delta x \\end{array} \\right]\n$$\n\n$$\n\\left(\\frac {\\partial}{\\partial t} + \\vec {u} \\cdot \\nabla\\right) \\vec {u} = - \\frac {1}{m n} \\vec {\\nabla} P + \\frac {1}{m} f _ {e x}\n$$\n\n$$\nn \\boxed {p (\\frac {\\partial}{\\partial t} + \\vec {u} \\cdot \\nabla) \\vec {u} + \\nabla p = n \\vec {f} _ {e x}} E u l e r i s e q u\n$$\n\n③ Adiabatic condition conservation f en Tropy\n\n$$\n\\text { per   particle } - s:\n$$\n\n$$\ns \\text { does   not   change   with   the   flow }\n$$\n\n$$\nS = N k \\left[ \\frac {5}{2} - \\ln (n \\lambda_ {T} ^ {3}) \\right], S = \\frac {5}{N}, \\lambda = \\sqrt {\\frac {2 \\pi k ^ {2}}{m k _ {B} T}}\n$$\n\n$$\n\\Rightarrow n T ^ {- 3 / 2} \\quad \\text { is } \\quad \\text { conserved }\n$$\n\n$$\nP = n k _ {B} T \\quad a \\quad T = \\frac {P}{n k _ {B}}\n$$\n\n$$\n\\Rightarrow n (\\frac {n}{p}) ^ {3 / 2} = n ^ {5 / 2} p ^ {- 3 / 2} \\quad \\text { is } \\quad \\text { conserved }\n$$\n\n$$\n\\therefore P p ^ {- 5 / 3} \\quad \\text { is } \\quad \\text { consnued }\n$$\n\n$$\n\\boxed {(\\frac {\\partial}{\\partial t} + \\vec {u} \\cdot \\nabla) (P P ^ {- r}) = 0} \\quad \\begin{array}{c} x = \\frac {5}{3} \\\\ A d i m b a t i c \\end{array}\n$$\n\na⃗₁ = Ⓤ₁ ⋅ ⋅\n(⃗₂, t) ⋅ ⋅ (⃗₂ ⓕ₃, t + a⃗₁)\nP₁ρ₁⁻ᵣ\nP₂ρ₂⁻ᵣ = P₁ρ₁⁻ᵣ\n\n5 unknown P, P,   $\\overline{u}$  & 5 equations.", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0005", "text": "Linearized equation\n\n$$\n\\rho = \\rho_ {0} + \\delta \\rho\n$$\n\n$$\nP = P _ {0} + \\delta P\n$$\n\n$$\n\\overrightarrow {u} = 0 + \\overrightarrow {u}\n$$\n\n$$\n\\frac {\\partial \\leq \\rho}{\\partial t} + \\rho_ {0} \\nabla \\cdot \\vec {u} = 0\n$$\n\n$$\n\\rho_ {0} \\frac {\\partial \\vec {u}}{\\partial t} + \\nabla \\delta P = 0\n$$\n\n$$\n\\frac {\\partial}{\\partial t} (P P ^ {- \\gamma}) = P _ {0} ^ {- \\gamma} \\frac {\\partial}{\\partial t} s P - \\gamma P _ {0} P _ {0} ^ {- \\gamma - 1} \\frac {\\partial}{\\partial t} s P = 0\n$$\n\n$$\n\\frac {\\partial}{\\partial t} s p - \\gamma \\frac {p _ {0}}{p _ {0}} \\frac {\\partial}{\\partial t} s p = 0\n$$\n\nSound waves.\n\n$$\n\\frac {\\partial^ {2} \\delta \\rho}{\\partial t ^ {2}} + \\rho_ {0} \\nabla \\cdot \\frac {\\partial \\vec {u}}{\\partial t} = 0\n$$\n\n$$\n\\frac {\\partial^ {2} \\delta P}{\\partial t ^ {2}} - \\nabla^ {2} \\delta P = 0\n$$\n\n$$\n\\frac {\\partial^ {2} \\delta \\rho}{\\partial t ^ {2}} - \\nabla^ {2} \\delta \\rho \\times \\frac {P _ {0}}{\\rho_ {0}} = 0\n$$\n\n$$\n\\Rightarrow \\frac {\\partial^ {2} \\delta \\rho}{\\partial t ^ {2}} - c ^ {2} \\nabla^ {2} \\delta \\rho = 0\n$$\n\n$$\nc = \\sqrt {\\frac {\\gamma P _ {0}}{\\rho_ {0}}} = \\sqrt {\\frac {\\gamma k _ {B} T}{m}}\n$$\n\n$$\n\\rightarrow \\boxed {\\frac {s p}{p _ {0}} = x \\frac {s p}{p _ {0}}}\n$$\n\n$$\nP = n k _ {B} T\n$$\n\n$$\nx = \\frac {5}{3}\n$$\n\n$$\n\\begin{array}{l} {c \\sim \\mathrm{thermal}} \\\\ {\\mathrm{velocityof}} \\\\ {\\mathrm{particle}} \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0006", "text": "$$\n<   \\vec {v} _ {i} > = 0 v _ {i} \\in \\mathcal {V} _ {i} ^ {2}, v _ {i} \\tag {98}\n$$\n\n* Diffusion\n\nestimate\n\n$$\n\\text { Net   flux: } 6 \\text {   direction, }\n$$\n\n$$\nj _ {x} = \\frac {1}{6} \\overline {{v _ {1}}} n _ {1} - \\frac {1}{6} \\overline {{v _ {2}}} n _ {2}\n$$\n\n$$\n= \\frac {1}{6} \\overline {{U}} (n _ {1} - n _ {2})\n$$\n\nA B\n1 2 x\n↔ ↔\n↑ mean f\n\nmean free path\n\nassume  $\\overline{U}$ is the same\n\nwhat  $n_1 = n_2$?\n\nn. density of particle coming\n\nfrom region.A\n\n$n_{2}$  for region B\n\n$$\nn _ {1} - n _ {2} = - \\partial_ {x} n \\cdot \\lambda\n$$\n\n$$\nj _ {x} = - \\frac {1}{6} \\overline {{\\sigma}} \\lambda \\partial_ {x} n\n$$\n\nL diffusion flux.\n\nin hydrodynamic limit\n\nwe set  $\\lambda = 0$ no diffusion\n\nin general\n\n$$\n\\boxed {j = - D \\nabla n}\n$$\n\n$$\nD \\simeq \\frac {\\lambda \\overline {{v}}}{6}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0007", "text": "$$\n\\text {   continuity   equation   (with   diffusion)   } \\frac {\\partial n}{\\partial t} + \\nabla \\cdot \\vec {j} = 0\n$$\n\n$$\n\\vec {j} = n \\vec {a} - D \\nabla n\n$$\n\n$$\n\\Rightarrow \\boxed {\\frac {\\partial n}{\\partial t} + \\nabla \\cdot n \\vec {u} - D \\nabla^ {2} n = 0}\n$$\n\n$$\n\\text { or } \\quad \\frac {\\partial \\rho}{\\partial t} + \\nabla \\cdot \\rho \\vec {u} - D \\nabla^ {2} \\rho = 0\n$$\n\n$$\n\\text { diffusion   equation } (\\text { set } \\vec {u} = 0)\n$$\n\n$$\n\\frac {\\partial n}{\\partial t} - D \\nabla^ {2} n = 0\n$$\n\n$$\n: 6 n (t = 0, \\vec {x}) = N S (\\vec {x}) \\quad (\\sqrt {\\text { particide }} - 1 \\vec {x} = 0)\n$$\n\n$$\n\\text { in... } k - s p a c e\n$$\n\n$$\n\\frac {\\partial \\tilde {n}}{\\partial t} + D k ^ {2} \\tilde {n} = 0\n$$\n\n$$\n\\tilde {n} = t, k) = e ^ {- D k ^ {2} t}\n$$\n\n$$\n\\tilde {n} (t, \\vec {x}) = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- D k ^ {2} +} e ^ {i \\vec {k} \\cdot \\frac {1}{x}}\n$$\n\n$$\n= \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e ^ {- D t (\\vec {k} - \\frac {i}{2} \\frac {\\vec {x}}{D t}) ^ {2}} e ^ {- \\frac {1}{4} \\frac {\\vec {x} ^ {2}}{D t}}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0008", "text": "$$\n\\Rightarrow \\boxed {n (\\vec {x}, t) = \\frac {N}{(4 \\pi D t) ^ {3 / 2}} e ^ {- \\frac {1 \\vec {x} t ^ {2}}{4 D t}}}\n$$\n\n$$\n\\int d x e ^ {- x ^ {2}} = \\sqrt {\\pi}\n$$\n\n$$\n\\begin{array}{l} \\langle x ^ {2} \\rangle = \\int d ^ {5} x \\frac {\\lambda}{x ^ {2}} \\frac {e ^ {- | x | ^ {2} / 4 D t}}{(4 \\pi D t) ^ {3 / 2}} \\\\ \\int d x x ^ {2} e ^ {- x ^ {2}} \\\\ = \\frac {1}{2} \\sqrt {\\pi} \\\\ = 4 D t \\times \\frac {3}{2} \\\\ = 6 D t \\\\ \\end{array}\n$$\n\n$$\n\\overline {{x}} \\equiv \\sqrt {<   x ^ {2} >} = \\sqrt {6 D t}\n$$\n\n$$\n\\text { Random   walk } \\& \\text { diffusion }\n$$\n\n$$\n\\text { each   stop } \\lambda\n$$\n\n$$\nN \\cdot \\text { step } \\quad \\text { later }\n$$\n\n$$\n\\overrightarrow {x} = \\textcircled {2} \\Delta \\overrightarrow {x _ {1}} + \\Delta \\overrightarrow {x _ {2}} + \\dots\n$$\n\n$$\n\\vert \\Delta x _ {1} \\vert = \\vert \\Delta x _ {2} \\vert = \\dots = x\n$$\n\n$$\nD i s t r i b u t i o n f \\overrightarrow {x}\n$$\n\n$$\nn (\\vec {x}, N) = \\frac {1}{(\\pi \\alpha) ^ {3 / 2}} e ^ {- x / \\alpha}\n$$\n\n$$\n\\alpha = ?\n$$\n\n$$\n\\langle \\frac {1}{x} ^ {2} \\rangle = \\Delta x _ {1} ^ {2} + \\Delta x _ {2} ^ {2} \\dots\n$$\n\n$$\n= N \\lambda^ {2} \\Rightarrow \\boxed {\\overline {{x}} = \\lambda \\sqrt {N}}\n$$\n\n$$\n= \\int d ^ {3} x \\vec {x} ^ {2} n = \\frac {3}{2} \\alpha \\Rightarrow \\alpha = \\frac {2 N \\lambda^ {2}}{3}\n$$\n\n$$\n\\Rightarrow \\text {   See   } *\n$$\n\n$$\nn (\\vec {x}) = \\frac {1}{(\\pi \\frac {2 N \\lambda^ {2}}{3}) ^ {3 / 2}} e ^ {- \\frac {\\vec {x} ^ {2}}{3} \\frac {2 N \\lambda^ {2}}{3}} = \\frac {e ^ {- \\frac {\\vec {x} ^ {2}}{3} \\frac {2 \\lambda^ {2}}{3 \\tau} t}}{(\\pi \\frac {2 \\lambda^ {2}}{3 \\tau} t) ^ {3 / 2}}\n$$\n\n$$\nD = \\frac {\\lambda^ {2}}{6 \\tau} = \\frac {\\lambda v}{6}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0009", "text": "Damping of sound wave due to diffusion\n\nLinearized :\n\n$$\n\\frac {\\partial \\rho}{\\partial t} + p _ {0} \\nabla \\cdot \\vec {u} - D \\nabla^ {2} \\rho = 0\n$$\n\n$$\n\\rho_ {0} \\frac {\\partial \\vec {u}}{\\partial t} + \\nabla \\delta P = 0\n$$\n\n$$\n\\frac {\\partial^ {2} t p}{\\partial t ^ {2}} + \\rho_ {0} \\nabla \\cdot \\frac {\\partial y}{\\partial t} - D \\nabla^ {2} \\frac {\\partial}{\\partial t} s p = 0\n$$\n\n$$\n\\frac {\\partial^ {2} \\rho}{\\partial t ^ {2}} - c ^ {2} \\nabla^ {2} \\rho - D \\nabla^ {2} \\frac {\\partial}{\\partial t} \\rho = 0\n$$\n\n$$\ns p = A e ^ {i (k x - \\omega t)}\n$$\n\n$$\n- \\omega^ {2} + c ^ {2} k ^ {2} + D k ^ {2} (- i \\omega) = 0\n$$\n\n$$\n\\text { small } D\n$$\n\n$$\nw ^ {2} = c ^ {2} k ^ {2} - i D k ^ {3} c = c ^ {2} k ^ {2} (1 - i \\frac {D k}{c})\n$$\n\n$$\nw = c k (1 - \\frac {1}{2} i \\frac {D k}{c})\n$$\n\n$$\ns p = A e ^ {i (k x - c k t)} e ^ {- t / \\tau}\n$$\n\n$$\n\\boxed { \\begin{array}{r l} {\\tau = \\frac {2}{D k ^ {2}}} & {= \\frac {1 2}{\\lambda \\sigma k ^ {2}} \\sim \\frac {1}{\\omega} \\frac {1}{\\lambda k}} \\\\ & {\\mathrm{L} _ {\\sim c}} \\end{array} }\n$$\n\nA sound wave can oscillate   $\\frac{1}{\\lambda k}$  times\n\n$$\n\\frac {1}{t} = \\frac {D k ^ {2}}{2} \\quad \\begin{array}{l l} & \\text { damping   rate } \\\\ & \\text { damping   coefficient } \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0010", "text": "$$\n\\begin{array}{l} \\begin{array}{l} \\text { Viscosity } - \\text { momentum } \\\\ (\\text { due   to   diffusion }) \\end{array} \\\\ F l _ {u x} \\text {   of   } P _ {x} \\text {   momentum   } \\\\ = F _ {\\text { orce   per   area }} F _ {x} \\\\ \\end{array}\n$$\n\nConduction\n\n$$\n\\begin{array}{l} F _ {x} = \\frac {i}{6} n \\overline {{v}} (\\cdot u _ {1 x} m - \\cdot u _ {2 x} m) \\\\ = - \\frac {1}{6} h m \\overline {{v}} \\lambda \\frac {\\partial u _ {x}}{\\partial y} \\\\ \\end{array}\n$$\n\n$$\nF _ {x} = - \\nu \\frac {\\partial u _ {x}}{\\partial y}\n$$\n\n$$\n= \\text { flux   of } x \\text { momentum   in }\n$$\n\n$$\ny - d i n e c t i o n\n$$\n\n$$\n\\begin{array}{r l} {P} & {= \\frac {1}{6} n \\overline {{v}} m \\lambda} \\\\ & {= \\sqrt {\\frac {2 m k _ {B} T}{3 \\sigma}}} \\end{array}\n$$\n\n$$\n= \\text { Force   in } x \\text {-direction on }\n$$\n\n$$\n\\text { a   surface   normal } \\pi_ {0} y.\n$$\n\n$$\n\\text { independent   of } n.\n$$\n\n$$\nP _ {2} \\text {   essence   } \\quad \\text {   tensor   } P _ {i j}:\n$$\n\n$$\n\\vec {F} _ {1} = (F _ {1 x}, F _ {1 y}, F _ {1 z}) = A (P _ {1 1}, P _ {1 2}, P _ {1 3})\n$$\n\n$$\n\\vec {F} _ {2} = A (P _ {2 1}, P _ {2 2}, P _ {2 3})\n$$\n\n$$\n\\vec {F} _ {3} = A (P _ {3 1}, P _ {3 2}, P _ {3 3})\n$$\n\nF₁ = force on surface\nnoimal x̂ x̂\n\nF₃\nF₁ F₂ x y\n\n$$\nP _ {i j} = \\underbrace {\\delta_ {j} P} _ {\\text { usual   pressure }} + \\underbrace {P _ {i j} ^ {\\prime}} _ {\\text { traceless   part }} \\quad \\& \\quad P _ {i j} ^ {\\prime} = - r (\\frac {\\partial u _ {i}}{\\partial x _ {j}} + \\frac {\\partial u _ {j}}{\\partial x _ {i}} - \\frac {2}{3} \\delta_ {i j} \\nabla \\cdot \\vec {u})\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 9, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 10"], "page_start": 10, "page_end": 10, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0011", "text": "$$\n\\text { Navien - Stokes   equation }\n$$\n\n$$\n\\begin{array}{l} \\Delta V \\rho (\\frac {\\partial}{\\partial t} + \\vec {u} \\cdot \\nabla) u _ {i} = \\text {   force   in   } i - d i r e c t e n \\\\ \\frac {d u _ {i}}{d t} \\\\ = + A \\sum_ {j} f _ {i j} (x + \\Delta x _ {j}) + P _ {i j} (x) \\\\ \\end{array}\n$$\n\nmass\n\n$$\n\\Rightarrow f (\\frac {\\partial}{\\partial t} + \\vec {u}, 0) u _ {i} + \\frac {\\partial P _ {i j}}{\\partial x _ {j}} = 0\n$$\n\n$$\n\\Rightarrow \\rho (\\frac {\\partial}{\\partial t} + \\vec {u} \\cdot \\nabla) \\vec {u} + \\nabla \\cdot (P - \\frac {\\nu}{3} \\nabla \\cdot \\vec {u}) - \\nu \\nabla^ {2} \\vec {u} = 0\n$$\n\n$$\nn = f _ {e x}\n$$\n\nAny quantity can be set to 1 by\n\nchouqing the unit.\n\n$$\n\\frac {\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\vec {u}) = 0\n$$\n\n$$\n\\left(\\frac {\\partial}{\\partial t} + \\vec {u} \\cdot \\nabla\\right) (p p ^ {- \\gamma}) = 0\n$$\n\nL\n- →\n→\n→\n→\n→\nu₀\n\nThe above these equations determine the\n\nflow in the pipe. quantities characters\n\nthe effect of viscosity. Input  $u_{0}$, L, V, ρ\n\n$$\n[ V ] = \\frac {\\text { force / area }}{\\text { velocity / length }} = \\frac {m \\sigma}{t L ^ {2}} \\frac {L}{\\sigma} = \\frac {m}{t L}\n$$\n\n$R \\gg 1$ turbulent\n\nR << 1 streamline\n\n$$\nR = \\frac {\\rho L u _ {0}}{v} \\quad \\text { dimensionless } \\quad \\text { Reynolds   number }\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 10, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 11"], "page_start": 11, "page_end": 11, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0012", "text": "Heat conduction - energy exchange without particle exchange.\n\n$$\n\\begin{array}{l} j _ {x} = \\frac {1}{6} (n _ {1} \\overline {{v}} _ {1} - n _ {2} \\overline {{v}} _ {2}) = 0 \\\\ n _ {1} \\approx n _ {2} \\quad \\overline {{v}} _ {1} \\approx \\overline {{v}} _ {2} \\\\ \\end{array}\n$$\n\n1\n2\nx\n\n$$\nf _ {x} = \\frac {1}{6} n \\overline {{v}} \\left(\\ell \\frac {k _ {B} T _ {1}}{2} - \\ell \\frac {k _ {B} T _ {2}}{2}\\right)\n$$\n\nL number of degrees of freedom\n\n$$\nl = 3 \\quad \\text { for   point   particle }\n$$\n\n$$\nT _ {1} - T _ {2} = - \\partial_ {x} T \\lambda\n$$\n\nE mean free path.\n\n$$\nf _ {x} = - \\frac {1}{1 2} \\lambda l n \\overline {{v}} k _ {B} \\partial_ {x} T\n$$\n\n$$\n\\boxed {\\vec {q} = - k \\nabla T}\n$$\n\n$$\n\\begin{array}{l} k = \\frac {l n \\overline {{v}} k _ {B} \\lambda}{1 2} \\\\ \\hat {L} = \\frac {1}{6} n \\overline {{v}} \\lambda c _ {v} \\\\ \\end{array}\n$$\n\nheat conductance\n\nspecific heat\n\npan particle\n$C_V = \\frac{lk_6}{2}$\n\nUsing   $\\overline{U}=\\sqrt{2k_{B}T/m}$\n\n$\\lambda = 1/n\\sigma$ scattering cross-section\n\n$$\nk = \\frac {c v}{3 \\sigma} \\boxed {\\frac {2 k _ {B} T}{m}}\n$$\n\nindependent of n\n\nEnergy conservation\n\n$$\n\\frac {\\partial u}{\\partial t} + \\nabla \\cdot \\vec {q} = 0\n$$\n\nheat conductance equation\n\nsince  $g = -K \\cap T$$u = n \\cap rT \\Rightarrow$\n\n$$\n\\text { with   heat   source } \\quad n c _ {v} \\frac {\\partial I}{\\partial t} - K \\nabla^ {2} T = W\n$$\n\n$$\n\\frac {\\partial T}{\\partial t} - \\frac {k}{n c _ {v}} \\nabla^ {2} T = 0\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 11, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 12"], "page_start": 12, "page_end": 12, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0013", "text": "* Speed of freezing\n\nA: T = 273 - ΔT\nice ↓x\n\nheat transport per unit area\n\nWater: T = 273\n\nper unit time  $g = k \\frac{\\Delta T}{x}$\n\nheat generated by freezing\n\nΔS - change of entropy per atom\n\n$l = \\Delta S T - \\text{latent heat per atom.}$\n\nheat generated per unit area per unit time\n\n$$\n\\frac {d x}{d t} \\pi l = K \\frac {\\Delta T}{x}   \\mathrm{numberdensity}\n$$\n\n$$\n\\Rightarrow \\frac {d x}{d t} = \\frac {k \\Delta T}{n l} \\frac {1}{x}\n$$\n\n$$\n\\Rightarrow x ^ {2} = 2 \\frac {k \\Delta T}{n e} t \\quad \\text { or } \\quad x = \\sqrt {\\frac {2 k \\Delta T}{n e}} \\cdot \\sqrt {t}\n$$\n\nNumerical estimate\n\n$\\Delta S :$$I_{\\text{doil}} \\text{ gas } S = k_B \\left( \\frac{5}{2} + \\ln \\frac{U}{\\lambda^2} \\right) \\text{ per atom}$\n\nassume   $\\frac{V_{water} = 2V_{ice}}{C_{v} \\sim k_{B}} \\Rightarrow \\Delta S = k_{B}$\n\n$$\nk = \\frac {c _ {v}}{3 \\sigma} \\sqrt {\\frac {2 k _ {B} T}{m}} = \\frac {k _ {B}}{\\sigma} \\sqrt {\\frac {k _ {B} T}{m}}\n$$\n\n$$\n\\sqrt {\\frac {2 k \\Delta T}{n l}} \\sim \\sqrt {\\frac {\\frac {k _ {B}}{\\sigma} \\sqrt {\\frac {k _ {B} T}{m}} \\cdot \\Delta T}{n k _ {B} T _ {0}}} = \\sqrt {\\frac {\\overline {{U}} \\Delta T}{\\sigma n T _ {0}}} \\sim \\sqrt {a \\overline {{U}} \\frac {\\Delta T}{T _ {0}}}\n$$\n\n$$\na \\sim 5 A ^ {0} \\quad \\overline {{v}} = 3 0 0 m / s \\quad \\sqrt {a \\overline {{v}}} = 0. 3 m m \\frac {1}{\\sqrt {\\sec}}\n$$\n\n$$\nx = 0. 3 m m \\sqrt {\\frac {4 T}{T}} \\sqrt {\\frac {t}{\\sec}}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 12, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 13"], "page_start": 13, "page_end": 13, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0014", "text": "Transport in a metal\n\nfraction with no collision\n\n* Drude model:\n\n$$\n\\Delta <   u > = (1 - \\frac {\\Delta t}{2}) <   u > + \\frac {\\Delta t}{2} <   0 >\n$$\n\n$$\n\\frac {d u}{d t} = - \\frac {1}{\\tau} u \\underbrace {\\text { collision }} _ {\\text { average   velocity }}\n$$\n\n$$\nv = v _ {0} e ^ {- t / \\tau}\n$$\n\nwith external electric\n\nI am mean-free time\n\nfield\n\n$\\tau = \\lambda / \\overline{\\sigma}$ from scattering with impurity\n\n$$\n\\frac {d u}{d t} = - \\frac {1}{\\tau} u + \\frac {e}{m} E\n$$\n\n$$\n\\begin{array}{c} \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ \\rightarrow \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ > \\\\ >< |content_end|>\n$$\n\nj = enu - change current density\n\n$$\nj + \\tau \\frac {d j}{d t} = \\frac {e ^ {2} n \\tau}{m} E\n$$\n\n$$\n\\text {   steady   state   } \\quad \\boxed {\\vec {j} = \\sigma \\vec {E}, \\sigma = \\frac {e ^ {2} n \\tau}{m}} \\text {   Ohm's   law.   }\n$$\n\nTemperature dependence of 5\n\nOnly T. depenon T.\n\nT ~ λ/√\nI_{no T}\ndependence\nimpurity density\nis fixed\nV = √(E_F/m) √\n= (π n^1/3/m)\n√ ∝ T^0 √\n⇒ T ∝ T^{-1/2}\nσ ∝ T^{-1/2} ×\n√\nE_F\nT^{-1/2}", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 13, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 14"], "page_start": 14, "page_end": 14, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0015", "text": "Hall effect\n\n$$\n\\frac {d \\vec {u}}{d t} = - \\frac {1}{2} \\vec {u} + \\frac {e}{m} \\vec {E} + \\frac {e}{m c} \\vec {u} \\times \\vec {B} (c g s)\n$$\n\n$$\n\\vec {v} = e n \\vec {u}\n$$\n\n$$\n\\frac {d \\vec {j}}{d t} = - \\frac {1}{\\tau} \\vec {j} + \\frac {e ^ {2} n}{m} \\vec {E} + \\frac {e}{m c} \\vec {j} \\times \\vec {B}\n$$\n\nsteady flow\n$\\vec{j} = \\frac{e^{2}n\\tau}{m}\\vec{E} + \\frac{e\\tau}{mc}\\vec{j} \\times \\vec{B}$\n\nLet  $\\overrightarrow{B} = (0,0,B)$$\\overrightarrow{j} \\times \\overrightarrow{B} = (j_B, -j_X B, 0)$\n\n$$\nj _ {3} = \\frac {e n \\tau}{m} E _ {z}\n$$\n\n$$\nw _ {c} = e B / m c \\quad ⓤ e ^ {\\circ}\n$$\n\n$$\n\\left( \\begin{array}{l} j _ {x} \\\\ j _ {y} \\end{array} \\right) = \\frac {\\sigma_ {0}}{1} \\left( \\begin{array}{l} E _ {x} \\\\ E _ {y} \\end{array} \\right) + \\left( \\begin{array}{l l} 0 & w _ {c} \\tau \\\\ - w _ {c} \\tau & 0 \\end{array} \\right) \\left( \\begin{array}{l} j _ {x} \\\\ j _ {y} \\end{array} \\right)\n$$\n\n$$\n\\text { cyclotron   frequency }\n$$\n\n$$\na \\quad \\boxed  \\begin{array}{l} (E _ {x} \\\\ E _ {y} \\end{array} ) = \\sigma_ {0} ^ {- 1} \\left( \\begin{array}{c c} 1 & - w _ {c} z \\\\ w _ {c} z & 1 \\end{array} \\right) \\binom{j _ {x}}{j _ {y}} = \\left( \\begin{array}{c c} p _ {x x} & p _ {x y} \\\\ p _ {y x} & p _ {y y} \\end{array} \\right) \\binom{j _ {x}}{j _ {y}}\n$$\n\n$$\n| \\rho_ {x y} | = | \\rho_ {y x} | = \\frac {\\omega_ {c} \\tau}{\\sigma_ {0}} = \\frac {e ^ {\\prime} B \\pi}{\\mu c} \\frac {m}{e ^ {+} n \\tau} = \\frac {B}{e c n}\n$$\n\n$$\n\\left| \\rho_ {x y} \\right| = \\frac {B}{e n c} = R _ {H} B \\quad R _ {H} = \\frac {1}{e n c}\n$$\n\nmeaning : if   $\\begin{pmatrix} j_{x} \\\\ j_{y} \\end{pmatrix} = \\begin{pmatrix} j_{x} \\\\ 0 \\end{pmatrix}$$\\quad E_{x}$$E_{y}$  =   $\\begin{pmatrix} p_{xx} \\\\ (x_{y}) \\end{pmatrix} j_{x}$\n\n$$\nE _ {y} = P _ {x y} j _ {x} = R _ {H} B j _ {x} \\quad \\uparrow^ {E _ {y}} j _ {x}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 14, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 15"], "page_start": 15, "page_end": 15, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0016", "text": "Boltzmann equation (fluid in   $(\\vec{x}, \\vec{k})$ )\n\n$$\ng (\\vec {r}, \\vec {k}, t): d N = g (\\vec {r}, \\vec {k}, t) \\frac {d ^ {3} \\vec {r} d ^ {3} k}{(2 \\pi) ^ {3}}\n$$\n\n1\n\nnumber of particles in   $d\\frac{3}{4}d\\frac{3}{k}$\n\nnumber of particle in a k level\n\nEquilibrium distribution\n\n$$\ng _ {0} = \\frac {1}{e ^ {\\mu (\\epsilon_ {k} - \\mu)} + 1} \\quad f e r m i o n\n$$\n\n$$\ng _ {0} = \\frac {1}{e ^ {\\beta (\\varepsilon_ {n} - \\mu)} - 1} \\quad b o s o n\n$$\n\n$$\ng _ {0} = e ^ {- \\beta (\\varepsilon_ {h} - \\mu)} \\quad \\text { classical } g _ {c s}\n$$\n\nIf   $g(\\vec{r},\\vec{k},t)\\neq g_{0}$  non-equilibrium distribution\n\nDiffusionless motion (hydrodynamic motion)\n\n$$\nd N (t) = d N (t - d t)\n$$\n\n$$\ng (\\vec {r}, \\vec {k}, t) \\underbrace {d ^ {3} r (t) d ^ {3} k (t)} = g (\\vec {r} - \\vec {\\sigma} _ {(k)} d t, \\vec {k} - \\vec {F} \\frac {d t}{k}, t - d t) x\n$$\n\n$$\nⓌ d ^ {3} r d ^ {3} k\n$$\n\nLiouville's Theorem\n\n$$\nd ^ {3} r d ^ {3} k = d ^ {3} \\vec {r} \\cdot d ^ {3} \\vec {k}\n$$\n\n$d^{3}\\vec{r}(t-d t)d^{3}\\vec{k}(t-d t)$$d^{3}r'dk'$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 15, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 16"], "page_start": 16, "page_end": 16, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0017", "text": "$$\ng (\\vec {r}, k, t) = g (\\vec {r} - \\vec {v} _ {k} d t, \\vec {k} - \\vec {F} \\frac {d t}{k}, + \\cdot d t)\n$$\n\n$$\n\\boxed {\\frac {\\partial g}{\\partial t} + \\vec {v} \\cdot \\frac {\\partial}{\\partial r} g + \\frac {1}{k} \\vec {F} \\cdot \\frac {\\partial}{\\partial k} g = 0}\n$$\n\n$$\n\\text { relaxation } = \\text { time   approximation }\n$$\n\n$$\n\\text { no   diffusion }\n$$\n\n$$\n\\begin{array}{c} \\text {(collission} \\\\ \\text {offeet)} \\end{array}\n$$\n\n$$\n\\begin{array}{c}d N _ {1}\\\\\\square\\\\\\square\\\\d N _ {2}\\end{array}\\rightarrow \\boxed {d N _ {1}} ^ {\\prime}\n$$\n\n$$\n\\frac {\\partial g}{\\partial t} + \\vec {v} \\cdot \\frac {\\partial}{\\partial r} g + \\frac {1}{r} \\vec {F} - \\frac {\\partial}{\\partial k} g = - \\frac {1}{\\tau} (g - g _ {0})\n$$\n\nout\n\n1.4\n\n$$\n\\text { Let } \\quad s g = g - g _ {0}\n$$\n\n$$\n\\text {   Steady   state   :   }\n$$\n\n$$\n- \\frac {1}{2} g\n$$\n\n$$\nⓞ u t - \\mathrm{lemm}\n$$\n\n$$\ns g = - \\tau \\vec {U} \\frac {\\partial}{\\partial r} g _ {0} - \\frac {T}{k} \\vec {F} \\cdot \\frac {\\partial}{\\partial k} g _ {0}\n$$\n\n$$\n+ \\frac {1}{\\sqrt {x}} - \\frac {1}{t} g _ {0}   i n - t e r m\n$$\n\n$$\n- \\tau \\vec {v} \\frac {\\partial}{\\partial \\vec {F}} g - \\frac {\\tau}{5} \\vec {F} \\frac {\\partial}{\\partial k} s g \\leftarrow O (\\frac {\\partial}{\\partial F} g, \\vec {F}) ^ {2}\n$$\n\n$$\nA _ {\\mathrm{assume}} \\frac {\\partial}{\\partial \\vec {r}} g _ {0} \\quad \\mathrm{and} \\vec {F} \\quad \\mathrm{is} \\quad 5. m a l l\n$$\n\n$$\ns g = O (\\frac {\\partial}{\\partial \\vec {r}} g _ {0}, \\vec {F})\n$$\n\n$$\n\\frac {\\partial f}{\\partial \\vec {r}} = T \\frac {\\partial f}{\\partial E} \\frac {\\partial \\beta (E - r)}{\\partial \\vec {r}}\n$$\n\n$$\n\\delta g = - \\tau \\vec {v} _ {(k)} \\frac {\\partial}{\\partial \\vec {r}} g _ {0} - \\frac {\\tau}{k} \\vec {F} \\cdot \\frac {\\partial}{\\partial k} g _ {0}\n$$\n\n$$\n= + \\frac {\\partial f}{\\partial t} (- \\nabla \\mu - \\frac {t - \\mu}{T} \\nabla T)\n$$\n\n$$\ng _ {0} (T (\\vec {r}), \\mu (\\vec {r}; \\vec {k})) \\quad f o r f e r m i o n g _ {0} = f = \\frac {1}{e ^ {\\beta (\\xi - \\mu) + 1}}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 16, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 17"], "page_start": 17, "page_end": 17, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0018", "text": "$$\n\\begin{array}{l} F _ {0} \\quad f e r m i o n g _ {0} = f = \\frac {1}{e ^ {\\beta (\\vec {r}) [ \\varepsilon_ {\\vec {k}} - \\mu (\\vec {r}) ]} + 1} \\\\ \\frac {\\partial f}{\\partial r} = \\frac {1}{\\beta} \\frac {\\partial f}{\\partial \\epsilon} \\quad \\frac {\\partial [ \\beta (\\epsilon - \\mu) ]}{\\partial r} \\\\ = T \\frac {\\partial f}{\\partial \\varepsilon} (- \\beta \\frac {\\partial \\mu}{\\partial \\vec {r}} - \\frac {\\varepsilon - \\mu}{T ^ {2}} \\frac {\\partial T}{\\partial \\vec {r}}) \\\\ = - \\frac {\\partial f}{\\partial \\epsilon} (\\nabla \\mu + \\frac {\\epsilon - \\mu}{T} \\nabla T) \\\\ \\frac {\\partial f}{\\partial k} = \\frac {\\partial f}{\\partial t} \\frac {\\partial E}{\\partial k} \\\\ = h \\vec {v} _ {k} \\frac {\\partial f}{\\partial t} \\\\ \\end{array}\n$$\n\n$$\ns g _ {(\\vec {r}, \\vec {k})} = (- \\frac {\\partial f}{\\partial \\epsilon}) \\quad = \\vec {v} \\left[ - \\nabla \\mu + \\vec {F} - \\frac {\\epsilon - \\mu}{T} \\nabla T \\right]\n$$\n\n$$\n\\delta g (\\vec {r}, \\vec {k}) = (- \\frac {\\partial f}{\\partial \\epsilon}) _ {(\\vec {r}, \\vec {k})} \\quad \\tau \\vec {v} _ {\\vec {k}} \\cdot [ - \\nabla \\mu (\\vec {r}) + \\vec {F} _ {(\\vec {r}, \\vec {k})} - \\frac {\\epsilon_ {\\vec {k}} - \\mu (\\vec {r})}{1 (\\vec {r})} \\nabla ]\n$$\n\n$$\n\\overrightarrow {F} (\\vec {r}, \\vec {k}) = e \\vec {E} (\\vec {r}) + \\frac {e}{c} \\vec {v} _ {(k)} \\times \\vec {B} (\\vec {r})\n$$\n\n$$\n\\text { One   we   know } \\quad \\vec {E} (\\vec {r}), \\vec {B} (\\vec {r}), T (\\vec {r}), \\mu (\\vec {r})\n$$\n\n$$\n\\text { we } \\quad k \\text { now } \\quad g (\\vec {r}, \\vec {k}) = g _ {0} + s g\n$$\n\n$$\n\\text { We } \\dots \\text { can } \\quad \\text { calculate } \\quad \\text { any   Thing }\n$$\n\n$$\n\\text { Also   works   for   boson   gas   if   we replace }\n$$\n\n$$\nf = \\frac {1}{e ^ {\\beta (t - \\mu)} + 1} \\quad b _ {0} f _ {b} = \\frac {1}{e ^ {\\beta (\\epsilon - \\mu)} - 1}\n$$\n\n$$\n\\text { or } \\quad \\text { classical } _ {\\text { gas }} \\quad \\text { by. } f _ {c} = \\frac {1}{e ^ {\\beta (t - \\mu)}}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 17, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 18"], "page_start": 18, "page_end": 18, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0019", "text": "Application: electric current\n\n$$\n\\frac {1}{B} = 0\n$$\n\n$$\ng _ {0} = g _ {0} + \\tau (- \\frac {\\partial f}{\\partial \\epsilon}) \\vec {v} _ {k} \\cdot [ + e \\vec {\\xi} + \\frac {\\epsilon - \\mu}{T} (- \\nabla T) ]\n$$\n\n$$\n\\vec {E} \\equiv \\vec {E} - \\frac {\\nabla^ {M}}{e}\n$$\n\n§9\n\nelectric current\n\n$$\nj _ {x} = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e v _ {x} (\\vec {k}) g (\\vec {k}) = \\int \\frac {d ^ {3} \\vec {k}}{(2 \\pi) ^ {3}} e v _ {x} \\cdot \\delta g\n$$\n\n$$\n= \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\tau e ^ {2} \\left(- \\frac {\\partial f}{\\partial \\varepsilon}\\right) v _ {x} \\vec {v} \\cdot \\vec {\\varepsilon}\n$$\n\n$$\n+ \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} e \\tau (- \\frac {\\partial f}{\\partial \\epsilon}) \\frac {\\epsilon - \\mu}{T} v _ {x} \\vec {v} \\cdot (- \\nabla T)\n$$\n\n$$\n= L ^ {1 1} \\varepsilon_ {x} + L ^ {1 2} (- \\partial_ {x} T) d ^ {3} k = 4 \\pi k ^ {2} d k\n$$\n\n$$\nL ^ {\\prime \\prime} = \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} + e ^ {2} (- \\frac {\\partial f}{\\partial \\epsilon}) v _ {x} ^ {2} = 2 \\pi k \\frac {d k ^ {2}}{k ^ {2}} d \\epsilon\n$$\n\n$$\n\\int d \\epsilon (- \\frac {\\partial f}{\\partial \\epsilon}) = 1\n$$\n\n$$\n= \\frac {1}{3} \\int \\frac {d ^ {3} k}{(2 \\pi) ^ {3}} \\tau e ^ {2} (- \\frac {\\partial f}{\\partial \\epsilon}) \\sigma^ {2}\n$$\n\n$$\n= \\frac {1}{3} \\frac {1}{(2 \\sqrt {7}) ^ {3}} 2 \\pi k _ {F} \\frac {2 m}{k ^ {2}} \\tau e ^ {2} \\sigma_ {F}\n$$\n\n$$\n= \\frac {1}{3} \\frac {1}{(2 \\pi) ^ {3}} \\frac {4 \\pi}{m} \\tau e ^ {2} k _ {F} ^ {3} n = \\frac {4}{3} \\pi k _ {F} ^ {3} \\frac {1}{(2 \\pi) ^ {3}}\n$$\n\n$$\n= \\frac {e ^ {2} \\tau n}{m} \\quad (\\text {   result   of   } D _ {\\text {   node   model   }}) \\checkmark\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 18, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 19"], "page_start": 19, "page_end": 19, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0020", "text": "Thermopower.\n\nT₁ j=0 T₀\n\n$$\n\\vec {E} = Q \\nabla T \\quad (j = 0)\n$$\n\n$$\nQ = \\frac {L ^ {1 2}}{L ^ {1 1}}\n$$\n\nT₁\nT₀\n\n$$\n\\approx \\frac {n ^ {1 / 3} e ^ {\\pi k _ {B} ^ {2}} T}{k ^ {2}} / \\frac {e ^ {2} \\pi n}{m}\n$$\n\nintin elezition spacing\n\n$$\n= \\frac {k _ {B}}{e} \\left(\\frac {m k _ {B} T}{n ^ {3 / 3} h ^ {2}}\\right) \\sim \\frac {k _ {B}}{e} \\frac {1}{(n ^ {1 / 3} \\lambda_ {T}) ^ {2}} \\sim \\frac {k _ {B}}{e} \\left(\\frac {\\ell}{\\lambda_ {T}}\\right) ^ {2}\n$$\n\n$$\nL _ {e} t \\quad Q = Q _ {0} \\frac {k _ {B}}{e}\n$$\n\nT₀ ← △V → T₁\n\n$$\ne L E = Q _ {0} k _ {B} (T _ {1} - T _ {0})\n$$\n\n$$\ni e V \\sim 1 2 6 0 K\n$$\n\n$$\n\\varepsilon = \\frac {1}{1 1 6 0} Q _ {0} (\\nabla \\cdot T) _ {\\frac {\\text { Kelvin }}{\\text { meter}}} (\\frac {\\text { Volts }}{\\text { meter }})\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 19, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 20"], "page_start": 20, "page_end": 20, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0021", "text": "Einstein relation:\n\nOne particle in a gas in   $V=\\frac{1}{2}Kx^{2}$  potential\n\nBoltzman distribution\n\n$$\nP (x) = \\cdot e ^ {- \\frac {\\beta}{2} K x ^ {2}}\n$$\n\nSimple line drawing of a curved surface with an arrow and a circular mark, no text or symbols present.\n\n$$\n\\langle \\frac {1}{2} k x ^ {2} \\rangle = \\frac {1}{2} k _ {B} T \\Rightarrow \\langle x ^ {2} \\rangle = \\frac {k _ {B} T}{k}\n$$\n\n$<x^{2}>$ from point view of diffusion\n\nWhen V=0\n\nProb. distribution\n\n$P(x,t)$ satisfies\n\ndiffusion equation\n\n$$\n\\frac {\\partial P}{\\partial t} - D \\frac {\\partial^ {2} P}{\\partial x ^ {2}} = 0\n$$\n\n$$\n\\langle x ^ {2} \\rangle = 2 D t (\\mathrm{in} 1 D)\n$$\n\n$$\n\\hat {I} \\text { diffusion   const. }\n$$\n\n$$\n\\langle x ^ {2} \\rangle = 2 d D t (\\text { in } d - d i m e n s i o n s)\n$$\n\n$$\n= N _ {t} \\lambda_ {0} \\leftarrow s t e p l e n g t h.\n$$\n\n$$\n\\hat {L} \\# f s t e p s\n$$\n\nWhen  $V \\neq 0$, particle experience a force\n\nF. F induces a drift\n\n$$\nu = \\eta F\n$$\n\n$\\overline{1}$ mobility\n\ndrift make  $<x^{2}$ finite\n\nWe require   $\\langle x^{2}\\rangle=\\frac{K_{B}1}{K}\\Rightarrow$\n\nin t = 0 limit\n\nfriction y related to diffusion\n\nF\n←\nu ←", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 20, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 21"], "page_start": 21, "page_end": 21, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0022", "text": "Diffusion with Force.\n\n$$\n(n _ {w} = N P (x))\n$$\n\n$$\n\\vec {j} = n \\vec {u} - D \\nabla n\n$$\n\nDiffusion equation: from   $\\frac{\\partial n}{\\partial t} + \\nabla \\cdot j = 0$\n\n$$\n\\vec {u} = \\eta \\vec {F}\n$$\n\n$$\n\\Rightarrow \\boxed {\\frac {\\partial n}{\\partial t} + \\eta \\nabla \\cdot (\\vec {F} n) - D \\nabla^ {2} n = 0}\n$$\n\nSteady state (equilibrium state)   $\\frac{\\partial n}{\\partial t}=0$\n\n$$\n\\eta \\nabla \\cdot (\\vec {F} n) - D \\nabla^ {2} n = 0\n$$\n\n$$\n\\sigma \\quad \\eta \\vec {F} n = D \\nabla n.\n$$\n\n$$\n\\because \\vec {F} = - \\nabla V \\Rightarrow D \\nabla n = - \\eta n \\nabla V\n$$\n\n$$\n\\nabla \\ln n = - \\frac {\\eta}{D} \\nabla V\n$$\n\n$$\n\\text { In   equilibrium,   we   should   have }\n$$\n\n$$\nn \\propto e ^ {- \\beta V} \\Rightarrow \\nabla \\ln n = - \\beta \\nabla V\n$$\n\n$$\n\\Rightarrow \\frac {\\eta}{D} = \\beta \\quad \\text { in } \\quad D = \\eta k _ {B} T \\quad \\text { Einstein   relation }\n$$\n\nfluctuation  $\\Leftrightarrow$ dissipation (diffusion  $\\Leftrightarrow$ friction)\n\n$$\n\\begin{array}{c c c c} \\uparrow & & \\\\ \\hline <   x ^ {2} > = 2 d D t & & \\uparrow & \\\\ & & D = \\eta k _ {B} T & \\\\ & & \\frac {\\partial n}{\\partial t} - D \\nabla^ {2} n = 0 & \\\\ & & & \\eta \\\\ & & & \\vec {u} = \\eta \\vec {F} \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 21, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 22"], "page_start": 22, "page_end": 22, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0023", "text": "Random walk of particles in Water viscosity of   $H_{2}O$  at   $20^{\\circ}C$ .\n\n$$\nv = 0. 0 1 \\dots P o i s e (d y n e, s e c / c m ^ {2})\n$$\n\n$$\nV = \\text { Pressure } / \\frac {d u}{d x} = d y n e / c m ^ {2} / \\frac {c m}{s e c} / c m = (d y n e / c m ^ {2}) \\cdot s e c\n$$\n\n$$\n\\vec {F} = ?\n$$\n\n$$\nF = p \\cdot a ^ {2}\n$$\n\n$$\n= a ^ {2}, \\nu \\cdot \\frac {d u}{d x} = a ^ {2} \\nu \\frac {u}{a} = a \\nu u\n$$\n\n$$\n\\vec {F} = \\frac {6 \\pi}{1 8} a v \\vec {u} \\Rightarrow \\eta = (6 \\pi a v)\n$$\n\n$$\nE _ {\\text { insTein's }} \\text { relation }\n$$\n\n$$\nD = k _ {B} T \\eta\n$$\n\n$$\n\\begin{array}{r l} {\\langle \\vec {x} ^ {2} \\rangle = 6 D t} & {= 6 k _ {B} T \\eta t} \\\\ & {= \\frac {k _ {B} T t}{\\pi a v}.} \\end{array}\n$$\n\nvalid only when\n-1\na >> λ\n\n$$\nk _ {B} = \\frac {\\langle \\vec {x} ^ {2} \\rangle \\pi a v}{T t}\n$$\n\n$$\nR = k _ {B} N _ {A}\n$$\n\nAvogadro's number\n\n$$\n\\sqrt {\\frac {x ^ {2}}{x}} = 8. 9 \\mu m \\sqrt {\\frac {t}{1 \\min} / (\\frac {a}{1 \\mu m})}\n$$\n\nFor gas, V ~   $\\overline{P}$  < average momentum   $V = \\frac{\\sqrt{2mk_{B}T}}{30}$\n\nσ ← scattering cross-section\n\n$$\n\\langle x ^ {2} \\rangle \\approx \\frac {k _ {B} T t}{a \\overline {{p}} / \\sigma} = \\frac {\\overline {{\\sigma}} k _ {B} T}{a \\overline {{p}}} t\n$$\n\n$$\n= \\frac {\\sigma \\overline {{u}}}{a} t = \\sigma \\frac {\\overline {{u}} t}{a}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 22, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 23"], "page_start": 23, "page_end": 23, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0024", "text": "Langevin equation - Dynamics in random walk\n\n$$\n\\frac {d V}{d t} = \\frac {F _ {e x}}{m} + \\frac {F ^ {\\prime}}{m}\n$$\n\nI collision force\n\nbut   $\\frac{d<v}{dt}=\\frac{E}{m}$  if   $<F'>=0$\n\nno friction   $\\Rightarrow$$<F'>=-\\gamma v$\n\n$$\n\\frac {d \\sigma}{d t} + \\gamma \\sigma = \\frac {F _ {s x}}{m} + \\frac {\\tilde {F}}{m} <   \\tilde {F} > = 0\n$$\n\nsteady state <0> =   $\\frac{1}{m\\gamma}$$\\overline{F_{2n}}$\n\ndescribe the true motion of\n\n$\\frac{1}{m_{2}}=\\eta$ mobility\n\nBrownian pmtile\n\n$<x^{2}>$\n\n$$\nx \\frac {d ^ {2} x}{d t ^ {2}} + \\gamma x \\frac {d x}{d t} = \\frac {\\tilde {F}}{m} x \\quad (\\text {   Let   } F _ {e x} = 0)\n$$\n\n$$\n<   x \\frac {d ^ {2} x}{d t ^ {2}} + \\delta x \\frac {d x}{d t} > = 0 <   \\hat {F} x > = 0\n$$\n\n$$\nI \\frac {d}{d t} \\times \\frac {d x}{d t} - \\left(\\frac {d x}{d t}\\right) ^ {2} = \\frac {1}{2} \\frac {d ^ {2}}{d t ^ {2}} x ^ {2} - (\\frac {d x}{d t}) ^ {2}\n$$\n\n$$\n\\frac {1}{2} \\frac {d ^ {2}}{d t ^ {2}} <   x ^ {2} > + \\frac {1}{2} \\gamma \\frac {d <   x ^ {3}}{d t} - \\left<   (\\frac {d x}{d t}) ^ {2} \\right> = 0\n$$\n\n$$\n\\text { Let } y = \\langle x ^ {2} \\rangle \\quad \\text { 个 } \\frac {k _ {B} T / 2}{m / 2}\n$$\n\n$$\n\\therefore \\frac {d ^ {2}}{d t ^ {2}} y = - \\gamma \\frac {d Y}{d t} + 2 \\frac {k _ {B} T}{m}\n$$\n\n$$\nI _ {\\mathrm{max}} \\neq 1 \\quad E ^ {\\prime} f i c t w i n ^ {\\prime \\prime} L ^ {\\prime} F o r e e ^ {\\prime \\prime}\n$$\n\n$$\nt \\rightarrow \\infty \\quad y \\frac {d y}{d t} = 2 \\frac {k _ {B} T}{m}\n$$\n\n$$\nY = \\angle x ^ {2} > = \\frac {2 k _ {B} T}{m y} t = 2 D t\n$$\n\n$$\nD = \\frac {k _ {B} T}{m \\gamma} = k _ {B} T \\eta\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 23, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 24"], "page_start": 24, "page_end": 24, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0025", "text": "In general\n\n$$\n<   x ^ {2} > = A + B e ^ {- \\gamma t} + \\frac {2 k _ {B} T}{m \\gamma} t\n$$\n\n$$\na + t = 0 <   x > = 0 <   x ^ {2} > = 8 t ^ {2}\n$$\n\n$$\n\\frac {d \\langle x ^ {2} \\rangle}{d t} = \\langle x ^ {2} \\rangle = 0 \\vert_ {t = 0}\n$$\n\n$$\nA + B = 0 \\quad A - \\gamma B + \\frac {2 k _ {B} T}{m \\gamma} = 0\n$$\n\n$$\n<   x ^ {2} > = \\frac {2 k _ {B} T}{m g} [ t - r ^ {- 1} (1 - e ^ {- r t}) ]\n$$\n\n$$\n= 2 k _ {B} T \\eta \\left[ t - m \\eta (1 - e ^ {- t / m \\eta}) \\right]\n$$\n\n$$\nt = \\infty : \\quad <   x ^ {2} > = 2 k _ {B} T y t - m \\eta\n$$\n\n$$\nt \\Rightarrow 0 \\quad , \\quad <   x ^ {2} > = 2 k _ {B} T y \\cdot \\frac {1}{2} \\frac {1}{(m y) ^ {a}} t ^ {2}\n$$\n\n$$\n= \\frac {k _ {B} I}{m} t ^ {2} \\quad \\frac {1}{2} m v ^ {2} = \\frac {1}{2} k _ {B} T\n$$\n\n$$\n<   v ^ {2} >:\n$$\n\n$$\nv \\frac {d v}{d t} + x v ^ {2} = \\frac {v E ^ {2}}{m}\n$$\n\n$$\n\\frac {1}{2} \\frac {d}{d t} <   v ^ {2} > + 8 (v ^ {2}) = \\frac {<   v F >}{m} \\quad <   v F > \\neq 0\n$$\n\n$$\n\\text { from } \\quad \\frac {d v}{d t} + \\gamma v = \\frac {\\tilde {F}}{m} \\Rightarrow \\sqrt {v = v _ {0} e ^ {- \\gamma t} + \\int_ {- \\infty} ^ {t} e ^ {- \\gamma (t - t ^ {\\prime})} \\frac {\\tilde {F} (t ^ {\\prime})}{m} d t ^ {\\prime}}\n$$\n\n$$\n\\frac {d v}{d t} = - \\gamma v _ {0} e ^ {- \\gamma t} + \\frac {\\widehat {E} (t)}{m} + \\int_ {- \\infty} ^ {t} - \\gamma e ^ {- \\gamma (t - t ^ {\\prime})} \\frac {\\widehat {F} (t + 1)}{m} d t ^ {\\prime}\n$$\n\n$$\n<   \\sigma \\widetilde {F} > = \\frac {1}{m} \\int_ {- \\infty} ^ {t} <   \\widetilde {F} (t) \\widetilde {F} (t ^ {\\prime}) > e ^ {- \\gamma (t - t ^ {\\prime})} d t ^ {\\prime}\n$$\n\n$$\n= \\frac {1}{m} \\int_ {0} ^ {\\infty} K (s) e ^ {- 8 s} d s\n$$\n\n$$\nK (s) = \\langle F (t + s) F (t) \\rangle\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 24, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 25"], "page_start": 25, "page_end": 25, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0026", "text": "$$\n\\frac {1}{2} \\frac {d}{d t} <   v ^ {2} > + \\gamma <   v ^ {2} > = \\frac {i}{m ^ {2}} K _ {\\gamma}\n$$\n\n$$\nK _ {r} = \\int_ {0} ^ {\\infty} K (s) e ^ {- r s} d s\n$$\n\n$$\n<   v ^ {2} > = \\cdot A e ^ {- 2 \\gamma t} + \\frac {1}{m ^ {2} \\gamma} k _ {\\gamma}\n$$\n\n$$\na t \\quad t \\rightarrow \\infty\n$$\n\n$$\n<   U ^ {2} > = \\frac {1}{m ^ {2} y} K y\n$$\n\n$$\n= \\frac {k _ {1 3} T}{m}\n$$\n\n$$\n\\text { random   force. }\n$$\n\n$$\n\\text { generate   flux   Tan   Fions }\n$$\n\n$$\n4 v\n$$\n\n$$\n\\Rightarrow m \\gamma = \\frac {1}{k _ {B} T} K _ {\\gamma}\n$$\n\n$$\n\\begin{array}{c} \\text { Correlation   of   random } \\\\ \\text { force   determine   friction } \\end{array}\n$$\n\n$$\n\\frac {1}{\\eta} = \\frac {1}{k _ {B} T} \\int_ {0} ^ {\\infty} <   F (s) F (0) > e ^ {- t / \\eta m} d s\n$$\n\n$$\n\\text {   个   friction   } \\cos f = \\frac {1}{4} v\n$$\n\n$$\n\\begin{array}{c} \\text {   Friction   f   a   dish   in   gas   } \\\\ F (t) \\quad \\begin{array}{c} \\text {   F } _ {0} \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {   } \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.}. \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {.} \\\\ \\text {..}\\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\text {.}. \\\\ \\end{array}\n$$\n\na²\n\n$$\n\\langle F (s) F (0) \\rangle = 0\n$$\n\n$$\n\\text { if } \\quad s > t _ {0}\n$$\n\n$$\n<   F _ {(0)} F _ {(0)} > = F _ {0} ^ {2} \\frac {t _ {0}}{t ^ {*}}\n$$\n\n$$\n\\text { assume } t _ {0} \\ll m y = \\tau \\text { relaxation   time }\n$$\n\nK(s)\nF₀²(t₀/τ*)\nt₀\ns\n\n$$\n\\int_ {0} ^ {\\infty} K (s) e ^ {- t / \\eta m} d s = \\frac {F _ {0} ^ {2} t _ {0} ^ {2}}{2 \\tau^ {*}} = \\frac {(2 m v _ {x}) ^ {2}}{2 \\tau^ {*}} = \\frac {2 m k _ {B} T}{\\tau^ {*}}\n$$\n\n$$\n\\frac {1}{y} = \\frac {2 m}{\\tau^ {+}} \\div 2 m n a ^ {2} \\overline {{v}} \\bigg | _ {\\mathrm{validif} a <   \\lambda} \\frac {\\tau^ {*} v}{1 / n a ^ {2} \\overline {{v}}}\n$$\n\n$$\n\\begin{array}{r l} {\\mathrm{h} _ {u} \\mathrm{d} i o -} & {: \\frac {1}{\\eta} = 6 \\pi v. a = \\pi n m \\overline {{v}} a \\lambda | _ {i f a > > \\lambda} a ^ {2} \\leftrightarrow a \\lambda} \\\\ {\\mathrm{dynamics}} & {\\quad I _ {\\frac {1}{6} n \\overline {{v}} m \\lambda} \\quad \\mathrm{if} a > > \\lambda} \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 25, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 26"], "page_start": 26, "page_end": 26, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0027", "text": "Power P8 amplitude A\n\nhanmonics\n\n$P \\propto  {A}^{2}$\n\n$$\n\\mathrm {L i g h t} \\quad P \\propto \\frac {\\vec {E} ^ {2}}{V (t)} \\quad P = \\frac {V ^ {2}}{R}\n$$\n\nTotal energy:\n\n$$\n\\begin{array}{l} E = \\int_ {d t} ^ {t _ {\\infty}} A _ {(t)} ^ {2} \\quad A (\\omega) = \\int d t A (H) e ^ {i \\omega t} \\\\ A (t) = \\int \\frac {d w}{2 \\pi} A (w) e ^ {- i w t} \\\\ = \\int \\frac {d w}{2 \\pi} A (w) A (- w) \\\\ = \\int_ {- \\infty} ^ {+ \\infty} \\frac {d w}{2 \\pi} | A (w) | ^ {2} = \\int_ {0} ^ {w} \\frac {d w}{\\pi} | A (w) | ^ {2} \\\\ \\end{array}\n$$\n\nTotal energy in   $[w, w+d\\omega]$$\\frac{d\\omega}{2h}|A+(w)|^{2}$\n\nPower in [w, w+dw] =   $\\frac{|A_{1}(w)|^{2}}{\\pi t\\omega}$  dw\n\n$$\n\\frac {\\left| A (\\omega) \\right| ^ {2}}{\\pi t _ {0}} = P o w e r s p e c t r u m\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 26, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 27"], "page_start": 27, "page_end": 27, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0028", "text": "$$\n\\text {   How   to   calculate   } \\frac {| A (w) | ^ {2}}{\\pi t _ {\\infty}}?\n$$\n\n(2)\n\n$$\nⓘ A (\\omega) = \\int d t A (t) e ^ {i \\omega t} \\quad b u t A (t) r a n d e m\n$$\n\n$$\nⓏ <   \\frac {| A (w) | ^ {2}}{\\pi t _ {\\infty}} > = \\frac {1}{\\pi t _ {\\infty}} <   \\int d t A (t) e ^ {i w t} \\int d t ^ {\\prime} A (t ^ {\\prime}) e ^ {- i w t ^ {\\prime}} >\n$$\n\n$$\n= \\frac {1}{\\pi t _ {\\infty}} \\int d t d t ^ {\\prime} \\underbrace {<   A (t) A (t ^ {\\prime}) > e ^ {i w (t - t ^ {\\prime})}} _ {\\equiv G (t - t ^ {\\prime})}\n$$\n\n$$\n= \\frac {1}{\\pi t _ {\\infty}} \\int d t d \\tilde {t} G (\\tilde {t}) e ^ {i \\omega \\tilde {t}}\n$$\n\n$$\n= \\frac {1}{\\pi} \\int_ {- \\infty} ^ {\\infty} d \\tilde {t} G (\\tilde {t}) e ^ {i \\omega \\tilde {t}} = \\frac {S (\\omega)}{\\pi}\n$$\n\n$$\n\\text { Power   spectrum } = \\langle \\frac {| A (\\omega) | ^ {2}}{\\pi t _ {\\omega}} \\rangle = \\frac {1}{\\pi} \\int d t G (t) e ^ {i \\omega t} = \\frac {S (\\omega)}{\\pi}\n$$\n\n$$\nG (t) = <   A (t) A (0) > \\dots \\text { correlation   function }\n$$\n\n$$\nS (\\omega) = \\int d t G (t) e ^ {i \\omega t}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 27, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 28"], "page_start": 28, "page_end": 28, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0029", "text": "Assume  $G(t) = c e^{-\\gamma |t|}$\n\n$$\nS (\\omega) = \\frac {c \\gamma / \\pi}{\\omega^ {2} + \\gamma^ {2}}\n$$\n\nA(t)\nV/γ\ncon\n\nG(t)\n1/8\nt\n\n| w | Gw |\n| --- | --- |\n| 0 | 0 |\n| Peak | High |\n| Low | 0 |\n| High | Decreasing |\n\nWhite noise\n\n$$\nS _ {(w)} = \\text { const. }\n$$\n\n1y → 0\n\nA(t)\nt\n\n$A(t)$ and  $A(f + \\Delta t)$ have no correlation.", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 28, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 29"], "page_start": 29, "page_end": 29, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0030", "text": "123\n\nShot noise\n\nΔt\nt\n\n$$\nI _ {i} = \\frac {e}{\\Delta t}, s _ {i}\n$$\n\n$$\nP (s: \\dots) = p\n$$\n\n$$\nP (S; = 0) = (1 - p)\n$$\n\n$$\n<   I _ {i} I _ {j} > = \\left\\{\\left(\\frac {P e}{\\Delta t}\\right) ^ {2} - i b i + j \\right.\n$$\n\n$$\n(\\frac {e}{\\Delta t}) ^ {2} p \\quad i f \\quad i = j\n$$\n\n$$\n= \\left\\{ \\begin{array}{l l} I ^ {2} & i f i \\pm j \\\\ \\frac {e}{d t} I & i f i = j \\end{array} \\right.\n$$\n\n$$\n\\text { Note: } \\pi h _ {\\mathrm{out}} \\frac {2}{i} (<   I; \\overline {{I}}; 2 - I ^ {2}) = \\frac {e}{s t} I\n$$\n\n$$\n\\Rightarrow \\int d t <   I (t) I (0) > = e I\n$$\n\n$$\n<   I (t) J (0) > = I ^ {2} + e I S (t) \\equiv G (t)\n$$\n\n$$\n\\text { Power   spectrum } = \\left\\langle \\frac {| 3 (\\omega) | ^ {2}}{\\pi + \\infty} \\right\\rangle\n$$\n\n$$\n= \\frac {S (w)}{T _ {1}}\n$$\n\n$$\nS (w) = \\int d t \\xi (t) e ^ {i w t} = I ^ {2} (2 \\pi) \\delta (w) + e I\n$$\n\n$$\n\\text { Power   spectrum } = 2 I ^ {2} \\int (w) + \\frac {e J}{\\pi} \\quad \\begin{array}{l} \\text { ind.   of   } w \\\\ \\text { depend   in   } e! \\end{array}\n$$\n\nSho# noise", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 29, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 30"], "page_start": 30, "page_end": 30, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0031", "text": "124\n\nNoise spectrum in Brownian motion\n\nLangevin equation:\n\n$$\n\\frac {d \\sigma}{d t} + \\gamma \\sigma = \\frac {F (t)}{m} <   F (t) > = 0\n$$\n\nIn w-space\n\n$$\n- i v _ {w} + \\gamma v _ {w} = \\frac {F _ {w}}{m}\n$$\n\n$$\nv _ {w} = \\frac {F _ {w}}{m (y - i w)}\n$$\n\nNow consider\n\n$$\n<   v _ {w} v _ {- w} > = \\int d t _ {1} d t _ {2} <   v (t _ {1}) v (t _ {2}) > e ^ {i w (t _ {1} - t _ {2})}\n$$\n\n$$\n= + \\infty \\int d \\tau <   v (\\tau) v (0) > e ^ {- i w \\tau}\n$$\n\nSimilarly\n\n$$\n\\langle F _ {w} F _ {- w} \\rangle = + \\infty \\int d \\tau \\langle F (z) F (0) \\rangle e ^ {i w z}\n$$\n\nNoise spectrum\n\n$$\n\\begin{array}{l} \\frac {S _ {(w)}}{\\pi} = \\frac {1}{\\pi} \\frac {\\langle \\theta_ {w} \\theta_ {- w} \\rangle}{t _ {w}} \\\\ = \\frac {1}{\\pi} \\frac {1}{m ^ {2} (y + w ^ {2})} \\frac {\\langle F _ {w} F _ {- w} \\rangle}{t _ {w}} \\\\ \\end{array}\n$$\n\n$$\n\\frac {S (w)}{\\pi} = \\frac {1}{\\pi} \\frac {1}{m ^ {2} (\\gamma^ {2} + w ^ {2})} \\int d \\tau <   F (t) F (0) > e ^ {- i w t}\n$$\n\nLet  $K_{1w} = \\int d\\tau < F(\\tau) F(0) > e^{\\tau w\\tau}$\n\n$$\n\\frac {S (\\omega)}{\\pi} = \\frac {1}{\\pi} \\frac {K (\\omega)}{m ^ {2} (\\gamma^ {2} + \\omega^ {2})}\n$$", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 30, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 31"], "page_start": 31, "page_end": 31, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0032", "text": "Let  $< F(t_{1})F(t_{2})> = 2\\pi K \\delta (t_{1} - t_{2})$\n\n$$\nK (w) = 2 \\pi K\n$$\n\n$$\n\\frac {S (w)}{\\pi} = \\frac {2 K}{m ^ {2} (\\gamma^ {2} + w ^ {2})}\n$$\n\nsee page 119\n\n$$\nm y = \\frac {1}{k _ {B} T} \\quad K _ {y}\n$$\n\n$$\n= \\frac {1}{k _ {B} T} \\pi K\n$$\n\n$$\nK _ {r} = \\int_ {0} ^ {\\infty} d \\tau <   F (\\tau) F (0) > e ^ {- r \\tau}\n$$\n\n$$\n= \\pi K _ {0}\n$$\n\n$$\nP _ {(w)} ^ {v} = \\frac {S _ {(w)}}{\\pi} = \\frac {2 k _ {B} T m \\gamma}{\\pi m ^ {2} (\\gamma^ {2} + w ^ {2})} = \\frac {2 k _ {B} T . \\gamma}{\\pi m (\\gamma^ {2} + w ^ {2})}\n$$\n\n$$\n\\langle U _ {(t)} ^ {2} \\rangle = \\int_ {- \\infty^ {2} \\pi} ^ {\\infty} \\frac {\\langle W _ {w} v - w \\rangle}{+ \\infty} = \\int_ {- \\infty} ^ {\\infty} d w \\quad \\frac {k _ {B} T \\gamma}{\\pi m (\\gamma^ {2} + w ^ {2})} = \\frac {k _ {B} T}{m}\n$$\n\nPower spectrum of velocity in determined\n\nby friction  $r$ and mass  $m$ (and temperature  $T$)", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 31, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 32"], "page_start": 32, "page_end": 32, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0010-ix-transport-bd5d1295-1c4d88dc:page-0033", "text": "$$\n\\text { Noise } \\quad \\text { in } \\quad \\text { RCL } \\quad \\text { circuit }\n$$\n\n$$\nV = \\frac {q}{c} = L \\frac {d I}{d t}\n$$\n\n$$\nI = - \\frac {d q}{d t} - \\frac {V}{R}\n$$\n\n$$\n\\mathrm{Let} \\quad x = L I\n$$\n\n$$\n\\frac {1}{c} \\frac {d ^ {q}}{d t} = \\frac {d ^ {2} x}{d t ^ {2}} = \\frac {1}{c} (- I - \\frac {V}{R})\n$$\n\n$$\n\\Rightarrow C \\frac {d ^ {2} x}{d t ^ {2}} = - \\frac {x}{L} - \\frac {1}{R} \\frac {d x}{d t}\n$$\n\n$$\n\\text { total   energy   of   the   system }\n$$\n\n$$\nE = \\frac {1}{2} L I ^ {2} + \\frac {1}{2} \\in V ^ {2}\n$$\n\n$$\n= \\frac {1}{2} \\frac {1}{L} x ^ {2} + \\frac {1}{2} C \\dot {x} ^ {2}\n$$\n\n$$\nR C L \\quad c i n c u i t = p a n T i c l e \\quad o f \\quad m a s s \\quad C (= m) _ {i n} \\frac {1}{2 L} x ^ {2} \\quad p o t o n t i o n\n$$\n\n$$\n\\text {   Friction   coefficient   } \\quad \\gamma_ {m} = \\frac {1}{R} \\quad \\text { or } \\quad \\boxed {\\gamma = \\frac {1}{R C}}\n$$\n\n$$\nL e + L = \\infty : \\boxed {R}\n$$\n\n$$\n\\text { power   spectrum   of } V (= \\dot {x})\n$$\n\n$$\nP _ {(w)} ^ {v} = \\frac {2 k _ {B} T \\frac {1}{R C}}{\\pi C [ (\\frac {1}{R C}) ^ {2} + w ^ {2} ]} = \\frac {2 k _ {B} T R}{\\pi (1 + w ^ {2} R ^ {2} c ^ {2})}\n$$\n\n$$\nP ^ {V} (f) = 2 \\pi P ^ {V} (\\omega) = 4 k _ {B} T R / (1 + \\omega^ {2} R ^ {2} (2) = 4 k _ {B} T R \\vert_ {\\epsilon = 0}\n$$\n\nV\nL → I\n8\nC\nR", "source": "mit-ocw", "source_doc_id": "0010-ix-transport-bd5d1295-1c4d88dc", "source_title": "Transport:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 32, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Transport:", "Page 33"], "page_start": 33, "page_end": 33, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0001", "text": "# Chapter 4\n\n# Boson systems\n\nA boson system is the simplest system in nature. It demonstrates a wide variety of physical phenomena, such as superfluidity, magnetism, crystals, etc. We can also use boson system to study many different phase transitions. Despite (or due to) its simplicity, a boson system may also be the deep fundamental structure that produces all the elementary particles including photons and electrons [Wen 2003b]. If this is true, a boson system will actually be a theory of everything.\n\nIn this chapter, we will study interacting bosons using a classical picture. We first develop a classical field theory that describes the bosons. Then we consider the collective vibration modes of the field. After quantizing those vibration modes, we gain a understanding of the low energy properties of the quantum interacting bosons.\n\n# 4.1 A first look at a free boson system\n\n-  $n$-boson Hamiltonian and  $n$-boson energy eigenstates for a free boson system.\n\nThere are two kinds of particles in nature, bosons and fermions. Photons and Hydrogen molecules are two examples of bosons. Photons hardly interact with each other. So the photon system is a non-interacting boson system, or a free boson system. Hydrogen molecules have a short range interaction. For a dilute Hydrogen gas there is little chance for two molecules to be close to each other. Thus the interaction between the Hydrogen molecules can also be ignored and we can treat the Hydrogen gas as a system of free bosons. In this section, we will study such free boson systems. To simplify our discussion even further, we will consider free bosons in one dimension. The generalization to higher dimension is often straight forward.\n\nTo construct a quantum theory for many bosons, let us start with the simplest case: the state with no particle. Such a state is called a vacuum state and is denoted by   $|0\\rangle$ . The energy of such a state is zero.\n\nThe next simplest state is a state with one particle. Actually there are many different one-particle states. Those states form a Hilbert space  $\\mathcal{H}_1$. One set of bases vectors for  $\\mathcal{H}_1$ is  $|x\\rangle$ which describe a particle at  $x$.  $|x\\rangle$'s are normalized according to\n\n$$\n\\langle x | x ^ {\\prime} \\rangle = \\delta (x - x ^ {\\prime}).\n$$\n\nA generic one-particle state   $|\\psi\\rangle$  is described by a complex wave function   $\\psi(x)$ :\n\n$$\n| \\psi \\rangle = \\int \\mathrm{d} x \\psi (x) | x \\rangle\n$$", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0002", "text": "Let us assume that the particle is relativistic and is described by the one-particle Hamiltonian\n\n$$\n\\hat {H} _ {1} = \\sqrt {- c ^ {2} \\partial_ {x} ^ {2} + m ^ {2} c ^ {4}} \\tag {4.1.1}\n$$\n\nwhere  $m$ is the mass of the particle and  $c$ the speed of light. The energy eigenstates of such a Hamiltonian are plane waves\n\n$$\n| k \\rangle = \\int \\mathrm{d} x \\mathrm{e} ^ {\\mathrm{i} k x} | x \\rangle\n$$\n\nwith energy   $E_{k} = \\sqrt{c^{2}k^{2} + m^{2}c^{4}}$ . Certainly the statistics is not important here. The one-particle states for a boson or a fermion are identical.\n\nFor a particle in three dimensions,   $\\hat{H}_{1}$  becomes\n\n$$\n\\hat {H} _ {1} = \\sqrt {c ^ {2} (- \\partial_ {x} ^ {2} - \\partial_ {y} ^ {2} - \\partial_ {z} ^ {2}) + m ^ {2} c ^ {4}}\n$$\n\nThe wave vector will have three components   $\\boldsymbol{k} = (k_{x}, k_{y}, k_{z})$  and the energy of a 3D plain wave state   $|k\\rangle$  will be   $E_{\\boldsymbol{k}} = \\sqrt{c^{2}\\boldsymbol{k}^{2} + m^{2}c^{4}}$ . If we take m to be the mass of the Hydrogen molecule,   $E_{k}$  will be the energy of single Hydrogen molecule. For a single massless photon, the energy can be obtained by taking m = 0 and is given by   $E_{\\boldsymbol{k}} = c|\\boldsymbol{k}|$ .\n\nThe two-particle states form a bigger Hilbert space  $\\mathcal{H}_2$. One set of bases vectors for  $\\mathcal{H}_2$ is  $|x_1x_2\\rangle$ with a understanding that  $|x_1x_2\\rangle$ and  $|x_2x_1\\rangle$ are the two names for the same physical state. So we have\n\n$$\n\\left| x _ {1} x _ {2} \\right\\rangle = \\left| x _ {2} x _ {1} \\right\\rangle \\tag {4.1.2}\n$$\n\nThe equivalence of   $\\left|x_{1}x_{2}\\right\\rangle$  and   $\\left|x_{2}x_{1}\\right\\rangle$  is very important. It means that there is only one state with one particle at   $x_{1}$  and one particle at   $x_{2}$. If   $\\left|x_{1}x_{2}\\right\\rangle$  and   $\\left|x_{2}x_{1}\\right\\rangle$  describe two different quantum states, then there are two different states with one particle at   $x_{1}$  and one particle at   $x_{2}$. In this case the system will be a system of non-identical particles. The condition that there is only a single state with one particle at   $x_{1}$  and one particle at   $x_{2}$  makes the particles in our system identical particles.\n\nA generic two-particle state is given by\n\n$$\n\\left| \\psi_ {\\text {two - particles}} \\right\\rangle = \\int_ {x _ {1} \\leqslant x _ {2}} \\mathrm{d} x _ {1} \\mathrm{d} x _ {2} \\psi (x _ {1}, x _ {2}) \\left| x _ {1} x _ {2} \\right\\rangle \\tag {4.1.3}\n$$\n\nNote that the integration is only over the region  $x_{1} \\leqslant x_{2}$ to avoid double counting, since  $|x_{1}x_{2}\\rangle$ and  $|x_{2}x_{1}\\rangle$ represent the same state. So the two-particle wave function  $\\psi(x_{1}, x_{2})$ is only defined for  $x_{1} \\leqslant x_{2}$.\n\nUsing eqn (4.1.2), we can extend the wave function  $\\psi(x_1, x_2)$ to the region with  $x_1 > x_2$ through the relation\n\n$$\n\\psi (x _ {1}, x _ {2}) = \\psi (x _ {2}, x _ {1})\n$$\n\nThis allows us to rewrite eqn (4.1.3) as\n\n$$\n| \\psi_ {\\mathrm{two-particles}} \\rangle = \\frac {1}{2} \\int \\mathrm{d} x _ {1} \\mathrm{d} x _ {2} \\psi_ {x _ {1}, x _ {2}} | x _ {1} x _ {2} \\rangle\n$$\n\nwhere the integration is over the whole 2D plane  $(x_{1}, x_{2})$. We see that the states of two identical particles can be described by symmetric wave functions  $\\psi(x_{1}, x_{2}) = \\psi(x_{2}, x_{1})$.\n\nA careful reader may note that so far we only specified that the two particles are identical particles. We did not specify if the two particles are bosons or fermions. So the above reasoning implies that both bosonic and fermionic identical particles are described by symmetric wave functions.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0003", "text": "But what determines the statistics of the identical particles? It turns out that the statistics is not determined by the symmetry or antisymmetry property of the wave function, but by the Hamiltonian that governs the dynamics of the two particles.\n\nIf we choose the Hamiltonian that acts on the two-particle state   $\\left|\\psi_{two-particles}\\right\\rangle$  to be the sum of two one-particle Hamiltonian (4.1.1)\n\n$$\n\\hat {H} _ {2} = \\sqrt {- c ^ {2} \\partial_ {x _ {1}} ^ {2} + m ^ {2} c ^ {4}} + \\sqrt {- c ^ {2} \\partial_ {x _ {2}} ^ {2} + m ^ {2} c ^ {4}} \\tag {4.1.4}\n$$\n\nthen the two identical particles will be bosons. Further more such a Hamiltonian also implies that there is no interaction between the two particles. So   $\\hat{H}_{2}$  describes our free 1D boson system with two bosons.\n\nWe note that   $\\hat{H}_{2}$  is invariant under the exchange   $x_{1} \\leftrightarrow x_{2}$ . So when it acts on a symmetric wave function   $\\psi(x_{1}, x_{2})$ ,   $\\hat{H}_{2}$  will generate another symmetric wave function. Since the identical particles (bosons or fermions) are always described by symmetric wave functions, the two-particle Hamiltonian for identical particles are always invariant under the exchange, so that the action of the Hamiltonian on the allowed wave functions can only generate allowed wave functions.\n\nThe energy eigenstates of  $\\hat{H}_2$ are plain waves  $\\psi(x_1, x_2) = \\mathrm{e}^{\\mathrm{i}(k_1x_1 + k_2x_2)} + \\mathrm{e}^{\\mathrm{i}(k_1x_2 + k_2x_1)}$ (which is symmetric under the exchange of  $x_1$ and  $x_2$) or\n\n$$\n\\left| k _ {1} k _ {2} \\right\\rangle = \\int_ {x _ {1} \\leqslant x _ {2}} \\mathrm{d} x _ {1} \\mathrm{d} x _ {2} \\left(\\mathrm{e} ^ {\\mathrm{i} (k _ {1} x _ {1} + k _ {2} x _ {2})} + \\mathrm{e} ^ {\\mathrm{i} (k _ {1} x _ {2} + k _ {2} x _ {1})}\\right) \\left| x _ {1} x _ {2} \\right\\rangle\n$$\n\nWe note that  $|k_1k_2\\rangle = |k_2k_1\\rangle$. So  $|k_1k_2\\rangle$'s are also redundant names:  $|k_1k_2\\rangle$ and  $|k_2k_1\\rangle$ are two names for the same plain wave state. The energy of the plain wave state is  $E_{k_1k_2} = \\epsilon_{k_1} + \\epsilon_{k_2}$ where\n\n$$\n\\epsilon_ {k} = \\sqrt {c ^ {2} k ^ {2} + m ^ {2} c ^ {4}}. \\tag {4.1.5}\n$$\n\nThe above discussion can be easily generalized to  $n$-particles. The  $n$-particle Hamiltonian have a form\n\n$$\n\\hat {H} _ {n} = \\sum_ {i = 1} ^ {n} \\sqrt {- c ^ {2} \\partial_ {x _ {i}} ^ {2} + m ^ {2} c ^ {4}} \\tag {4.1.6}\n$$\n\nSuch a Hamiltonian determines the statistics of the particles to be bosonic. The energy eigenstates are   $|k_{1}k_{2}\\cdots k_{n}\\rangle$  with energy   $\\sum_{i=1}^{n}\\epsilon_{k_{i}}$ . The different orders of   $k_{1}k_{2}\\cdots k_{n}$  in   $|k_{1}k_{2}\\cdots k_{n}\\rangle$  correspond to the same state, for example   $|k_{1}k_{2}k_{3}\\rangle = |k_{2}k_{1}k_{3}\\rangle = |k_{3}k_{1}k_{2}\\rangle$ .\n\n# 4.2 **A brief look at Fermi statistics\n\n- Identical particles can always be described by symmetric wave functions. The statistics of the identical particles is determined by  $n$-particle Hamiltonians. This provides a unified way to understand Bose, Fermi, and fractional statistics.\n\nWe have stressed that both bosons and fermions can be described by symmetric wave functions. The statistics of the identical particles are determined by the many-particle Hamiltonian. The particular two-particle Hamiltonian (4.1.4) gives rise to Bose statistics. A curious reader may wonder what kind of two-particle Hamiltonian gives rise to Fermi statistics. As an example, let me just write a two-particle Hamiltonian that gives rise to Fermi statistics in 2D (in non-relativistic", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0004", "text": "(x,y)\nΘ\ny\nx\n\nlimit):\n\n$$\n\\begin{array}{l} \\hat {H} _ {2} ^ {\\mathrm{ferm}} = - \\frac {(\\partial_ {x _ {1}} + \\mathrm{i} a _ {x}) ^ {2} + (\\partial_ {y _ {1}} + \\mathrm{i} a _ {y}) ^ {2}}{2 m} - \\frac {(\\partial_ {x _ {2}} - \\mathrm{i} a _ {x}) ^ {2} + (\\partial_ {y _ {2}} - \\mathrm{i} a _ {y}) ^ {2}}{2 m} \\\\ a _ {x} = \\frac {y _ {1} - y _ {2}}{(x _ {1} - x _ {2}) ^ {2} + (y _ {1} - y _ {2}) ^ {2}}, \\quad a _ {y} = - \\frac {x _ {1} - x _ {2}}{(x _ {1} - x _ {2}) ^ {2} + (y _ {1} - y _ {2}) ^ {2}}, \\tag {4.2.1} \\\\ \\end{array}\n$$\n\nWe note that   $\\hat{H}_{2}^{ferm}$  is still invariant under the exchange   $x_{1} \\leftrightarrow x_{2}$ . Such a two-particle Hamiltonian when acting on symmetric wave functions   $\\psi(x_{1}, y_{1}, x_{2}, y_{2}) = \\psi(x_{2}, y_{2}, x_{1}, y_{1})$  describes two fermions in two dimensions.\n\n$\\hat{H}_2^{\\mathrm{ferm}}$ can be simplified by the following transformation\n\n$$\n\\psi (x _ {1}, y _ {1}, x _ {2}, y _ {2}) = \\mathrm{e} ^ {\\mathrm{i} \\Theta (x _ {1} - x _ {2}, y _ {1} - y _ {2})} \\tilde {\\psi} (x _ {1}, y _ {1}, x _ {2}, y _ {2})\n$$\n\n$$\n\\hat {H} _ {2} ^ {\\text { ferm }} = \\mathrm{e} ^ {\\mathrm{i} \\Theta (x _ {1} - x _ {2}, y _ {1} - y _ {2})} \\tilde {\\hat {H}} _ {2} ^ {\\text { ferm }} \\mathrm{e} ^ {- \\mathrm{i} \\Theta (x _ {1} - x _ {2}, y _ {1} - y _ {2})} \\tag {4.2.2}\n$$\n\nwhere   $\\Theta(x,y)$  is the angle between the vector   $(x,y)$  and the positive x direction (see Fig. 4.1). For positive x and y,   $\\Theta(x,y)=\\arctan\\left(\\frac{y}{x}\\right)$ . Although   $\\Theta(x,y)$  is discontinuous on the positive x-axis with a discontinuity of   $2\\pi$ , the function   $\\mathrm{e}^{\\mathrm{i}\\Theta(x,y)}$  is a smooth function of   $(x,y)$  (except at   $(x,y)=(0,0)$ ). Using the relation\n\n$$\n\\mathrm{e} ^ {\\mathrm{i} \\Theta (x, y)} \\partial_ {x} \\mathrm{e} ^ {- \\mathrm{i} \\Theta (x, y)} = \\mathrm{i} \\frac {y}{x ^ {2} + y ^ {2}}, \\qquad \\mathrm{e} ^ {\\mathrm{i} \\Theta (x, y)} \\partial_ {y} \\mathrm{e} ^ {- \\mathrm{i} \\Theta (x, y)} = - \\mathrm{i} \\frac {x}{x ^ {2} + y ^ {2}},\n$$\n\nwe find that the transformed Hamiltonian  $\\hat{H}_2^{\\mathrm{ferm}}$ has a simple form\n\n$$\n\\tilde {\\dot {H}} _ {2} ^ {\\mathrm{ferm}} = - \\frac {1}{2 m} (\\partial_ {x _ {1}} ^ {2} + \\partial_ {y _ {1}} ^ {2}) - \\frac {1}{2 m} (\\partial_ {x _ {2}} ^ {2} + \\partial_ {y _ {2}} ^ {2}).\n$$\n\nFrom  $\\mathrm{e}^{\\mathrm{i}\\Theta (x,y)} = -\\mathrm{e}^{\\mathrm{i}\\Theta (-x, - y)}$, we can show that the transformed wave function is antisymmetric  $\\tilde{\\psi} (x_1,y_1,x_2,y_2) = -\\tilde{\\psi} (x_2,y_2,x_1,y_1)$.\n\nSo the simple two-particle Hamiltonian   $\\hat{H}_{2}^{ferm}$  when acting on antisymmetric wave functions   $\\tilde{\\psi}(x_{1},y_{1},x_{2},y_{2})$  describes two fermions in two dimensions. This way we recover the usual result that fermions are described by antisymmetric wave functions.\n\nIn the standard way to understand fermions, fermions are defined as particles described by antisymmetry wave functions. Through the above discussion, we see that this standard understanding of fermions did not capture the essence of Fermi statistics. This is because fermions can be described by both symmetric wave functions (with a complicated many-particle Hamiltonian) or antisymmetric wave functions (with a simpler many-particle Hamiltonian).\n\nI personally believe that symmetric wave functions plus complicated many-particle Hamiltonian is a correct way to understand fermions, at least physically. The standard understanding using", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0005", "text": "antisymmetric wave function is very formal and misleading despite its mathematical simplicity. The confusion cause by the standard understanding is reflected in the following conversation:\n\nA: I have two fermions. One at   $x_{1}$  and the other at   $x_{2}$ . I wonder what is the amplitude of such a state.\n\nB: Well it depends on how you say it. If you say one fermion at   $x_{1}$  and one fermion at   $x_{2}$ , the amplitude will be   $\\psi$ . If you say one fermion at   $x_{2}$  and one fermion at   $x_{1}$ , the amplitude will be   $-\\psi$ .\n\nA: This is ridiculous. The two ways of saying mean exactly the same thing. How come it leads to two different results.\n\nB: Well, it is not that ridiculous. You know that two wave functions differ by a total phase factor   $e^{i\\theta}$  actually describe the same quantum state. So the amplitudes   $\\psi$  and   $-\\psi$  actually correspond to the same physical state. There is no contradiction.\n\nA: But then why is the minus sign important? Why does the minus sign characterize the Fermi statistics? Saying one particle at   $x_{2}$  and the other at   $x_{1}$  may very well leads to an amplitude   $e^{i\\theta}\\psi$  instead of   $-\\psi$ . The phase   $\\theta$  should have no physical meaning, less to determine the statistics of the identical particles.\n\nB: Well, we should look at the wave function of identical particles   $\\psi(\\boldsymbol{x}_{1},\\boldsymbol{x}_{2},..., \\boldsymbol{x}_{n})$  as a whole. Imposing an arbitrary exchange phase, say   $\\psi(\\boldsymbol{x}_{1},\\boldsymbol{x}_{2},..., \\boldsymbol{x}_{n}) = \\mathrm{e}^{\\mathrm{i}\\theta}\\psi(\\boldsymbol{x}_{2},\\boldsymbol{x}_{1},..., \\boldsymbol{x}_{n})$ , may result in a discontinuous many-particle wave function   $\\psi(\\boldsymbol{x}_{1},\\boldsymbol{x}_{2},..., \\boldsymbol{x}_{n})$ . Only when   $\\theta = 0$  or   $\\pi$  can we have a continuous wave function.\n\nA: But the continuity of the wave function should not be essential. If the identical particles are defined on a lattice, the continuity of the wave function will be meaningless. On the other hand, identical particles on lattice still have well defined statistics (and even fractional statistics in 2D).\n\nI hope that I have made my point. The statistics of identical particles is a very tricky subject. Using exchange symmetry of many-particle wave function to understand statistics is formal and misleading. It misses the essence of statistics. If we understand the statistics that way, the origin of statistics will appear to be very mysterious. Such a understanding does not tell us how to make identical particles with different statistics. It does not encourage us to think how to make identical particles with different statistics. It suggests that the statistics is fundamental and is given. We just have to accept it.\n\nIn contrast, the description of identical particle using symmetric wave function and encoding the statistics in the many-particle Hamiltonian leads to completely different picture. I believe it is a more correct picture that captures more essence of statistics despite its mathematical complexity. Within such a picture, statistics of identical particles is a dynamical property determined by many-particle Hamiltonian. We can change the Hamiltonian to obtain different statistics. We can also naturally obtain fractional statistics in two dimensions. Such an understanding tells us how to make different statistics. We can also have phase transitions that change the statistics of particles. We will have a more detailed discussion of Fermi statistics later.\n\n# Problem 4.2.1\n\nBy rescaling   $a_{x}$  and   $a_{y}$  in eqn (4.2.1), we can obtain the Hamiltonian that describes two particles with fractional statistics. The following Hamiltonian, when acting on symmetric wave functions, describes two such particles confined by a harmonic potential   $\\frac{K}{2}(x^{2} + y^{2})$ :\n\n$$\n\\hat {H} _ {2} ^ {\\mathrm{frac}} = - \\frac {1}{2 m} (\\partial_ {x _ {1}} + i a _ {x}) ^ {2} - \\frac {1}{2 m} (\\partial_ {y _ {1}} + i a _ {y}) ^ {2} - \\frac {1}{2 m} (\\partial_ {x _ {2}} - i a _ {x}) ^ {2} - \\frac {1}{2 m} (\\partial_ {y _ {2}} - i a _ {y}) ^ {2}\n$$\n\n$$\n+ \\frac {K}{2} (x _ {1} ^ {2} + y _ {1} ^ {2} + x _ {2} ^ {2} + y _ {2} ^ {2})\n$$\n\n$$\na _ {x} = \\frac {\\theta}{\\pi} \\frac {y _ {1} - y _ {2}}{(x _ {1} - x _ {2}) ^ {2} + (y _ {1} - y _ {2}) ^ {2}}, \\quad a _ {y} = - \\frac {\\theta}{\\pi} \\frac {x _ {1} - x _ {2}}{(x _ {1} - x _ {2}) ^ {2} + (y _ {1} - y _ {2}) ^ {2}}, \\tag {4.2.3}\n$$\n\nwhere  $\\theta$ is the statistical angle.  $\\theta = 0$ correspond to bosons and  $\\theta = \\pi$ correspond to fermions.\n\n(a) Find the ground state energy of  $\\hat{H}_2^{\\mathrm{frac}}$ for  $\\theta = 0$,  $\\pi / 2$, and  $\\pi$.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0006", "text": "```mermaid\ngraph TD\nA[\"①\"] -->|x| B[\"②\"]\nB -->|y| A\nC[\"②\"] --> D[\"②\"]\nD --> E[\"①\"]\n```\n\n- (b) The ground state energy for two bosons/fermions in the harmonic potential   $\\frac{K}{2}(x^{2}+y^{2})$  can also be obtained by filling energy levels. Check the correctness of your results in (a) against the results obtained by filling energy levels.\n- (c) Can we still use a transformation similar to eqn (4.2.2) to encode the fractional statistics in the exchange symmetry of the wave function?\n\n# Problem 4.2.2\n\n- (a) Find a 3-particle Hamiltonian that acts on symmetric wave functions and describes Fermi statistics. (Hint: You may generalize eqn (4.2.1)).\n- (b) Find a transformation that transform symmetric wave function to antisymmetric wave function. Find the transformed 3-particle Hamiltonian. (Try to choose the 3-particle Hamiltonian in (a) such that the transformed 3-particle Hamiltonian is a sum of three one-particle Hamiltonians.)\n\n# 4.3 A second look at the free boson system\n\n- The bosons states with different numbers of bosons can all be labeled by occupation numbers.\n- A free boson system is equivalent to collection of harmonic oscillators.\n\nIn the section 4.1, we have viewed vacuum as an empty stage. The bosons are actors on the stage. In this picture, the existence and the origin of identical particles are very mysterious. To appreciate this point, let us consider a state with particle 1 at x and particles 2 at y. After exchanging the two particles we get another state with particle 1 at y and particles 2 at x (see Fig. 4.2). If the two states are different, then the two particles are not identical. If the two states before and after the exchange are actually the same state, then the two particles are called identical particles\n\nBut why the two states have to be the same? It appears that we can always follow the trajectories of the particles in time history to distinguish the two states before and after the exchange. This is the source of the mystery of identical particles, if we view particles are somethings placed in an empty vacuum. We wonder where do identical particles come from? Why do they have to exist? The mystery of identical particles is one of most fundamental mystery of our nature. It reflects certain deep structures in physics law and the properties of our vacuum. But what is the message? What does the existence of identical particle tell us about the physics laws and the properties of vacuum? In this section, we will try to provide an answer to those fundamental questions. We will use the 1D free boson system to illustrate our points.\n\nWe assume the bosons live on circle of length  $L$. In this case, the wave vectors  $k$ are quantized:\n\n$$\nk = \\kappa_ {n} \\equiv \\frac {2 \\pi}{L} n\n$$\n\nwhere  $n$ is an integer. The total Hilbert space of arbitrary number of bosons is formed by the no-boson state, one-boson states, two-boson states, etc:\n\n$$\n\\mathcal {H} = \\mathcal {H} _ {0} \\oplus \\mathcal {H} _ {1} \\oplus \\mathcal {H} _ {2} \\dots = \\{| 0 \\rangle , | k _ {1} \\rangle , | k _ {1} k _ {2} \\rangle , \\dots \\}\n$$", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0007", "text": "| k | ε_k |\n|---|---|\n| -4 | ● |\n| -3 | ● |\n| -2 | ● |\n| -1 | ● |\n| 0 | ● |\n| 1 | ● |\n| 2 | ● |\n| 3 | ● |\n| 4 | ● |\n\nInstead of using   $k_{i}$  that describes the momentum of each boson, we can use a set of the occupation numbers   $n_{\\kappa_{n}}$  to label each bosonic state in   $H \\left| \\cdots n_{\\kappa_{-1}} n_{\\kappa_{0}} n_{\\kappa_{1}} n_{\\kappa_{2}} \\cdots \\right\\rangle$ .   $n_{\\kappa_{n}}$  is the number of bosons in the level with momentum   $k = \\kappa_{n}$  (see Fig. 4.3). Such an level will be called k-level. The vacuum state   $|0\\rangle$  is given by the state with all   $n_{\\kappa_{n}} = 0$ :   $|0\\rangle = |\\cdots 0000 \\cdots\\rangle$ . The one-boson state   $|k_{1}\\rangle$  is given by the state with all   $n_{\\kappa_{n}} = 0$ , except   $n_{k_{1}} = 1$ . A more general state is represented in Fig. 4.3. In this way, all the states in H are labeled by   $|\\cdots n_{\\kappa_{-1}} n_{\\kappa_{0}} n_{\\kappa_{1}} n_{\\kappa_{2}} \\cdots\\rangle$  with   $n_{\\kappa_{n}} = 0, 1, 2, \\cdots$ . From the relation between the two ways of labeling states,   $|k_{1} k_{2} \\cdots\\rangle$  and   $|\\cdots n_{\\kappa_{-1}} n_{\\kappa_{0}} n_{\\kappa_{1}} n_{\\kappa_{2}} \\cdots\\rangle$ , one can show that   $\\sum_{i} \\epsilon_{k_{i}} = \\sum_{n} n_{\\kappa_{n}} \\epsilon_{\\kappa_{n}}$ . So if the bosons are free, the state   $|\\cdots n_{\\kappa_{-1}} n_{\\kappa_{0}} n_{\\kappa_{1}} n_{\\kappa_{2}} \\cdots\\rangle$  is an energy eigenstate with an energy\n\n$$\nE _ {\\text { tot }} = \\sum_ {n} n _ {\\kappa_ {n}} \\epsilon_ {\\kappa_ {n}}. \\tag {4.3.1}\n$$\n\nWe like to show that the above free boson system can be viewed as a collection of harmonic oscillators. The different oscillators in the collection are labeled by   $\\kappa_{n}$ . The eigenstates of the oscillator   $\\kappa_{n}$  are labeled by   $n_{\\kappa_{n}}$ . The energy of   $|n_{\\kappa_{n}}\\rangle$  is   $(n_{\\kappa_{n}}+\\frac{1}{2})\\hbar\\omega_{\\kappa_{n}}$  where   $\\omega_{\\kappa_{n}}$  is the oscillation angular frequency of the oscillator   $\\kappa_{n}$ . If we put the oscillators together, a state of the collection with the oscillator   $\\kappa_{n}$  in the   $n_{\\kappa_{n}}$ th excited state can be denoted as   $|\\cdots n_{\\kappa_{-1}}n_{\\kappa_{0}}n_{\\kappa_{1}}n_{\\kappa_{2}}\\cdots\\rangle$ . The energy of such a state is   $\\sum_{n}(n_{\\kappa_{n}}+\\frac{1}{2})\\hbar\\omega_{\\kappa_{n}}$ . We see that if we choose   $\\hbar\\omega_{\\kappa_{n}}=\\epsilon_{\\kappa_{n}}$ , then the above energy will reproduce the boson energy   $\\sum_{n}n_{\\kappa_{n}}\\epsilon_{\\kappa_{n}}$  apart from an overall constant   $\\frac{1}{2}\\sum_{n}\\epsilon_{\\kappa_{n}}$ . Also the oscillator states and the many-boson states have an one-to-one correspondence. Thus the free many-boson system can be viewed as a collection of harmonic oscillators.\n\n# 4.4 A vibrating-string picture of 1D boson system\n\n- A 1D boson system is equivalent to quantized vibrating string.\n- A classical vibrating string provides a classical picture of 1D bosons.\n\nThe vibrating string has a more formal name - 1D field theory.\n\nThe collection of the oscillators with frequency   $\\omega_{\\kappa_{n}} = \\epsilon_{\\kappa_{n}} = \\sqrt{c^{2}\\kappa_{n}^{2} + m^{2}c^{4}}$  can be shown to be the vibration modes of a string. This leads to a vibrating-string picture or a field theory of the bosons. Let us consider a string whose dynamics is described by the following wave equation\n\n$$\n\\ddot {h} (x, t) = c ^ {2} \\partial_ {x} ^ {2} h (x, t) - m ^ {2} c ^ {4} h (x, t) \\tag {4.4.1}\n$$", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0008", "text": "where  $h(x,t)$ is the vibration amplitude of the string and we assume the string form a loop of length  $L$:\n\n$$\nh (x, t) = h (x + L, t).\n$$\n\nTo show that the vibration modes of the above wave equation reproduce the collection of the oscillators labeled by   $\\kappa_{n}$ , we rewrite the wave equation as\n\n$$\n\\dot {h} (x, t) = \\sqrt {- c ^ {2} \\partial_ {x} ^ {2} + m ^ {2} c ^ {4}} p (x, t)\n$$\n\n$$\n- \\dot {p} (x, t) = \\sqrt {- c ^ {2} \\partial_ {x} ^ {2} + m ^ {2} c ^ {4}} h (x, t) \\tag {4.4.2}\n$$\n\nIntroducing a complex amplitude\n\n$$\n\\phi (x, t) = \\frac {h (x , t) + \\mathrm{i} p (x , t)}{\\sqrt {2}} \\tag {4.4.3}\n$$\n\nthe wave equation (4.4.2) becomes\n\n$$\n\\mathrm{i} \\dot {\\phi} (x, t) = \\sqrt {- c ^ {2} \\partial_ {x} ^ {2} + m ^ {2} c ^ {4}} \\phi (x, t) \\tag {4.4.4}\n$$\n\nwhich has a form of Schrödinger equation! Now we do mode expansion by writing\n\n$$\n\\phi (x, t) = \\sum_ {\\kappa_ {n}} \\phi_ {\\kappa_ {n}} (t) L ^ {- 1 / 2} \\mathrm{e} ^ {\\mathrm{i} \\kappa_ {n} x}\n$$\n\nThe wave equation (4.4.4) becomes\n\n$$\n\\mathrm{i} \\dot {\\phi} _ {\\kappa_ {n}} (t) = \\sqrt {c ^ {2} \\kappa_ {n} ^ {2} + m ^ {2} c ^ {4}} \\phi_ {\\kappa_ {n}} (t) \\tag {4.4.5}\n$$\n\nLet  $X_{\\kappa_n}$ and  $P_{\\kappa_n}$ be the real and imaginary parts of  $\\phi_{\\kappa_n} = X_{\\kappa_n} + \\mathrm{i}P_{\\kappa_n}$. We rewrite eqn (4.4.5) as\n\n$$\n\\dot {X} _ {\\kappa_ {n}} (t) = \\sqrt {c ^ {2} \\kappa_ {n} ^ {2} + m ^ {2} c ^ {4}} P _ {\\kappa_ {n}} (t)\n$$\n\n$$\n\\dot {P} _ {\\kappa_ {n}} (t) = - \\sqrt {c ^ {2} \\kappa_ {n} ^ {2} + m ^ {2} c ^ {4}} X _ {\\kappa_ {n}} (t) \\tag {4.4.6}\n$$\n\nWe recognize that the equation (4.4.6) is the equation of motion of an oscillator with  $X_{\\kappa_n}$ as the coordinate and  $P_{\\kappa_n}$ as the momentum. There is one oscillator for every  $\\kappa_n = 2\\pi n / L$. The mass of the oscillator is  $M_{\\kappa_n} = 1 / \\sqrt{c^2\\kappa_n^2 + m^2c^4}$ and the spring constant is  $K_{\\kappa_n} = \\sqrt{c^2\\kappa_n^2 + m^2c^4}$. So the total (classical) energy of the oscillators is\n\n$$\n\\begin{array}{l} E _ {\\mathrm{tot}} = \\sum_ {\\kappa_ {n}} \\left(\\frac {1}{2 M _ {\\kappa_ {n}}} P _ {\\kappa_ {n}} ^ {2} + \\frac {1}{2} K _ {\\kappa_ {n}} X _ {\\kappa_ {n}} ^ {2}\\right) \\\\ = \\int \\mathrm{d} x \\phi^ {*} (x) \\sqrt {- c ^ {2} \\partial_ {x} ^ {2} + m ^ {2} c ^ {4}} \\phi (x) \\\\ = \\int \\mathrm{d} x \\left(\\frac {1}{2} \\dot {h} \\frac {1}{\\sqrt {- c ^ {2} \\partial_ {x} ^ {2} + m ^ {2} c ^ {4}}} \\dot {h} + \\frac {1}{2} h \\sqrt {- c ^ {2} \\partial_ {x} ^ {2} + m ^ {2} c ^ {4}} h\\right) \\tag {4.4.7} \\\\ \\end{array}\n$$\n\nThe expression of the energy and the equation of motion (4.4.1) or (4.4.4) provide a complete description of the oscillators as a classical system. Such a system of vibrating string is also called a classical field theory (in one dimension) where h or   $\\phi$  is the field. From the order of the time derivative, we find that eqn (4.4.1) is a coordinate-space equation of motion while eqn (4.4.4) is a phase-space equation of motion.\n\nFor oscillator   $\\kappa_{n}$ , the oscillation frequency is   $\\omega_{\\kappa_{n}} = \\sqrt{K_{\\kappa_{n}} / M_{\\kappa_{n}}} = \\sqrt{c^{2}\\kappa_{n}^{2} + m^{2}c^{4}}$ . We see that   $\\hbar\\omega_{\\kappa_{n}} = \\epsilon_{\\kappa_{n}}$  (note that   $\\hbar = 1$ ). So indeed the vibration modes of the string give rise to the collection", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0009", "text": "(a)\n\n(b)\n\n(c)\n\nFigure 4.4: Three pictures of a two-boson state where both bosons have the same momentum   $k = 2\\frac{2\\pi}{L}$ . (a) The momentum picture, (b) the occupation picture, and (c) the vibrating-string picture. (a)\n\n(b) Figure 4.5: (a) A boson gas and (b) the same boson gas as a vibrating string.\n\nof the oscillators that describes the bosons with dispersion relation   $\\epsilon_{k} = \\sqrt{c^{2}k^{2} + m^{2}c^{4}}$  (see Fig. 4.4).\n\nThe above result is quite amazing. The quantum theory of a vibrating string described by eqn (4.4.1) is the same as the quantum theory of bosons described by eqn (4.1.6)! So instead of using the picture Fig. 4.5a to describe a quantum boson gas, we can also use the picture Fig. 4.5b to describe the same quantum boson gas. The picture Fig. 4.5b represents a field theory description of the boson gas.\n\nThe vibrating-string picture of the boson gas only works in 1D. In 2D, we need to replace string by membrane: a boson gas in 2D can also be regarded a vibrating membrane. Similarly in d-dimensions, a boson gas is equivalent to a vibrating d-brane (a d-dimensional membrane).\n\n# Problem 4.4.1\n\nFollowing the example of Fig. 4.4, draw three pictures for a two-boson state with one boson carrying momentum  $k = 4\\pi / L$ and the other  $k = 6\\pi / L$.\n\n# 4.5 *The second quantized description of free bosons\n\n- Quantization of the vibrating string.\n- A Hamiltonian without fixing the number of bosons.\n- Boson creation and annihilation operators.\n\nThe vibrating string discussed in the last section is a classical theory. Such a classical theory does not directly describe the quantum boson gas. Only quantized vibrating string describe the quantum bosons. To quantize the vibrating string, we note that a vibrating string is a collection of oscillators described by   $(X_{\\kappa_{n}}, P_{\\kappa_{n}})$ . Since   $(X_{\\kappa_{n}}, P_{\\kappa_{n}})$  is a canonical coordinate-momentum pair, a quantized theory can be obtained by replacing   $(X_{\\kappa_{n}}, P_{\\kappa_{n}})$  by a pair of operators   $(\\hat{X}_{\\kappa_{n}}, \\hat{P}_{\\kappa_{n}})$  that satisfy\n\n$$\n[ \\hat {X} _ {\\kappa_ {n}}, \\hat {P} _ {\\kappa_ {n}} ] = \\mathrm{i}\n$$", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0010", "text": "So after quantization, the classical modes   $\\phi_{\\kappa_{n}}$  and the classical field   $\\phi(x)$  all become operators:\n\n$$\n\\phi_ {\\kappa_ {n}} \\rightarrow \\hat {\\phi} _ {\\kappa_ {n}} = \\frac {1}{\\sqrt {2}} (\\hat {X} _ {\\kappa_ {n}} + \\mathrm{i} \\hat {P} _ {\\kappa_ {n}})\n$$\n\n$$\n\\phi (x) \\rightarrow \\hat {\\phi} (x) = \\sum_ {\\kappa_ {n}} L ^ {- 1 / 2} \\mathrm{e} ^ {\\mathrm{i} x \\kappa_ {n}} \\hat {\\psi} _ {\\kappa_ {n}}\n$$\n\nOne can check that   $\\hat{\\phi}_{\\kappa_{n}}$  and   $\\hat{\\phi}(x)$  satisfy the following algebra\n\n$$\n[ \\hat {\\phi} _ {\\kappa_ {n}}, \\hat {\\phi} _ {\\kappa_ {m}} ^ {\\dagger} ] = \\delta_ {\\kappa_ {n} \\kappa_ {m}}, \\qquad [ \\hat {\\phi} _ {\\kappa_ {n}}, \\hat {\\phi} _ {\\kappa_ {m}} ] = 0. (4. 5. 1)\n$$\n\nand\n\n$$\n[ \\hat {\\phi} (x), \\hat {\\phi} ^ {\\dagger} (x ^ {\\prime}) ] = \\delta (x - x ^ {\\prime}), \\qquad [ \\hat {\\phi} (x), \\hat {\\phi} (x ^ {\\prime}) ] = 0,\n$$\n\nThe Hamiltonian of the quantized vibrating string can be obtained from the classical energy  $E_{\\mathrm{tot}} = \\sum_{\\kappa_n}\\frac{1}{2}\\epsilon_{\\kappa_n}(P_{\\kappa_n}^2 +X_{\\kappa_n}^2)$ (see eqn (4.4.7))\n\n$$\n\\hat {H} _ {0} = \\sum_ {\\kappa_ {n}} \\frac {1}{2} \\epsilon_ {\\kappa_ {n}} (\\hat {P} _ {\\kappa_ {n}} ^ {2} + \\hat {X} _ {\\kappa_ {n}} ^ {2})\n$$\n\n$$\n= \\sum_ {\\kappa_ {n}} \\epsilon_ {\\kappa_ {n}} (\\hat {\\phi} _ {\\kappa_ {n}} ^ {\\dagger} \\hat {\\phi} _ {\\kappa_ {n}} + \\frac {1}{2}) \\tag {4.5.2}\n$$\n\nwhere   $\\epsilon_{k}$  is given by eqn (4.1.5). Let   $n_{\\kappa_{n}}$  be the eigenvalues of   $\\hat{\\phi}_{\\kappa_{n}}^{\\dagger}\\hat{\\phi}_{\\kappa_{n}}$ . The eigenstates of   $\\hat{H}_{0}$  has a form   $|\\cdots n_{\\kappa_{-1}}n_{\\kappa_{0}}n_{\\kappa_{1}}n_{\\kappa_{2}}\\cdots\\rangle$  which has an eigenvalue   $\\sum_{\\kappa_{n}}n_{\\kappa_{n}}\\epsilon_{\\kappa_{n}}$ . So the above Hamiltonian can also be regarded as the Hamiltonian of the free quantum many-boson system. Such a description of the bosons is called the second quantized description.\n\nFrom eqn (4.5.1) and eqn (4.5.2), we see that   $\\hat{\\phi}_{\\kappa_{n}}^{\\dagger}$  and   $\\hat{\\phi}_{\\kappa_{n}}$  are the raising and the lowing operators of the oscillator   $\\kappa_{n}$ . Since the eigenvalues of   $\\hat{\\phi}_{\\kappa_{n}}^{\\dagger}\\hat{\\phi}_{\\kappa_{n}}$  correspond to occupation numbers, the total boson number operator is given by\n\n$$\n\\hat {N} = \\sum_ {\\kappa_ {n}} \\hat {\\phi} _ {\\kappa_ {n}} ^ {\\dagger} \\hat {\\phi} _ {\\kappa_ {n}} = \\int \\mathrm{d} x \\hat {\\phi} ^ {\\dagger} (x) \\hat {\\phi} (x)\n$$\n\nThus we may interpret   $\\hat{\\phi}^{\\dagger}(x)\\hat{\\phi}(x)$  as the boson number-density operator. One can also show that\n\n$$\n[ \\hat {N}, \\hat {\\phi} ^ {\\dagger} ] = \\hat {\\phi} ^ {\\dagger}, \\qquad [ \\hat {N}, \\hat {\\phi} ] = - \\hat {\\phi}.\n$$\n\nwhere   $\\hat{\\phi}$  is   $\\hat{\\phi}(x)$  or   $\\hat{\\phi}_{\\kappa_{n}}$ . Thus   $\\hat{\\phi}^{\\dagger}$  increases the boson number by one while   $\\hat{\\phi}$  decreases the boson number by one. For this reason we also call   $\\phi^{\\dagger}$  the creation operator and   $\\phi$  the annihilation operator. In contract to the Hamiltonian (4.1.6) in the particle picture which only acts on n-particle states, the Hamiltonian (4.5.2) in the second quantized description acts on states with any numbers of bosons.\n\n# 4.6 Vacuum as a dynamical medium and the Casimir effect\n\n- The differences between the two views of bosons: the particle picture and the vibrating-string picture.\n- Vacuum is not empty. It is a dynamical medium just like any materials encountered in condensed matter physics.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 9, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 10"], "page_start": 10, "page_end": 10, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0011", "text": "We have discussed two ways to view a many-boson system: the particle picture (which include both the momentum picture   $|k_{1}k_{2}\\cdots\\rangle$  and the occupation picture   $|\\cdots n_{\\kappa_{-1}}n_{\\kappa_{0}}n_{\\kappa_{1}}n_{\\kappa_{2}}\\cdots\\rangle$ ) and the vibrating-brane picture. The essence of the particle picture is the assumption that the vacuum is empty. In this picture, bosons are thing placed on the empty vacuum. However, in the vibrating-brane picture, the vacuum is regarded as a dynamical non-empty medium. The vibration of such a medium gives rise to bosons in the vacuum. In the last section, we have stressed the mathematical equivalence of the two pictures. However, the two pictures are not equivalent in physical sense.\n\nFirst, the vibrating-brane picture (or the field theory picture) provides an origin and an explanation of identical particles. The particles arising from the vibrations of a d-brane are naturally and always identical bosons. Actually, it is impossible to obtain non-identical particles from the vibrations of the brane. In contrast, particles do not have to be identical particles in the particle picture. In this case, the existence and the appearance of identical particles is very mysterious.\n\nWe can also turn our argument around. The very existence of identical particles in our vacuum suggests that we should not view particles as things placed in an empty vacuum. We should instead view our vacuum as a 3-brane and the particles as the vibrations of the 3-brane. Our vacuum is not empty. It is a dynamical medium whose collective motion give rise to elementary particles. So our vacuum just like a material studied on condensed matter physics. The theory of elementary particles is actually a theory about one material – the vacuum material.\n\nHere we would like to remark that the vibrating-brane only reproduce a particular kind of identical particles – scaler bosons. The photons in our vacuum are actually vector bosons (or gauge bosons) due to its two polarizations and electrons are fermions. Those particles cannot arise from vibrating 3-branes. However, as we will see later in this book, the above philosophy is still correct. We should not view our vacuum as an empty stage. We should view it as a dynamical medium whose collective motion can even give rise to photons and electrons. The vacuum material has a more complicated internal structure than the simple 3-brane. It is this more complicated internal structure that leads to photons, electrons, gluons, quarks,[Wen 2002, 2003b] and possibly all other elementary particles observed in our vacuum. In this chapter, for simplicity, we treat photons as massless scalar bosons and regard them as vibrations of a 3-brane.\n\nThe above discussion about the advantage of the vibrating-brane picture sounds philosophical. Actually, the vibrating-brane picture has a measurable consequence – Casimir effect.[Casimir 1948] The Casimir effect has been observed in our vacuum, confirming that the vibrating-brane picture is a correct picture while the particle picture is an incorrect picture for bosons.\n\nTo understand the Casimir effect, we start with the energy of vacuum. In the particle picture, the vacuum is just a empty stage or a reference point. We naturally assign a zero energy to it and define the reference point of energy. In contrast, the vacuum energy in the vibrating-brane picture is naturally non-zero and is given by the zero-energies   $\\frac{1}{2}\\hbar\\omega_{k}=\\frac{1}{2}\\epsilon_{k}$  of the oscillators. For 1D free boson system, the vacuum energy is given by\n\n$$\nU _ {\\mathrm{vac}} = \\sum_ {\\kappa_ {n}} \\frac {1}{2} \\epsilon_ {\\kappa_ {n}}\n$$\n\nwhere   $\\epsilon_{k}$  is the dispersion of the bosons. One may say that   $U_{vac}$  is just a constant term in the total energy which define the reference point of the energy. Such a constant term has no measurable effect. Indeed,   $U_{vac}$  cannot be measured directly. However, if we change the dispersion   $\\epsilon_{k}$  and/or change the distribution of quantized momentum   $\\kappa_{n}$ , then the change in   $U_{vac}$  has physical effects and can be measured.\n\nBefore discussing how to calculate the change of   $U_{vac}$ , let us calculate   $U_{vac}$  itself. At first sight, the calculation of   $U_{vac}$  appear to be very simple since   $U_{vac} = \\frac{1}{2} \\sum_{\\kappa_n} \\epsilon_{\\kappa_n} = +\\infty$ . Certainly, such a simple result of infinity is meaningless. The infinite   $U_{vac}$  is the famous infinity problem that plague all forms of field theories. It is so annoying that it once led people to abandon field theories. One can use this infinity problem to argue that the vibrating-brane picture for bosons is wrong.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 10, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 11"], "page_start": 11, "page_end": 11, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0012", "text": "Figure 4.6: According to quantum gravity, a continuous string cannot be a physical reality. A real string may look more like a sequence discrete beads.\n\n(a)\n\n(b) Figure 4.7: (a) A boson gas between two hard walls. (b) The same boson gas is described by a vibrating string with boundary condition   $h(0) = h(L) = 0$ .\n\nActually, the problem of the infinity is not the problem of the vibrating-brane picture but the problem of regarding space (or the vibrating-brane) as a continuous manifold. As has been argued in section 2.2, continuous manifold simply does not exist in our universe. It is meaningless and impossible to have two points separated by a distance less the Planck length  $l_{P}$. Similarly, a wave vector larger than  $1 / l_{P}$ is also meaningless. So it is more correct to view the string as a sequence discrete beads (see Fig. 4.6). I would like to stress that this is a quantum gravity effect (see section 2.2). Our vibrating-brane picture is meaningful only for wave vectors less then  $\\Lambda \\sim 1 / l_{P}$ if we take into account the quantum gravity effect. The momentum scale  $\\Lambda$ is called the cut-off scale. So we should limit the Wave vector summation  $\\sum_{\\kappa_n}$ to over the meaningful wave vectors  $|\\kappa_n| < \\Lambda$. Therefore, the vacuum energy is really finite\n\n$$\nU _ {\\mathrm{vac}} = \\sum_ {\\kappa_ {n} = 0, \\pm 2 \\pi / L, \\pm 4 \\pi / L, \\dots , \\pm \\Lambda} \\frac {1}{2} \\epsilon_ {\\kappa_ {n}}\n$$\n\nCertainly, we are not sure that the energy levels should suddenly disappear for  $k$ above  $\\Lambda$ as implied by the above formula. We may very well have a softer cut-off\n\n$$\nU _ {\\mathrm{vac}} = \\sum_ {\\kappa_ {n} = 0, \\pm 2 \\pi / L, \\pm 4 \\pi / L, \\dots} \\frac {1}{2} \\epsilon_ {\\kappa_ {n}} \\mathrm{e} ^ {- | \\kappa_ {n} | / \\Lambda}\n$$\n\nBut what is the right way to calculate   $U_{vac}$ ? The physics at short distance is still unknown to us (since we still do not have a theory of quantum gravity). So it is not clear what is the correct way to cut-off the wave vector summation   $\\sum_{\\kappa_{n}}$ . However, we will see later that the low energy and long distance effects do not depend on how we cut-off the momentum summation. This allows us to make prediction without a complete understanding of the theory. The ignorance of the short distance physics does not prevent us from gaining a (partial) understanding of long distance physics.\n\nLet us consider a 1D mass m boson system between two hard walls (see Fig. 4.7a). One wall is at x = 0 and the other at x = L. Such a boson system is described by a string (4.4.1) that satisfy the boundary condition   $h(0) = h(L) = 0$  (see Fig. 4.7b).  $^{1}$  In contrast to the periodic boundary condition   $h(0) = h(L)$ , the string vibration modes for the hard-wall case has a form   $h(x) \\propto \\sin(n\\pi x/L)$  and are labeled by   $\\tilde{\\kappa}_{n} = n\\pi/L$ ,   $n = 0, 1, 2, \\cdots$ . In this case, the vacuum energy\n\n$^{1}$ The connection between the hard wall and the boundary condition   $h(0)=h(L)=0$  will be discussed in section 4.7.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 11, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 12"], "page_start": 12, "page_end": 12, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0013", "text": "L\na\n\nis different from that for periodic boundary condition\n\n$$\nU _ {\\mathrm{vac}} ^ {\\mathrm{hw}} (L) = \\sum_ {\\tilde {\\kappa} _ {n} = 0, \\pi / L, 2 \\pi / L, \\dots} \\frac {1}{2} \\epsilon_ {\\tilde {\\kappa} _ {n}} \\mathrm{e} ^ {- | \\tilde {\\kappa} _ {n} | / \\Lambda}\n$$\n\nFor massless particles,  $\\epsilon_{k} = c|k|$.  $U_{\\mathrm{vac}}^{\\mathrm{hw}}(L)$ can be calculated easily. We find\n\n$$\nU _ {\\mathrm{vac}} ^ {\\mathrm{hw}} (L) = \\frac {c \\pi}{8 L \\sinh^ {2} (\\frac {c \\pi}{2 L \\Lambda})} = \\frac {\\Lambda^ {2} L}{4 c \\pi} - \\frac {c \\pi}{2 4 L} + O (\\Lambda^ {- 2}) \\tag {4.6.1}\n$$\n\nNow let us add the third hard walls at  $x = a$ (see Fig. 4.8). The third hard wall modifies the distribution of the vibration modes which changes the vacuum energy. Using the result (4.6.1), we find the modified vacuum energy to be\n\n$$\nU _ {\\mathrm{vac}} ^ {\\mathrm{hw}} (a, L) = \\frac {\\Lambda^ {2} L}{4 c \\pi} - \\frac {c \\pi}{2 4 (L - a)} - \\frac {c \\pi}{2 4 a} + O (\\Lambda^ {- 2})\n$$\n\nWhen L is very large (i.e.   $L \\gg a$ ), we find that the total vacuum energy depends on the separation between the two walls at x = 0 and a:   $U_{\\mathrm{vac}}^{\\mathrm{hw}}(a, L) = -\\frac{c\\pi}{24a} + \\mathrm{Const}$ . This causes an attractive force between the two walls separated by a distance a\n\n$$\nF = \\frac {\\pi \\hbar c}{2 4 a ^ {2}}\n$$\n\nThis force between two walls in vacuum is called the Casimir effect.[Casimir 1948] It is interesting to note that the force does not depend on the cut-off.\n\nThe above result is for massless bosons in 1D. For massless photons in 3D, the attractive force between two plates separated by  $a$ is\n\n$$\nF = \\frac {\\pi^ {2} \\hbar c}{2 4 0 a ^ {4}} A\n$$\n\nwhere A is the area of the plates. Such a force was measured by Spamaay, 1958 and Lamoreaux, 1997, indicating that our vacuum is really a dynamical medium. The study of our vacuum and its elementary particles is really a material science.\n\n# 4.7 Classical field theory for non-relativistic free bosons\n\n- Phase-space Lagrangian for the classical field theory that describes free bosons.\n\nIn the section 4.4, we discussed the classical field theory (or the vibrating-string picture) of relativistic bosons. The same calculation also apply to a d-dimensional non-relativistic bosons in a potential U. Such non-relativistic bosons have a dispersion\n\n$$\n\\epsilon_ {\\pmb {k}} = \\frac {\\pmb {k} ^ {2}}{2 m} + U\n$$", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 12, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 13"], "page_start": 13, "page_end": 13, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0014", "text": "and are described by an n-particle Hamiltonian\n\n$$\n\\hat {H} _ {n} = \\sum_ {i = 1} ^ {n} (- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x} _ {i}} ^ {2} + U) \\tag {4.7.1}\n$$\n\nFollowing the discussion in the last section, we can show that the quantum system (4.7.1) is related to a classical system - a classical field theory or a vibrating brane. The quantized vibrating brane describes the quantum system (4.7.1). The classical system $^2$ is described by a phase-space equation of motion\n\n$$\n\\mathrm{i} \\dot {\\phi} (\\boldsymbol {x}, t) = \\left(- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x}} ^ {2} + U\\right) \\phi (\\boldsymbol {x}, t) \\tag {4.7.2}\n$$\n\nand a total energy\n\n$$\nE _ {\\mathrm{tot}} = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\phi^ {*} \\left(- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x}} ^ {2} + U\\right) \\phi , \\tag {4.7.3}\n$$\n\nwhere the complex field   $\\phi$  encodes both the amplitude and the velocity of the vibration (see eqn (4.4.3) and eqn (4.4.2)). Both the phase-space equation of motion (4.7.2) and the total energy (4.7.3) can be derived from the following phase-space Lagrangian\n\n$$\nL = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\mathrm{i} \\phi^ {*} \\dot {\\phi} - E _ {\\text {tot}} = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\left(\\mathrm{i} \\phi^ {*} \\dot {\\phi} - \\phi^ {*} \\left(- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x}} ^ {2} + U\\right) \\phi\\right). \\tag {4.7.4}\n$$\n\nWe note that the potential U in the classical energy (4.7.3) or in the phase-space Lagrangian (4.7.4) may have a spatial dependence   $U = U(\\boldsymbol{x})$ . A hard wall in a certain region can be realized by an infinite potential   $U(\\boldsymbol{x})$  in that region. In order for the total energy (4.7.3) to be finite, an infinite potential in a region will force   $\\phi(\\boldsymbol{x})$  to be zero in that region. This is why a hard wall can be represented by the boundary condition   $\\phi(\\boldsymbol{x}) = 0$  or   $h(\\boldsymbol{x}) = 0$ .\n\n# Problem 4.7.1\n\nDerive eqn (4.7.2) and eqn (4.7.3) from the phase-space Lagrangian (4.7.4).\n\n# 4.8 *The second quantized description of interacting bosons\n\n- The many-body Hamiltonian that describes interacting bosons.\n- The boson density operator.\n\nNow let us turn to a more complicated problem of interacting bosons. Let   $V(\\boldsymbol{x})$  be the potential energy of two bosons separated by x. The Hamiltonian that describe n interacting bosons in d-dimension is\n\n$$\n\\hat {H} _ {n} = \\sum_ {i} (- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x} _ {i}} ^ {2} + U) + \\sum_ {i <   j} V (\\boldsymbol {x} _ {i} - \\boldsymbol {x} _ {j}) \\tag {4.8.1}\n$$\n\n$^{2}$ The classical field theory (the vibrating brane) is also described by a coordinate-space equation of motion\n\n$\\ddot{h} = -\\frac{1}{2m}\\partial_{x}^{2} + U^{2}h$\n\nwhere h is the amplitude of the vibration and a Lagrangian\n\n[L = \\int \\mathrm{d}^{d}\\pmb {x}\\quad \\frac{1}{2}\\dot{h}\\frac{1}{-\\frac{1}{2m}\\partial_{\\pmb{x}}^{2} + U}\\dot{h} -\\frac{1}{2} h\\quad -\\frac{1}{2m}\\partial_{\\pmb{x}}^{2} + U\\quad h.]", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 13, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 14"], "page_start": 14, "page_end": 14, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0015", "text": "We know that the free bosons have a second quantized description (4.5.2) which describes 0-boson, 1-boson, 2-boson, and general n-boson situations with a single Hamiltonian. What is the second quantized description (or the oscillator description) of the interacting bosons?\n\nAgain the second quantized description is written in terms of the boson creation/annihilation operators   $\\hat{\\phi}$  and   $\\hat{\\phi}^{\\dagger}$ . To obtain the form of the second quantized Hamiltonian for interacting bosons, we first note that the total potential energy can be rewritten in terms of the boson density\n\n$$\n\\sum_ {i <   j} V (\\pmb {x} _ {i} - \\pmb {x} _ {j}) = \\int \\mathrm{d} ^ {d} \\pmb {x} \\mathrm{d} ^ {d} \\pmb {x} ^ {\\prime} \\frac {1}{2} n (\\pmb {x}) V (\\pmb {x} - \\pmb {x} ^ {\\prime}) n (\\pmb {x} ^ {\\prime})\n$$\n\nSince the boson density operator is given by  $\\hat{n} (\\pmb {x}) = \\hat{\\phi}^{\\dagger}(\\pmb {x})\\hat{\\phi} (\\pmb {x})$ as discussed in section 4.5, the interaction Hamiltonian is given by\n\n$$\n\\begin{array}{l} \\hat {H} _ {\\mathrm{int}} = \\int \\mathrm{d} ^ {d} \\pmb {x} \\mathrm{d} ^ {d} \\pmb {x} ^ {\\prime} \\frac {1}{2} \\hat {n} (\\pmb {x}) V (\\pmb {x} - \\pmb {x} ^ {\\prime}) \\hat {n} (\\pmb {x} ^ {\\prime}) \\\\ = \\frac {1}{2 \\mathcal {V}} \\sum_ {\\boldsymbol {q}, \\boldsymbol {k}, \\boldsymbol {k} ^ {\\prime}} \\hat {\\phi} _ {\\boldsymbol {k} + \\boldsymbol {q}} ^ {\\dagger} \\hat {\\phi} _ {\\boldsymbol {k}} V _ {\\boldsymbol {q}} \\hat {\\phi} _ {\\boldsymbol {k} ^ {\\prime} - \\boldsymbol {q}} ^ {\\dagger} \\hat {\\phi} _ {\\boldsymbol {k} ^ {\\prime}} \\\\ \\end{array}\n$$\n\nwhere\n\n$$\nV _ {\\pmb {q}} = \\int \\mathrm{d} ^ {d} \\pmb {x} \\mathrm{e} ^ {- \\mathrm{i} \\pmb {q} \\cdot \\pmb {x}} V (\\pmb {x})\n$$\n\nHere we have assumed that our system is a d-dimensional cube of volume   $V = L^{d}$ . The wave vectors q, k,   $k'$  are quantized, i.e. their components are   $2\\pi/L$  times integers. The summation   $\\sum_{q,k,k'}$  sums over those quantize wave vectors.   $\\hat{\\phi}_{k}$  is given by\n\n$$\n\\hat {\\phi} (\\boldsymbol {x}) = \\sum_ {\\boldsymbol {k}} L ^ {- d / 2} \\mathrm{e} ^ {\\mathrm{i} \\boldsymbol {k} \\cdot \\boldsymbol {x}} \\hat {\\phi} _ {\\boldsymbol {k}} \\tag {4.8.2}\n$$\n\nand satisfies the algebra\n\n$$\n[ \\hat {\\phi} _ {\\pmb {k}}, \\hat {\\phi} _ {\\pmb {k ^ {\\prime}}} ^ {\\dagger} ] = \\delta_ {\\pmb {k k ^ {\\prime}}}, \\qquad [ \\hat {\\phi} _ {\\pmb {k}}, \\hat {\\phi} _ {\\pmb {k ^ {\\prime}}} ] = 0.\n$$\n\nPutting   $\\hat{H}_{int}$  and   $\\hat{H}_{0}$  in eqn (4.5.2) together, we find the following Hamiltonian\n\n$$\n\\hat {H} = \\sum_ {\\boldsymbol {k}} \\tilde {\\epsilon} _ {\\boldsymbol {k}} (\\hat {\\phi} _ {\\boldsymbol {k}} ^ {\\dagger} \\hat {\\phi} _ {\\boldsymbol {k}} + \\frac {1}{2}) + \\frac {1}{2 \\mathcal {V}} \\sum_ {\\boldsymbol {q}, \\boldsymbol {k}, \\boldsymbol {k} ^ {\\prime}} \\hat {\\phi} _ {\\boldsymbol {k} + \\boldsymbol {q}} ^ {\\dagger} \\hat {\\phi} _ {\\boldsymbol {k}} V _ {\\boldsymbol {q}} \\hat {\\phi} _ {\\boldsymbol {k} ^ {\\prime} - \\boldsymbol {q}} ^ {\\dagger} \\hat {\\phi} _ {\\boldsymbol {k} ^ {\\prime}} \\tag {4.8.3}\n$$\n\ndescribes the interacting bosons. Naively, one expects   $\\tilde{\\epsilon}_{k}$  describes the boson dispersion and should be taken to be   $\\tilde{\\epsilon}_{k} = \\frac{k^{2}}{2m} + U$ . As we will see below, this naive expectation is incorrect. We need to choose a different   $\\tilde{\\epsilon}_{k}$  in order to reduce the single-boson dispersion   $\\frac{k^{2}}{2m} + U$ .\n\nTo determine the proper form of  $\\tilde{\\epsilon}_{\\pmb{k}}$, let us discuss a few known eigenstates of  $\\hat{H}$ in eqn (4.8.3). One eigenstate  $|0\\rangle$ of  $\\hat{H}$ is defined by the algebraic relation\n\n$$\n\\hat {\\phi} _ {\\boldsymbol {k}} | 0 \\rangle = 0 \\tag {4.8.4}\n$$\n\nSuch a state is an eigenstate of the boson number operator with zero eigenvalue  $\\hat{N} |0\\rangle = 0$. Thus  $|0\\rangle$ is the vacuum state with no bosons. The energy of  $|0\\rangle$ is  $E_0 = \\sum_{\\pmb{k}}\\frac{1}{2}\\tilde{\\epsilon}_{\\pmb{k}}$. Another class of the eigenstates is given by\n\n$$\n| \\pmb {k} \\rangle = \\hat {\\phi} _ {\\pmb {k}} ^ {\\dagger} | 0 \\rangle\n$$\n\nTo show  $|\\pmb{k}\\rangle$ is an eigenstate, we commute  $\\hat{\\phi}$ through  $\\hat{\\phi}^{\\dagger}$ in  $\\hat{H}$ using the commutation relation (4.8.2) to put all  $a$ to the right of  $a^{\\dagger}$. Such a procedure is called normal ordering. This allows us to rewrite  $\\hat{H}$ as\n\n$$\n\\hat {H} = \\sum_ {\\pmb {k}} (\\epsilon_ {\\pmb {k}} \\hat {\\phi} _ {\\pmb {k}} ^ {\\dagger} \\hat {\\phi} _ {\\pmb {k}} + \\frac {1}{2} \\tilde {\\epsilon} _ {\\pmb {k}}) + \\frac {1}{2 \\mathcal {V}} \\sum_ {\\pmb {q}, \\pmb {k}, \\pmb {k} ^ {\\prime}} V _ {\\pmb {q}} \\hat {\\phi} _ {\\pmb {k} + \\pmb {q}} ^ {\\dagger} \\hat {\\phi} _ {\\pmb {k} ^ {\\prime} - \\pmb {q}} ^ {\\dagger} \\hat {\\phi} _ {\\pmb {k}} \\hat {\\phi} _ {\\pmb {k} ^ {\\prime}} (4. 8. 5)\n$$", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 14, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 15"], "page_start": 15, "page_end": 15, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0016", "text": "where  $\\epsilon_{\\pmb{k}}$ is given by\n\n$$\n\\epsilon_ {\\boldsymbol {k}} = \\tilde {\\epsilon} _ {\\boldsymbol {k}} + \\frac {V (0)}{2} \\tag {4.8.6}\n$$\n\nand   $V(0)$  is   $V(\\boldsymbol{x})$  at x = 0. Using eqn (4.8.5), we easily see that   $|k\\rangle$  is an eigenstate of   $\\hat{H}$  with eigenvalue   $E_{\\boldsymbol{k}} = \\epsilon_{\\boldsymbol{k}} + E_{0}$ .   $|k\\rangle$  is a state with one boson that carries a momentum k. The excitation energy of such a one-boson state is   $E_{\\boldsymbol{k}} - E_{0} = \\epsilon_{\\boldsymbol{k}}$ . We see that the single-boson dispersion is given by   $\\epsilon_{k}$ . To reproduce the dispersion   $\\epsilon_{\\boldsymbol{k}} = \\frac{|\\boldsymbol{k}|^{2}}{2m} + U$ , we need to choose   $\\tilde{\\epsilon}_{k}$  to be   $\\tilde{\\epsilon}_{\\boldsymbol{k}} = \\frac{|\\boldsymbol{k}|^{2}}{2m} + U - \\frac{V(0)}{2}$ .\n\nA generic state described by a collection of boson occupation numbers   $\\{n_{k}\\}$  can be created by the boson creation operators   $\\hat{\\phi}_{k}^{\\dagger}$  from the vacuum state   $|0\\rangle$ :\n\n$$\n| \\{n _ {\\boldsymbol {k}} \\} \\rangle \\propto \\prod_ {\\boldsymbol {k}} (\\hat {\\phi} _ {\\boldsymbol {k}} ^ {\\dagger}) ^ {n _ {\\boldsymbol {k}}} | 0 \\rangle\n$$\n\nSuch a state is an eigenstate of the total boson number operator   $\\hat{N} = \\sum_{k} \\hat{\\phi}_{k}^{\\dagger} \\hat{\\phi}_{k}$ . The total boson number of   $|\\{n_{k}\\}\\rangle$ . The state   $|\\{n_{k}\\}\\rangle$  also carry a definite total momentum   $P = \\sum_{k} kn_{k}$ . However, for interacting bosons the state   $|\\{n_{k}\\}\\rangle$  is not an energy eigenstate. In general, it is hard to calculate energy eigenstates.\n\nTo summarize, if we choose   $\\epsilon_{k} = \\frac{k^{2}}{2m} + U$ , eqn (4.8.5) and eqn (4.8.1) will describe the same interacting boson system! Since   $\\hat{H}$  is not quadratic in   $\\hat{\\phi}_{k}$  and   $\\hat{\\phi}_{k}^{\\dagger}$ ,   $\\hat{H}$  describes a collection of anharmonic quantum oscillators. So interacting bosons are described by anharmonic oscillators. When   $\\epsilon_{k} = \\frac{k^{2}}{2m} + U$ , eqn (4.8.5) can be written in a more compact form\n\n$$\n\\begin{array}{l} \\hat {H} = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\hat {\\phi} ^ {\\dagger} (\\boldsymbol {x}) \\left(- \\frac {\\partial_ {\\boldsymbol {x}} ^ {2}}{2 m} + U\\right) \\hat {\\phi} (\\boldsymbol {x}) \\\\ + \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\mathrm{d} ^ {d} \\boldsymbol {x} ^ {\\prime} \\frac {1}{2} V (\\boldsymbol {x} - \\boldsymbol {x} ^ {\\prime}) \\hat {\\phi} ^ {\\dagger} (\\boldsymbol {x}) \\hat {\\phi} ^ {\\dagger} (\\boldsymbol {x} ^ {\\prime}) \\hat {\\phi} (\\boldsymbol {x} ^ {\\prime}) \\hat {\\phi} (\\boldsymbol {x}) \\tag {4.8.7} \\\\ \\end{array}\n$$\n\nwhere  $\\hat{\\phi} (\\pmb {x})$ satisfies the following algebra\n\n$$\n[ \\hat {\\phi} (\\boldsymbol {x}), \\hat {\\phi} ^ {\\dagger} (\\boldsymbol {x} ^ {\\prime}) ] = \\delta (\\boldsymbol {x} - \\boldsymbol {x} ^ {\\prime}), \\quad [ \\hat {\\phi} (\\boldsymbol {x}), \\hat {\\phi} (\\boldsymbol {x} ^ {\\prime}) ] = 0,\n$$\n\n# Problem 4.8.1\n\nShow that   $[\\hat{N},\\hat{H}]=0$ . So the total boson number is conserved and the Hamiltonian is invariant under the   $U(1)$  transformation generated by   $\\hat{N}\\colon\\hat{H}=\\mathrm{e}^{\\mathrm{i}\\theta\\hat{N}}\\hat{H}\\mathrm{e}^{-\\mathrm{i}\\theta\\hat{N}}$ .\n\n# Problem 4.8.2\n\nFind the total momentum operator  $\\hat{P}$ of the bosons in terms of  $\\hat{\\phi}_{k}$. Show that  $[\\hat{P},\\hat{H}] = 0$ and hence the total momentum is conserved.\n\n# Problem 4.8.3\n\nIt is too hard to find the eigenstates and eigenvalues of   $\\hat{H}$  (4.8.5) in the 2-boson sector. Here we simplify the problem by limiting ourselves to 1D and consider only there k-levels:   $k = 0, \\pm 2\\pi/L$ . Find the eigenstates and the eigenvalues of   $\\hat{H}$  in the 2-boson sector with the above simplification.\n\n# Problem 4.8.4\n\nShow eqn (4.8.5).", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 15, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 16"], "page_start": 16, "page_end": 16, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0017", "text": "# 4.9 Classical field theory of interacting bosons\n\n- A classical picture for interacting bosons.\n\nThe free boson system (4.7.1) is easy to solve. We do not need a classical vibrating brane picture to understand and to visualize the behavior of free bosons. However, an interacting boson system described by\n\n$$\n\\hat {H} _ {n} = \\sum_ {i} (- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x} _ {i}} ^ {2} + U) + \\sum_ {i <   j} V (\\boldsymbol {x} _ {i} - \\boldsymbol {x} _ {j}) \\tag {4.9.1}\n$$\n\n(or eqn (4.8.5)) is entirely a different matter. The interacting Hamiltonian eqn (4.9.1) is so hard to solve that we have no clue what do the low energy eigenstates and the eigenvalues look like. So how can we understand the properties of the interacting bosons without being able to diagonalize the Hamiltonian eqn (4.9.1)?\n\nOne way to understand interacting bosons is to find the corresponding classical system and study its low energy collective motions. Then we can quantize those low energy classical motions to obtain low energy quantum properties. This way, we can gain a understanding of the low energy properties of the quantum interacting boson system.\n\nTo obtain the corresponding classical system for interacting bosons, let us first find out how to represent the boson density in the classical field theory. We know that if the potential U by   $\\delta U$ , the change in the total energy of the bosons will be   $\\delta E_{tot} = N\\delta U$  where N is the total number of the bosons. From eqn (4.7.3) we see that   $\\delta E_{tot} = \\int d^{d}x\\phi^{*}\\phi\\delta U$ . So the total number of the bosons is given by   $N = \\int d^{d}x\\phi^{*}\\phi$  in field theory and the boson density is\n\n$$\nn (\\pmb {x}) = \\phi^ {*} (\\pmb {x}) \\phi (\\pmb {x})\n$$\n\nThe total potential energy can be rewritten in terms of the boson density\n\n$$\n\\sum_ {i <   j} V (\\pmb {x} _ {i} - \\pmb {x} _ {j}) = \\int \\mathrm{d} ^ {d} \\pmb {x} \\mathrm{d} ^ {d} \\pmb {x} ^ {\\prime} \\frac {1}{2} n (\\pmb {x}) V (\\pmb {x} - \\pmb {x} ^ {\\prime}) n (\\pmb {x} ^ {\\prime})\n$$\n\nThis allows us to guess that the classical field theory for interacting bosons to have a modified total energy\n\n$$\nE _ {\\mathrm{tot}} = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\phi^ {*} \\left(- \\frac {\\partial_ {\\boldsymbol {x}} ^ {2}}{2 m} + U\\right) \\phi + \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\mathrm{d} ^ {d} \\boldsymbol {x} ^ {\\prime} | \\phi (\\boldsymbol {x}) | ^ {2} \\frac {V (\\boldsymbol {x} - \\boldsymbol {x} ^ {\\prime})}{2} | \\phi (\\boldsymbol {x} ^ {\\prime}) | ^ {2} \\tag {4.9.2}\n$$\n\nand hence a modified Lagrangian\n\n$$\n\\begin{array}{l} L = \\int \\mathrm{d} ^ {d} \\pmb {x} \\mathrm{i} \\phi^ {*} \\dot {\\phi} - E _ {\\mathrm{tot}} = \\int \\mathrm{d} ^ {d} \\pmb {x} \\left(\\mathrm{i} \\phi^ {*} \\dot {\\phi} - \\phi^ {*} \\left(- \\frac {1}{2 m} \\partial_ {\\pmb {x}} ^ {2} + U\\right) \\phi\\right) \\\\ - \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\mathrm{d} ^ {d} \\boldsymbol {x} ^ {\\prime} | \\phi (\\boldsymbol {x}) | ^ {2} \\frac {V (\\boldsymbol {x} - \\boldsymbol {x} ^ {\\prime})}{2} | \\phi (\\boldsymbol {x} ^ {\\prime}) | ^ {2} \\tag {4.9.3} \\\\ \\end{array}\n$$\n\nThe corresponding equation of motion can be obtained from  $L$ and is given by\n\n$$\n\\begin{array}{l} \\mathrm{i} \\dot {\\phi} (\\pmb {x}, t) = \\left(- \\frac {1}{2 m} \\partial_ {\\pmb {x}} ^ {2} + U _ {\\mathrm{eff}} (\\pmb {x})\\right) \\phi (\\pmb {x}, t) \\\\ U _ {\\text { eff }} (\\boldsymbol {x}) = U + \\int \\mathrm{d} ^ {d} \\boldsymbol {x} ^ {\\prime} V (\\boldsymbol {x} - \\boldsymbol {x} ^ {\\prime}) | \\phi (\\boldsymbol {x} ^ {\\prime}) | ^ {2} \\tag {4.9.4} \\\\ \\end{array}\n$$\n\nWe note that eqn (4.9.4) is a non-linear equation. So the interacting bosons are described by a non-harmonic field theory.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 16, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 17"], "page_start": 17, "page_end": 17, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0018", "text": "# Problem 4.9.1\n\nFollowing the discussion in section 2.3.2 to derive the equation of motion (4.9.4) from the Lagrangian (4.9.3).\n\n# Problem 4.9.2\n\n# Symmetry and conservation:\n\n- (a) Derive the equation of motion (4.9.4) from the Lagrangian (4.9.3).\n- (b) The Lagrangian (4.9.3) is invariant under a  $U(1)$ transformation  $\\phi \\to \\mathrm{e}^{\\mathrm{i}\\theta}\\phi$. Show that the particle number is conserved for such a system, i.e.  $\\frac{\\mathrm{d}}{\\mathrm{d}t}\\int \\mathrm{d}^d x|\\phi|^2 = 0$.\n- (c) We include a term  $\\int \\mathrm{d}^d x g(\\phi + \\phi^*)$ in the Lagrangian to break the  $U(1)$ invariance. Show that the particle number is no longer conserved for the new system.\n\n# Problem 4.9.3\n\nIn this section we have “guessed” the classical field theory (4.9.3) or (4.9.4) for the interaction bosons. In fact the classical wave equation (4.9.4) can be “derived” using the equation-of-motion approach described in section 2.6.1 from the second quantized Hamiltonian (4.8.7). Find the operator equation of motion for   $\\hat{\\phi}(\\boldsymbol{x},t)\\equiv\\mathrm{e}^{\\mathrm{i}\\hat{H}}\\hat{\\phi}(\\boldsymbol{x})\\mathrm{e}^{-\\mathrm{i}\\hat{H}}$  using   $\\partial_{t}\\hat{\\phi}(\\boldsymbol{x},t)=\\mathrm{i}[\\hat{H},\\hat{\\phi}(\\boldsymbol{x},t)]$ . Replace   $\\hat{\\phi}$  and   $\\hat{\\phi}^{\\dagger}$  by   $\\langle\\hat{\\phi}\\rangle=\\phi$  and   $\\langle\\hat{\\phi}^{\\dagger}\\rangle=\\phi^{*}$  to obtain the corresponding classical equation of motion.\n\n# 4.10 A quantum phase transition in interacting boson system\n\nInstead of directly studying the very difficult quantum interacting boson system (4.9.1), in the next a few sections, we will study the corresponding classical field theory (or the non-harmonic brane) described by eqn (4.9.3). The classical field theory is much easier to deal with. The physical properties of the classical field theory will give us some good ideas about the physical properties of the quantum interacting bosons.\n\nSince it encodes both coordinates and momenta, the complex field   $\\phi$  describes a classical state in the classical field theory. That is each different complex function   $\\phi(\\boldsymbol{x})$  correspond to different classical state. So looking for a classical ground state is equivalent to looking for a complex function that minimize the total energy (4.9.2).\n\nWe note that a constant function minimizes the kinetic energy term   $\\int d^{d}x\\phi^{*}\\frac{-\\partial_{x}^{2}}{2m}\\phi=\\int d^{d}x\\frac{\\partial_{x}\\phi^{*}\\partial_{x}\\phi}{2m}$ . So if the interaction V is not too large, we may assume the function that minimize the total energy is a constant. In this case the energy density becomes\n\n$$\nu = U | \\phi | ^ {2} + \\frac {1}{2} \\bar {V} | \\phi | ^ {4}\n$$\n\nwhere  $\\bar{V} \\equiv \\int \\mathrm{d}^d x V(x)$.\n\nWhen   $\\bar{V} < 0$ , we note that the total energy   $E_{tot} = \\int d^{d} x u$  is not bound from below and has no minimum. So the ground state of boson gas with attractive interaction is well defined.  $^{3}$  When   $\\bar{V} > 0$ , we find that the total energy is minimized by the following field (or the classical state)\n\n$$\n\\phi_ {\\text { grnd }} = \\left\\{ \\begin{array}{l l} 0, & U > 0 \\\\ \\sqrt {- U / \\bar {V}}, & U <   0 \\end{array} \\right. \\tag {4.10.1}\n$$\n\nThis field corresponds to the classical ground state of the system. The ground state energy density is\n\n$$\nu = \\left\\{ \\begin{array}{l l} 0, & U > 0 \\\\ - \\frac {1}{2} U ^ {2} / \\bar {V}, & U <   0 \\end{array} \\right. \\tag {4.10.2}\n$$\n\n$^{3}$ Here we just pointed out a mathematical inconsistency of a particular mathematical description of an attractive boson gas. It is interesting to think physically what really going to happen to a chamber of boson gas with an attractive interaction?", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 17, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 18"], "page_start": 18, "page_end": 18, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0019", "text": "Remember that U is the potential experienced by a boson and   $\\bar{V}$  characterize the interaction between two bosons. eqn (4.10.2) tells that how the ground state energy density depends on those parameters. Because the ground state is the state at zero temperature, how ground state energy depends on those parameters will tell us if there are quantum phase transitions  $^{4}$  or not. As we change the parameters that characterize a system, if the ground state energy changes smoothly, we will say that there is no quantum phase transition. If the ground state energy encounter a singularity, the singularity will represent a quantum phase transition. From eqn (4.10.2) we see that there is a quantum phase transition at U = 0 if   $\\bar{V} > 0$ . Since it is the second order derivative   $\\partial^{2}u/\\partial U^{2}$  that has a discontinuity, the phase transition is a second order phase transition. The phase for U > 0 contains no bosons since it cost energy have a boson. The phase for U < 0 has a non-zero boson density   $n = -U/\\bar{V}$ . This is because for negative U the system can lower its energy by having more bosons. But if there are too many bosons, there will be large cost of interaction energy due to the repulsive interaction between the bosons. So the density   $n = -U/\\bar{V}$  is a balance between the potential energy due to U and the interaction energy due to   $\\bar{V}$ . Later we will see that the phase with non-zero   $\\phi$  is a superfluid phase. It is also called the boson condensed phase.\n\nWe would like to remark that the eqn (4.10.2) is really the ground state energy density of the classical field theory. It is an approximation of the real ground state energy density of the interaction boson system. So we are not sure if the real ground state energy density contains a singularity or not. Even if the singularity does exist, we are not sure if it is the same type as described by eqn (4.10.2). A more careful study indicates that the real ground state energy density of the interaction boson system does have a singularity that is of the same type as in eqn (4.10.2) if the dimensions of the space is 2 or above. In 1D, the real ground state energy density has a singularity at U = 0 but the form of the singularity is different from that in eqn (4.10.2).\n\n# 4.11 Continuous phase transition and symmetry\n\n- Two mechanisms for phase transitions.\n- A continuous phase transition is a symmetry breaking transition.\n- The concept of order parameter.\n\nWe know that ground state energy (4.10.2) is obtained by minimizing the energy functional (4.9.2). U and   $\\bar{V}$  are the parameters in the energy functional. Can we have a more general and a deeper understanding when the minimum of an energy functional has a singular dependence on the parameters in the energy functional?\n\nLet us consider a simpler question: when the minimum of an energy function has a singular dependence on the parameters in the energy function? To be concrete, let us consider a real function parameterized by  $a$,  $b$,  $c$,  $d$ (where  $d > 0$):\n\n$$\nE _ {a b c d} (x) = a x + b x ^ {2} + c x ^ {3} + d x ^ {4} \\tag {4.11.1}\n$$\n\nLet  $E_0(a, b, c, d)$ be the minimum of  $E_{abcd}(x)$. How can the minimum  $E_0(a, b, c, d)$ to have a singular dependence on  $a, b, c, d$, knowing that the function  $E_{abcd}(x)$ itself has no singularity.\n\nOne mechanism for generating singularity in   $E_{0}(a,b,c,d)$  is through the “minima switching” as shown in Fig. 4.9. When   $E_{abcd}(x)$  has multiple local minima, a singularity in the global minimum   $E_{0}(a,b,c,d)$  is generated when the global minimum switch from a local minimum to another. The singularities generated by “minimum-switching” always correspond to first order phase transitions since the first order derivative of the ground state energy   $E_{0}$  is discontinuous at the singularities.\n\n$^{4}$ A quantum phase transition, by definition, is a phase transition at zero temperature.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 18, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 19"], "page_start": 19, "page_end": 19, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0020", "text": "a > 0\nE_a\nE_A\nE_B x\n\nE_a\na < 0\nE_A\nE_B\nx\n\na\nE_B E_A\n\nb > 0\nE_b\nE_0\nx\n\nb < 0\nE_b\nE_0\nx\n\nE₀\nb\n\nWhen energy function has a symmetry, there can be another mechanism for generating singularities in the ground state energy. The energy function   $E_{abcd}(x)$  has a   $x \\rightarrow -x$  symmetry if a = c = 0. For such a symmetric energy function, the single minimum at the symmetric point x = 0 for positive b splits into two minima at   $x_{0}$  and   $-x_{0}$  as b decreases below 0 (see Fig. 4.10). The shifting from the single minimum to one of the two minima generate the singularity in ground state energy   $E_{0}(b)$  at b = 0. We will call such a mechanism “minimum-splitting”. “Minimum-splitting” always generate continuous phase transitions. This is because the minima before and after the transition are connected continuously (see Fig. 4.11).\n\nWe would like to stress that the   $x \\rightarrow -x$  symmetry in the energy function   $E_{0b0d}(x)$  is crucial for the existence of the continuous transition caused by the “minimum-slitting”. Even a small symmetry breaking term, such as the ax term, will destroy the continuous transition by changing it into a smooth cross-over or a first order phase transition.\n\nWhen the energy function has a symmetry, one may expect that the minimum (i.e. the ground\n\nx\nb", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 19, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 20"], "page_start": 20, "page_end": 20, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0021", "text": "E(φ)\nφ\n\nstate) also has the symmetry. In our example Fig. 4.10, when b > 0 the ground state of   $E_{b}$  (given by x = 0) indeed has the   $x \\rightarrow -x$  symmetry (see Fig. 4.10a). However, after the phase transition (i.e. when b < 0), the new ground state no longer have the   $x \\rightarrow -x$  symmetry despite the energy function continues to have the same symmetry. Under the   $x \\rightarrow -x$  transformation, the new ground state is changed into another degenerate ground state (see Fig. 4.10b). This phenomenon of ground state having less symmetry than the energy function is called spontaneous symmetry breaking. Our example suggests that the continuous phase transition (caused by the “minimum-splitting”) always changes the symmetry of the ground state. So such a transition is also called symmetry breaking transition.\n\nOur simple example (4.11.1) reflects a general phenomenon. The picture described above also applies to more general energy functional such as eqn (4.9.2). It is a deep insight by Landau [Landau 1937; Landau and Lifschitz 1958] that the singularity in the ground energy (or the free energy) is intimately related to the spontaneous symmetry breaking. This leads to a general theory of phase and phase transition based on symmetry and symmetry breaking. Within such a theory, we can introduce an order parameter to characterize different phases. The order parameter must transform non-trivially under the symmetry transformation. In our example Fig. 4.10, we may choose x or   $x^{3}$  as the order parameter, since they both change signs under   $x \\rightarrow -x$  transformation. In the symmetry unbroken phase, the order parameter x = 0. In the symmetry breaking phase, the order parameter is nonzero   $x \\neq 0$ . The continuous phase transition is characterized by the order parameter acquiring a non-zero value. Landua's symmetry breaking theory is so general and so successful that for a long time it was believed that all continuous phase transitions are described by symmetry breaking.\n\nIn our field theory description of interacting bosons, the energy functional (4.9.2) has many symmetries, which include a  $U(1)$ symmetry  $\\phi \\to \\mathrm{e}^{\\mathrm{i}\\theta}\\phi$ and a translation symmetry  $\\pmb{x} \\to \\pmb{x} + \\pmb{a}$. The phase  $\\phi = 0$ (for  $U > 0$) is invariant under both the transformations  $\\phi \\to \\mathrm{e}^{\\mathrm{i}\\theta}\\phi$ and  $\\pmb{x} \\to \\pmb{x} + \\pmb{a}$. Thus the  $\\phi = 0$ break no symmetries. The phase  $\\phi = \\sqrt{-U / V}$ (for  $U < 0$) is invariant under the translation  $\\pmb{x} \\to \\pmb{x} + \\pmb{a}$ but not the  $U(1)$ transformation  $\\phi \\to \\mathrm{e}^{\\mathrm{i}\\theta}\\phi$. So the  $\\phi \\neq 0$ phase break the  $U(1)$ symmetry spontaneously. Under the  $U(1)$ transformation,  $\\phi = \\sqrt{-U / V}$ is changed to  $\\phi = \\mathrm{e}^{\\mathrm{i}\\theta}\\sqrt{-U / V}$ which corresponds to one of the infinitely many degenerate ground states (see Fig. 4.12). According to Landau's symmetry breaking theory, the two phases,  $\\phi \\neq 0$ and  $\\phi = 0$, having different symmetries, must be separated by a phase transition.\n\nAfter understanding the above two mechanism which generate first order phase transitions and continuous phase transitions, one may wonder “is there a third mechanism for the singularity in the ground state energy?” If you do find the third mechanism, it will represent a new type of phase transitions beyond Landau’s symmetry breaking theory!\n\n# Problem 4.11.1", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 20, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 21"], "page_start": 21, "page_end": 21, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0022", "text": "Show that if we include a term   $h \\int \\mathrm{d}^{d} x (\\phi + \\phi^{*})$  in the energy functional (4.9.2) to explicitly break the   $U(1)$  symmetry,  $^{5}$  the continuous transition cause by changing U will change into a smooth cross-over no matter how small h is.\n\n# Problem 4.11.2\n\nAdding   $h \\int \\mathrm{d}^{d} x (\\phi + \\phi^{*})$  term in eqn (4.9.2) completely breaks the   $U(1)$  symmetry and destroys the continuous phase transition. Show that adding   $h_{2} \\int \\mathrm{d}^{d} x (\\phi^{2} + c.c.)$  term in eqn (4.9.2) breaks the   $U(1)$  symmetry down to a   $Z_{2}$  symmetry, i.e. the resulting energy functional is still invariant under   $\\phi \\rightarrow -\\phi$ . Study the phase and the phase transition in the resulting   $Z_{2}$  symmetric system. Show that the   $Z_{2}$  symmetry in the energy functional allows a continuous phase transition.\n\n# 4.12 Collective modes - sound waves\n\n- Small fluctuations around the ground state have a wave-like dynamics.\n- The fluctuations around the symmetry breaking ground state have a linear dispersion for small  $\\pmb{k}$. Those fluctuations are called sound waves.\n\nFor our interacting boson system (4.9.2), the (classical) ground states in the symmetric phase   $\\phi_{grnd} = 0$  and in the symmetry breaking phase   $\\phi_{grnd} = \\sqrt{-U/\\bar{V}}$  are very different. As a result, the collective excitations above the ground states are also very different.\n\nThe collective fluctuations around the ground state are described   $\\delta\\phi = \\phi - \\phi_{grnd}$ . The equation of motion for   $\\delta\\phi$  that describes the classical dynamics of the fluctuations can be obtained by substituting   $\\phi = \\delta\\phi + \\phi_{grnd}$  into eqn (4.9.4).\n\nTo simplify our calculation, we assume the interaction potential   $V(\\boldsymbol{x} - \\boldsymbol{x}')$  is short ranged and approximate it by   $V(\\boldsymbol{x} - \\boldsymbol{x}') = g\\delta(\\boldsymbol{x} - \\boldsymbol{x}')$ . The wave equation (4.9.4) and the energy (4.9.2) are simplified to\n\n$$\n\\mathrm{i} \\dot {\\phi} (\\boldsymbol {x}, t) = \\left(- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x}} ^ {2} + U + g | \\phi (\\boldsymbol {x}) | ^ {2}\\right) \\phi (\\boldsymbol {x}, t) \\tag {4.12.1}\n$$\n\nand\n\n$$\nE _ {\\text { tot }} = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\phi^ {*} \\left(- \\frac {\\partial_ {\\boldsymbol {x}} ^ {2}}{2 m} + U + \\frac {g}{2} | \\phi | ^ {2}\\right) \\phi . \\tag {4.12.2}\n$$\n\nThe Lagrangian (4.9.3) is simplified to\n\n$$\nL = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\left(\\mathrm{i} \\phi^ {*} \\dot {\\phi} - \\phi^ {*} (- \\frac {\\partial_ {\\boldsymbol {x}} ^ {2}}{2 m} + U) \\phi - \\frac {g}{2} | \\phi | ^ {4}\\right). \\tag {4.12.3}\n$$\n\nLet us first discuss the equation of motion of   $\\delta\\phi$  in the symmetry breaking phase   $\\phi_{grnd} = \\sqrt{-U/g}$  for U < 0 (see eqn (4.10.1) and note that   $\\bar{V} = g$ ). Substituting   $\\phi = \\delta\\phi + \\sqrt{-U/g}$  into eqn (4.12.1), we find\n\n$$\n\\mathrm{i} \\dot {\\delta} \\phi = \\left(- \\frac {1}{2 m} \\partial_ {\\pmb {x}} ^ {2} - U\\right) \\delta \\phi - U \\delta \\phi^ {*}\n$$\n\nSince we are interested in low lying fluctuations, we can assume   $\\delta\\phi$  to be small. So in the above equation, we have only kept the terms linear in   $\\delta\\phi$ . Separating   $\\delta\\phi$  into real and imaginary part:\n\n$^{5}$ Changing the symmetry of the energy functional (or the Hamiltonian) is called explicit symmetry breaking, which should not be confused with the spontaneous symmetry breaking. A spontaneous symmetry breaking refers a change in the symmetry of the ground state with the symmetry of the energy functional (or Hamiltonian) unchanged.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 21, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 22"], "page_start": 22, "page_end": 22, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0023", "text": "$\\phi = \\frac{1}{\\sqrt{2}} (\\delta h + \\mathrm{i}\\delta p)$, we rewrite the above equation as\n\n$$\n- \\delta \\dot {p} = \\left(- \\frac {1}{2 m} \\partial_ {\\pmb {x}} ^ {2} - 2 U\\right) \\delta h\n$$\n\n$$\n\\delta \\dot {h} = - \\frac {1}{2 m} \\partial_ {\\pmb {x}} ^ {2} \\delta p\n$$\n\nFrom  $\\delta \\ddot{h} = -\\frac{1}{2m}\\partial_{\\pmb{x}}^{2}\\dot{\\delta} p$, we find\n\n$$\n\\delta \\ddot {h} = \\frac {1}{2 m} \\partial_ {\\boldsymbol {x}} ^ {2} \\left(- \\frac {1}{2 m} \\partial_ {\\boldsymbol {x}} ^ {2} - 2 U\\right) \\delta h \\tag {4.12.4}\n$$\n\nWe see that the equation of motion that describe the weak fluctuations is a linear wave equation. The solutions of eqn (4.12.4) are of form  $\\delta h = \\mathrm{Re}(C\\mathrm{e}^{\\mathrm{i}\\pmb{k}\\cdot \\pmb{x} - \\mathrm{i}\\omega_{\\pmb{k}}t})$. The dispersion relation of the wave is\n\n$$\n\\omega_ {\\boldsymbol {k}} = \\sqrt {\\frac {| \\boldsymbol {k} | ^ {2}}{2 m} \\left(\\frac {| \\boldsymbol {k} | ^ {2}}{2 m} - 2 U\\right)} = \\sqrt {\\frac {| \\boldsymbol {k} | ^ {2}}{2 m} \\left(\\frac {| \\boldsymbol {k} | ^ {2}}{2 m} + 2 n g\\right)} \\tag {4.12.5}\n$$\n\nwhere the   $n = |\\phi_{grnd}|^{2}$  is the boson density. For small k, we find a linear dispersion\n\n$$\n\\omega_ {\\pmb {k}} = v | \\pmb {k} |, \\qquad v = \\sqrt {\\frac {- U}{m}} = \\sqrt {\\frac {g n _ {0}}{m}}\n$$\n\nwhere v is the velocity of the fluctuating wave. Such a wave in the symmetry breaking phase is called the sound wave in the superfluid.\n\nIn the symmetric phase  $\\phi_{\\mathrm{grnd}} = 0$ for  $U > 0$, the equation of motion of  $\\delta \\phi = \\phi - \\phi_{\\mathrm{grnd}} = \\phi$ is\n\n$$\n\\mathrm{i} \\dot {\\delta} \\phi = \\left(- \\frac {1}{2 m} \\partial_ {\\pmb {x}} ^ {2} + U\\right) \\delta \\phi\n$$\n\nif we ignore the higher order terms in  $\\delta \\phi$. The solutions have a form  $\\delta \\phi = C\\mathrm{e}^{\\mathrm{i}\\pmb{k}\\cdot \\pmb{x} - \\mathrm{i}\\omega_k t}$. The dispersion relation of the wave is\n\n$$\n\\omega_ {\\boldsymbol {k}} = \\frac {| \\boldsymbol {k} | ^ {2}}{2 m} + U \\tag {4.12.6}\n$$\n\n# 4.13 Quantized collective modes - phonons\n\n- Waves = collection of oscillators.\n- Quantized waves = quantized oscillators = free phonons.\n- Phonons are a new type of bosons, completely different from the original interacting bosons that form the superfluid.\n- The emergence of phonons is the simplest example that completely new types of particles can emerge from collective fluctuations.\n\nWe know that a wave with a dispersion   $\\omega_{k}$  can be viewed as a collection of oscillators.  $^{6}$  Assuming the space to be a d-dimensional cube of volume   $V = L^{d}$ , then the wave vectors of the wave are quantized:   $\\boldsymbol{k} = \\frac{2\\pi}{L}(n_{1}, n_{2}, \\cdots)$ . Each quantized wave vector labels an oscillator. The frequency of the oscillator k is   $\\omega_{k}$ .\n\n$^{6}$ We will show explicitly how a wave is related to a collection of oscillators in section 4.14.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 22, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 23"], "page_start": 23, "page_end": 23, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0024", "text": "V(φ)\nφ\ngapless\nmode\n\nAfter identifying the wave (the collections modes) as a collection of oscillators, the quantum theory for the collective modes can then be obtained by quantize those oscillators. The eigenstates of the oscillator k is given by   $|n_{k}^{ph}\\rangle$  with energy   $n_{k}^{ph}\\omega_{k}$ . So the energy eigenstates for the collection of the oscillator are labeled by a set of integers   $\\{n_{k}^{ph}\\}$ :   $|\\{n_{k}^{ph}\\}\\rangle$ , where k runs over all quantized wave vectors. The energy of such a state   $|\\{n_{k}^{ph}\\}\\rangle$  is   $\\sum_{k} n_{k}^{ph}\\omega_{vk}$  where   $\\sum_{k}$  sums over all the quantized wave vectors.\n\nIn the symmetry breaking (the superfluid) phase for U < 0,   $\\omega_{k}$  is given by eqn (4.12.5). In the symmetric phase for U > 0,   $\\omega_{k}$  is given by eqn (4.12.6). The low energy eigenstates are labeled by a set of integers   $\\{n_{k}^{ph}\\}$ . The energies of those eigenstates are   $\\sum_{k} n_{k}^{ph} \\omega_{vk}$ . This way we obtain the low energy eigenstates and their eigenvalues of the very complicated interacting bosons system described by eqn (4.9.1) (or eqn (4.8.5)). Through the classical picture, we are able to obtain the low lying energy eigenstates and their eigenvalues of the Hamiltonian eqn (4.9.1)!\n\nIf we interpreted   $n_{k}^{ph}$  as the occupation number of a kind of bosons at the k-level, then the collection of the quantum oscillators can also be viewed as the system of free bosons with a dispersion   $\\omega_{k}$  (see section 4.3). This is quite amazing. We start with an interacting bosons. At the end, we find the low energy excitations of the interacting boson system are described by a free boson system. The distinguish the two kinds of bosons, we will call the free bosons that describe the low lying excitations emergent bosons.\n\nBut what are the emergent bosons? In the symmetric phase, the emergent bosons have the dispersion   $\\omega_{k} = \\frac{|k|^{2}}{2m} + U$ , which is exactly the same is the dispersion of the original bosons. In fact, in the symmetric phase, the emergent bosons are the original bosons. This is because the ground state for the symmetry phase,   $|\\{n_{k}^{ph} = 0\\}\\rangle$  is the state with no original bosons.   $n_{k}^{ph}$  in this case is identical to the occupation number   $n_{k}$  of the original bosons. For the low energy excitations, only few   $n_{k}^{ph}$ 's are non-zero, which correspond to a dilute gas of the original bosons. In this limit, the interactions between the original bosons can be ignore and the original bosons become the free emergent bosons.\n\nHowever, in the symmetry breaking phase, the emergent bosons have a linear dispersion for small k's, which is very different from the original bosons. The occupation numbers   $n_{k}^{ph}$  of the emergent bosons is not related to the occupation numbers   $n_{k}$  of the original bosons. In fact, since   $\\phi_{grnd} \\neq 0$  and the original bosons have a finite density, the interaction between the original bosons cannot be ignored. In this case, the occupation numbers   $n_{k}$  of the original bosons are not even well defined, i.e. the energy eigenstates do not have a definite occupation numbers   $n_{k}$  (although they do have a definite occupation numbers   $n_{k}^{ph}$  for the emergent bosons). Since the emergent bosons are completely different from the original bosons, we will give the emergent bosons a new name: phonons.\n\nWe note that phonons are gapless excitations above the superfluid ground state. We like to", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 23, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 24"], "page_start": 24, "page_end": 24, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0025", "text": "pointed out that Gapless excitations are very rare in nature and in condensed matter systems. Therefore, if we see gapless excitations, we should to ask why do they exist?\n\nOne mechanism for gapless excitations is spontaneously breaking of a continuous symmetry. As discussed in section 4.11, the superfluid phase spontaneously break the  $U(1)$ symmetry: the energy functional has the  $U(1)$ symmetry:  $E_{\\mathrm{tot}}[\\phi (\\pmb {x})] = E_{\\mathrm{tot}}[\\mathrm{e}^{\\mathrm{i}\\theta}\\phi (\\pmb {x})]$ while the ground state does not:  $\\phi_{\\mathrm{grnd}}\\neq \\mathrm{e}^{\\mathrm{i}\\theta}\\phi_{\\mathrm{grnd}}$. Only in this case, do the gapless excitations exist.\n\nIntuitively if a symmetry (continuous or discrete) is spontaneously broken, then the ground states must be degenerate (see Figs. 4.10b and 4.12). The different ground states are related by the symmetry transformations. So the spontaneous breaking of continuous symmetry gives rise to a continuous manifold of degenerate ground states. The fluctuations between the degenerate ground states correspond to the gapless excitations (see Fig. 4.13). Nambu and Goldstone have proved a general theorem: if a continuous symmetry is spontaneously broken in a phase, the phase must contain gapless excitations [Nambu 1960; Goldstone 1961]. Those gapless excitations are usually called the Nambu-Goldstone modes. The gapless phonon in the superfluid is a Nambu-Goldstone mode. In the next section, we will give a explicit discussion of the relation between the spontaneous  $U(1)$ symmetry breaking and the gapless phonons.\n\n# 4.14 *An oscillator picture for the sound wave\n\n- A derivation of the oscillator picture for the sound wave allows us to express the physical quantities of the original bosons, such as boson density, in terms of the oscillator variables. This will allow us to calculate physical properties of interacting bosons using simple oscillators.\n\nThe sound waves in the symmetry breaking phase are described by a collection of oscillators. To show the explicit relation between the sound waves and a collection of oscillators, we start with the field theory Lagrangian (4.12.3) for interacting bosons. Since the sound waves are described by the small fluctuations  $\\delta \\phi = \\phi -\\phi_{\\mathrm{grnd}}$, we can rewrite (4.12.3) in terms of  $\\delta \\phi$ to obtain the Lagrangian for the sound waves. However, to make the  $U(1)$ symmetry:  $\\phi \\rightarrow \\mathrm{e}^{\\mathrm{i}\\varphi}\\phi$ more explicit, we will instead use  $(\\theta ,\\rho)$ to describe the fluctuations around the ground state  $\\phi_{\\mathrm{grnd}} = \\sqrt{-U / g}$.  $(\\theta ,\\delta n)$ are defined through\n\n$$\n\\phi (\\pmb {x}, t) = \\sqrt {n _ {0} + \\delta n (\\pmb {x} , t)} \\mathrm{e} ^ {\\mathrm{i} \\theta (\\pmb {x}, t)}\n$$\n\nwhere   $n_{0} = -U/g$  is the boson density in the ground state.\n\nTo the quadratic order in  $(\\theta, \\delta n)$, eqn (4.12.3) becomes\n\n$$\nL = \\int d ^ {d} \\pmb {x} [ - (n _ {0} + \\delta n) \\partial_ {t} \\theta - \\frac {n _ {0} (\\partial_ {\\pmb {x}} \\theta) ^ {2}}{2 m} - \\frac {(\\partial_ {\\pmb {x}} \\delta n) ^ {2}}{8 m n _ {0}} - \\frac {g}{2} \\delta n ^ {2} ]. \\tag {4.14.1}\n$$\n\nThe invariance of eqn (4.12.3) under the  $U(1)$ transformation  $\\phi \\to \\mathrm{e}^{\\mathrm{i}\\varphi}\\phi$ implies that eqn (4.14.1) is invariant under  $\\theta (\\pmb {x},t)\\rightarrow \\theta (\\pmb {x},t) + \\varphi$. This is why the Lagrangian (4.14.1) contains no  $\\theta^2$ term. The absence of the  $\\theta^2$ term, as implied by the  $U(1)$ symmetry, will leads to gapless excitations.\n\nIntroducing  $^{7}$\n\n$$\n\\delta \\varphi = \\frac {1}{2 \\sqrt {n _ {0}}} \\delta n + \\mathrm{i} \\sqrt {n _ {0}} \\theta\n$$\n\nwe can rewrite eqn (4.14.1) as\n\n$$\nL = \\int \\mathrm{d} ^ {d} \\boldsymbol {x} \\left(\\mathrm{i} \\delta \\varphi^ {*} \\delta \\dot {\\varphi} - \\delta \\varphi^ {*} \\left(- \\frac {\\partial_ {\\boldsymbol {x}} ^ {2}}{2 m} + g n _ {0}\\right) \\delta \\varphi - g n _ {0} \\mathrm{Re} (\\delta \\varphi^ {2})\\right). \\tag {4.14.2}\n$$\n\n$^{7}$ δφ defined this way is equal to δφ up to the linear order in δφ: δφ = δφ + O(δφ²).", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 24, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 25"], "page_start": 25, "page_end": 25, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0026", "text": "where we have dropped some total time derivative terms. The equal coefficient in the front of the two terms   $\\delta\\varphi^{*}\\delta\\varphi$  and   $\\mathrm{Re}(\\delta\\varphi^{2})$  is a consequence of the   $U(1)$  symmetry.\n\nLet us expand the above Lagrangian in terms of k-modes. We can expand   $\\phi(\\boldsymbol{x})$  as\n\n$$\n\\delta \\varphi (\\pmb {x}) = \\sum_ {\\pmb {k}} \\mathcal {V} ^ {- 1 / 2} \\mathrm{e} ^ {\\mathrm{i} \\pmb {k} \\cdot \\pmb {x}} \\delta \\varphi_ {\\pmb {k}}\n$$\n\nIn terms of   $\\varphi_{k}$ , the Lagrangian for the sound wave takes a form\n\n$$\nL = \\sum_ {\\pmb {k}} \\left(\\mathrm{i} \\delta \\varphi_ {\\pmb {k}} ^ {*} \\delta \\dot {\\varphi} _ {\\pmb {k}} - \\delta \\varphi_ {\\pmb {k}} ^ {*} \\left(\\frac {| \\pmb {k} | ^ {2}}{2 m} + g n _ {0}\\right) \\delta \\varphi_ {\\pmb {k}} - \\frac {g n _ {0}}{2} (\\delta \\varphi_ {- \\pmb {k}} \\delta \\varphi_ {\\pmb {k}} + \\delta \\varphi_ {- \\pmb {k}} ^ {*} \\delta \\varphi_ {\\pmb {k}} ^ {*})\\right). (4. 1 4. 3)\n$$\n\nIf we ignore the term  $\\frac{gn_0}{2} (\\delta \\varphi_{-k}\\delta \\varphi_k + \\delta \\varphi_{-k}^*\\delta \\varphi_k^*)$, then each term in the sum  $\\sum_{k}$\n\n$$\nL _ {\\boldsymbol {k}} = \\mathrm{i} \\delta \\varphi_ {\\boldsymbol {k}} ^ {*} \\delta \\dot {\\varphi} _ {\\boldsymbol {k}} - \\delta \\varphi_ {\\boldsymbol {k}} ^ {*} \\left(\\frac {| \\boldsymbol {k} | ^ {2}}{2 m} + g n _ {0}\\right) \\delta \\varphi_ {\\boldsymbol {k}} \\tag {4.14.4}\n$$\n\nwill describe a decoupled harmonic oscillator where the real part of   $\\delta\\varphi_{k}$  corresponds to the coordinate and the imaginary part of   $\\delta\\varphi_{k}$  corresponds to the momentum of the oscillator. The term   $\\frac{g}{2}(\\delta\\varphi_{-k}\\delta\\varphi_{k}+\\delta\\varphi_{-k}^{*}\\delta\\varphi_{k}^{*})$  couples the oscillator k to the oscillator -k. So eqn (4.14.3) describes a collection of coupled oscillators.\n\nBut we can choose a different set of variables to obtain a set of decoupled oscillators. Since the mixing is between the k-mode and the -k-mode only, let us introduce\n\n$$\na _ {\\pmb {k}} = u _ {\\pmb {k}} \\varphi_ {\\pmb {k}} + v _ {\\pmb {k}} \\delta \\varphi_ {- \\pmb {k}}\n$$\n\nOne can show that, up to a total time derivative term,\n\n$$\n\\sum_ {\\boldsymbol {k}} a _ {\\boldsymbol {k}} ^ {*} \\dot {a} _ {\\boldsymbol {k}} = \\sum_ {\\boldsymbol {k}} \\varphi_ {\\boldsymbol {k}} ^ {*} \\dot {\\varphi} _ {\\boldsymbol {k}}\n$$\n\nif\n\n$$\n| u _ {\\pmb {k}} | ^ {2} - | v _ {\\pmb {k}} | ^ {2} = 1, \\qquad u _ {\\pmb {k}} ^ {*} v _ {\\pmb {k}} - u _ {- \\pmb {k}} v _ {- k} ^ {*} = 0. \\tag {4.14.5}\n$$\n\nOne can also show that\n\n$$\n\\sum_ {\\boldsymbol {k}} E _ {\\boldsymbol {k}} a _ {\\boldsymbol {k}} ^ {*} a _ {\\boldsymbol {k}} = \\sum_ {\\boldsymbol {k}} \\left(\\delta \\varphi_ {\\boldsymbol {k}} ^ {*} \\left(\\frac {| \\boldsymbol {k} | ^ {2}}{2 m} + g n _ {0}\\right) \\delta \\varphi_ {\\boldsymbol {k}} + \\frac {g n _ {0}}{2} (\\delta \\varphi_ {- \\boldsymbol {k}} \\delta \\varphi_ {\\boldsymbol {k}} + \\delta \\varphi_ {- \\boldsymbol {k}} ^ {*} \\delta \\varphi_ {\\boldsymbol {k}} ^ {*})\\right) \\tag {4.14.6}\n$$\n\nif one chooses\n\n$$\nu _ {\\pmb {k}} = \\sqrt {\\frac {\\frac {| \\pmb {k} | ^ {2}}{2 m} + g n _ {0}}{2 E _ {\\pmb {k}}} + \\frac {1}{2}}, v _ {\\pmb {k}} = \\sqrt {\\frac {\\frac {| \\pmb {k} | ^ {2}}{2 m} + g n _ {0}}{2 E _ {\\pmb {k}}} - \\frac {1}{2}},\n$$\n\n$$\nE _ {\\boldsymbol {k}} = \\sqrt {\\left(\\frac {| \\boldsymbol {k} | ^ {2}}{2 m} + g n _ {0}\\right) ^ {2} - (g n _ {0}) ^ {2}}. \\tag {4.14.7}\n$$\n\nSo in terms of   $a_{k}$ , the Lagrangian has a form\n\n$$\nL = \\sum_ {\\pmb {k}} \\left(\\mathrm{i} a _ {\\pmb {k}} ^ {*} \\dot {a} _ {\\pmb {k}} - E _ {\\pmb {k}} a _ {\\pmb {k}} ^ {*} a _ {\\pmb {k}}\\right)\n$$\n\nwhich describes a collection of decoupled oscillators.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 25, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 26"], "page_start": 26, "page_end": 26, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e:page-0027", "text": "To show that the term\n\n$$\nL _ {\\pmb {k}} = \\mathrm{i} a _ {\\pmb {k}} ^ {*} \\dot {a} _ {\\pmb {k}} - E _ {\\pmb {k}} a _ {\\pmb {k}} ^ {*} a _ {\\pmb {k}}\n$$\n\ndescribes a single harmonic oscillator, we introduce   $a_{\\mathbf{k}} = 2^{-1/2}(X_{\\mathbf{k}} + \\mathrm{i}P_{\\mathbf{k}})$  and rewrite   $L_{k}$  as (up to a total time derivative term)\n\n$$\nL _ {\\pmb {k}} = P _ {\\pmb {k}} \\dot {X} _ {\\pmb {k}} - \\left(\\frac {P _ {\\pmb {k}} ^ {2}}{2 M _ {\\pmb {k}}} + \\frac {K _ {\\pmb {k}} X _ {\\pmb {k}} ^ {2}}{2}\\right)\n$$\n\nwhere   $M_{k} = 1/E_{k}$  and   $K_{k} = E_{k}$ . We see that   $L_{k}$  is a phase-space Lagrangian that describes an oscillator of mass   $M_{k}$  and spring constant   $K_{k}$ . The oscillation frequency is\n\n$$\n\\omega_ {\\pmb {k}} = \\sqrt {\\frac {K _ {\\pmb {k}}}{M _ {\\pmb {k}}}} = E _ {\\pmb {k}} = \\sqrt {\\left(\\frac {| \\pmb {k} | ^ {2}}{2 m} + g n _ {0}\\right) ^ {2} - (g n _ {0}) ^ {2}}\n$$\n\nwhich agrees with eqn (4.12.5).\n\nWe note that in the expression of   $E_{k}$  (4.14.7), the first   $gn_{0}$  comes from the coefficient in front of the   $\\delta\\varphi^{*}\\delta\\varphi$  term in eqn (4.14.2). The second   $gn_{0}$  comes from the coefficient in front of the   $\\mathrm{Re}(\\delta\\varphi\\delta\\varphi)$  term in eqn (4.14.2). We see that the equal coefficient in front of   $\\delta\\varphi^{*}\\delta\\varphi$  and   $\\mathrm{Re}(\\delta\\varphi^{2})$  makes   $E_{k}\\to0$  as   $k\\to0$ . The   $U(1)$  symmetry protects the gapless excitations in the superfluid phase.\n\n# Problem 4.14.1\n\nShow eqn (4.14.5) and eqn (4.14.6).\n\n# Problem 4.14.2\n\nAdding   $h_{2}\\int\\mathrm{d}^{d}\\boldsymbol{x}\\left(\\phi^{2}+c.c.\\right)$  term to eqn (4.12.3) breaks the   $U(1)$  symmetry down to a   $Z_{2}$  symmetry. The resulting system still has two phases, one breaks the   $Z_{2}$  symmetry and the other does not. Show that the collective excitations in both phases have an energy gap. Show that the energy gap of the excitations approach to zero as we approach the continuous phase transition between the two phases.", "source": "mit-ocw", "source_doc_id": "0011-quantum-theory-for-interacting-bosons-29b81015-843f1f4e", "source_title": "Chapter 4: Boson systems", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 26, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 4: Boson systems", "Page 27"], "page_start": 27, "page_end": 27, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0001", "text": "# Chapter 1\n\n# Statistical Ensembles\n\n# 1.1 Principle of statistical physics and ensembles\n\nKey points:\n\n- All possible states appear with an equal probability.\n\nStatistical systems are complex systems. We do not know all the information that is needed to completely characterize the systems. For example, the liter of gas may contain   $10^{22}$  atoms. The completely characterize such a system we need to known the three components of the velocity for each atom and the three components of the position for each atom. It is impossible to obtain   $6 \\times 10^{22}$  real numbers to completely characterize the gas.\n\nHowever, not knowing all the information needed to characterize gas does not prevent us to develop a theory of gas. This is because we are only interested in some average properties of gas such as the pressure, volume, temperature. Those properties do not depend on every little details of each atoms. Not knowing every thing about the atoms does not prevent us from calculating those properties. This is the kind of problems facing statistical physics. In statistical physics we try to understand the properties of a complex system without know all the information of the systems. This is possible since the properties we are interested in do not depend on the details of the system.\n\nIn statistical physics there is only one principle: All possible states appear with an equal probability. Let us explain what do we mean by the above statement. Suppose we know certain properties of a complex systems. But those properties do not characterize the system completely. That means the system has a numbers of states that all have the same properties. Thus even after knowing those properties, we still do not know, among those possible states, which state the system is in. According to the principle of statistical physical, we say all the possible states are equally likely.\n\nBut the system can only be in one state at a given time. What do we mean by “all the possible states are equally likely”? There are two points of view. In the first point of view, we may imagine we have many copies of the system, all have the same properties. But each copy may be in a different possible states. Then “equally likely” means that each possible state appear the same number of times among the copies of the system. The copies of the system is called ensemble. We have to have an ensemble to even define the probabilities. Under the first interpretation, statistical physical is a science that deal with ensembles, rather than individual systems.\n\nThe second point of view only apply to the situation where the properties of the system is independent of time. In this case we interpret “equally likely” as that all the possible states appear", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0002", "text": "for the same amount of time during a long period of time. The second point of view is related to the first point of view if we view the system at different times as the different copies of the system.\n\nThe second point of view may be equivalent to the first point of view. The two points of view are equivalent only when the system can visit all the possible states, many times, during the long period of time. This is the ergodic hypothesis. Not all systems are ergodic. In this class, we will take the first point of view. We regard the statistical physics as a theory for ensembles. We will apply the theory for ensembles to individual systems, assuming the systems are ergodic.\n\n# 1.2 Microcanonical ensemble\n\nA microcanonical ensemble is an ensemble formed by isolated systems. All the systems in the ensemble have the same energy (and possibly some other properties). Here by “same energy” we really mean all systems has an energy which lies within a small window between E and   $E + \\Delta E$ .\n\n# 1.2.1 Number of states and entropy\n\nA simple example: N spins in magnetic field. Energy:   $E_{\\uparrow} = \\epsilon_{0}/2$  and   $E_{\\downarrow} = -\\epsilon_{0}/2$ .\n\nHow many states with energy E (ie with energy between   $E - \\epsilon_{0}$  and E)? How many states with   $M = \\frac{E}{\\epsilon_{0}} + \\frac{N}{2}$  up-spins and N - M down-spins? There are total of   $2^{N}$  states\n\n$$\n(1 + 1) ^ {N} = 1 ^ {0} 1 ^ {N} + N 1 ^ {1} 1 ^ {N - 1} + \\ldots + C _ {N} ^ {M} 1 ^ {M} 1 ^ {N - M} + \\ldots\n$$\n\n$$\n(\\uparrow + \\downarrow) ^ {N} = \\uparrow^ {0} \\downarrow^ {N} + N \\uparrow^ {1} \\downarrow^ {N - 1} + \\dots + C _ {N} ^ {M} \\uparrow^ {n} \\downarrow^ {N - M} + \\dots \\tag {1.2.1}\n$$\n\nwhere\n\n$$\nC _ {N} ^ {M} = \\frac {N !}{M ! (N - M) !} \\tag {1.2.2}\n$$\n\n$C_N^M$ is the number of ways to pick  $M$ objects from  $N$ objects. We find\n\n$$\n\\text { number   of   states   with } M \\uparrow \\operatorname{spins} = C _ {N} ^ {M} \\tag {1.2.3}\n$$\n\nThe entropy is a function of energy E which is defined as\n\n$$\nS (E) = k _ {B} \\ln (\\text {number of states}) = k _ {B} \\ln C _ {N} ^ {E / \\epsilon_ {0}} \\tag {1.2.4}\n$$\n\nWhen  $n$ is large\n\n$$\n\\ln (n!) = \\left(n + \\frac {1}{2}\\right) \\ln (n + 1) - (n + 1) + \\frac {1}{2} \\ln (2 \\pi) + \\dots \\tag {1.2.5}\n$$\n\nThus (see Fig. 1.1)\n\n$$\n\\begin{array}{l} k _ {B} ^ {- 1} S (E) = \\ln C _ {N} ^ {M} \\\\ \\approx N \\ln N - M \\ln M - (N - M) \\ln (N - M) \\\\ = - M \\ln (\\frac {M}{N}) - (N - M) \\ln (\\frac {N - M}{N}) \\\\ = N (- f _ {\\uparrow} \\ln f _ {\\uparrow} - f _ {\\downarrow} \\ln f _ {\\downarrow}) \\tag {1.2.6} \\\\ \\end{array}\n$$", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 1, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 2"], "page_start": 2, "page_end": 2, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0003", "text": "| E/E₀ = ε/ε₀ | S(E)/N |\n| ----------- | ------ |\n| -0.5        | 0      |\n| 0           | 1      |\n| 0.5         | 0      |\n\nwhere   $f_{\\uparrow}$  (or   $f_{\\downarrow}$ ) is the probability for a spin to be up (or down). Using   $f_{\\uparrow} = \\frac{M}{N} = \\frac{1}{2} + \\frac{E}{E_0}$  and   $f_{\\downarrow} = \\frac{M}{N} = \\frac{1}{2} - \\frac{E}{E_0}$  where   $E_0 = N \\epsilon_0$ , we find\n\n$$\nk _ {B} ^ {- 1} S (E) = N \\left[ - \\left(\\frac {1}{2} + \\frac {E}{E _ {0}}\\right) \\ln \\left(\\frac {1}{2} + \\frac {E}{E _ {0}}\\right) - \\left(\\frac {1}{2} - \\frac {E}{E _ {0}}\\right) \\ln \\left(\\frac {1}{2} - \\frac {E}{E _ {0}}\\right) \\right] \\tag {1.2.7}\n$$\n\nClearly, from the definition, the physical meaning of the entropy is\n\n$$\n\\text { number   of   states   with   energy } E = e ^ {S (E) / k _ {B}} \\tag {1.2.8}\n$$\n\n# 1.2.2 Concept of temperature\n\nTo introduce the concept of temperature, let us put two systems of spins together. System 1 has   $N_{1}$  spins and System 2 has   $N_{2}$  spins. Let   $\\tilde{E}_{1,2}$  be the energies of the two systems at the beginning. The total energy is   $E = \\tilde{E}_{1} + \\tilde{E}_{2}$ . If we allow the two systems to exchange their energy, then the spins in the two systems may wondering around and sample all the possible states with total energy E. The question is what is the probability for system 1 to have an energy   $E_{1}$\n\nThe number states with system 1 having an energy  $E_{1}$ is\n\n$$\nN (E _ {1}) = e ^ {k _ {B} ^ {- 1} S _ {1} (E _ {1})} e ^ {k _ {B} ^ {- 1} S _ {2} (E - E _ {1})} = e ^ {k _ {B} ^ {- 1} [ S _ {1} (E _ {1}) + S _ {2} (E - E _ {1}) ]} \\tag {1.2.9}\n$$\n\nEvery possible states are equally possible. Probability for system 1 to have an energy   $E_{1}$\n\n$$\nP (E _ {1}) \\propto e ^ {k _ {B} ^ {- 1} [ S _ {1} (E _ {1}) + S _ {2} (E - E _ {1}) ]} \\tag {1.2.10}\n$$\n\nFrom Fig. 1.2, we see that when  $N \\to \\infty$,  $P(E_1)$ is almost like a  $\\delta$-function. We can say for sure that the energy of system 1 has such a value  $\\bar{E}_1$ that it maximizes the total entropy  $S_1(E_1) + S_2(E - E_1)$, or\n\n$$\nS _ {1} ^ {\\prime} (\\bar {E} _ {1}) = S _ {2} ^ {\\prime} (E - \\bar {E} _ {1}) \\tag {1.2.11}\n$$\n\nIf   $\\tilde{E}_{1}$  at the beginning is not equal to   $\\bar{E}_{1}$ , then after we bring the two spin systems together,   $E_{1}$  will shift from   $\\tilde{E}_{1}$  to   $\\bar{E}_{1}$ . We see that Eq. (1.2.11) is a condition for equilibrium. It is also maximum entropy condition. We have derived the second law of thermodynamics: as an isolated system approach to the equilibrium state, its entropy always increase (if we define the entropy as in Eq. (1.2.4)).\n\nIf we define the temperature as\n\n$$\n\\frac {1}{T} = \\beta k _ {B} = \\frac {\\partial S (E)}{\\partial E} \\tag {1.2.12}\n$$", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 2, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 3"], "page_start": 3, "page_end": 3, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0004", "text": "| ε/ε₀ | β     | T     |\n|------|-------|-------|\n| -0.5 | High  | Low   |\n| 0.0  | Low   | 0     |\n| 0.5  | High  | Low   |\n\nthen the equilibrium condition Eq. (1.2.11) becomes\n\n$$\nT _ {1} = T _ {2} \\tag {1.2.13}\n$$\n\nFor our spin system\n\n$$\n\\frac {1}{T} = \\beta k _ {B} = k _ {B} \\frac {1}{\\epsilon_ {0}} \\ln \\left(\\frac {\\frac {1}{2} \\epsilon_ {0} + \\epsilon}{\\frac {1}{2} \\epsilon_ {0} - \\epsilon}\\right) \\tag {1.2.14}\n$$\n\nwhere   $\\epsilon = E/N$  is the average energy per spin.\n\n# 1.2.3 Curie's law\n\nFor a spin-1/2 system in magnetic field B,   $\\epsilon_{0}=g\\mu_{B}B$ . The total magnetic energy is MB where M is the magnetic moment. The energy per spin is   $\\epsilon=MB/N$ . From Eq. (1.2.14), we find a relation between the B-field induced magnetic moment M and the temperature T\n\n$$\n\\frac {1}{T} = \\frac {k _ {B}}{g \\mu_ {B} B} \\ln \\left(\\frac {g \\mu_ {B} N - 2 \\mathcal {M}}{g \\mu_ {B} N + 2 \\mathcal {M}}\\right)\n$$", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 3, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 4"], "page_start": 4, "page_end": 4, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0005", "text": "| T (K) | experiment | Curie law |\n|-------|------------|-----------|\n| 0     | 0.007      | 0.008     |\n| 50    | 0.008      | 0.007     |\n| 100   | 0.006      | 0.005     |\n| 150   | 0.004      | 0.003     |\n| 200   | 0.003      | 0.002     |\n| 250   | 0.002      | 0.001     |\n| 300   | 0.001      | 0.001     |\n\nFor  $B \\ll \\frac{k_{B}T}{g\\mu_{B}}$, we have  $\\mathcal{M} \\ll g\\mu_{B}N$ and\n\n$$\n\\mathcal {M} = \\frac {g ^ {2} \\mu_ {B} ^ {2} N}{4 k _ {B} T} B\n$$\n\nWe find magnetic susceptibility  $\\chi = \\frac{g^2\\mu_B^2N}{4k_BT}\\propto 1 / T$ . This is the Curie's law (see Fig. 1.4).\n\n# 1.2.4 Properties of entropy\n\n# Entropy is an extensive quantity\n\nFrom\n\n$$\nk _ {B} ^ {- 1} S (E) = N \\left[ - \\left(\\frac {1}{2} + \\frac {\\epsilon}{\\epsilon_ {0}}\\right) \\ln \\left(\\frac {1}{2} + \\frac {\\epsilon}{\\epsilon_ {0}}\\right) - \\left(\\frac {1}{2} - \\frac {\\epsilon}{\\epsilon_ {0}}\\right) \\ln \\left(\\frac {1}{2} - \\frac {\\epsilon}{\\epsilon_ {0}}\\right) \\right] \\tag {1.2.15}\n$$\n\nwe see that entropy is proportional to N, the size of system. Thus S is extensive quantity. In contrast,   $\\epsilon$ , as the average energy per spin, is intensive quantity. The total energy E is a extensive quantity and the temperature T is an intensive quantity.\n\n# Entropy and energy window\n\nFrom the definition of entropy\n\n$$\nS (E, \\Delta E) = k _ {B} \\ln (\\text { number   of   states   with   energy   between } E \\text { and } E + \\Delta E) \\tag {1.2.16}\n$$\n\nwe see that entropy also depend on the energy window   $\\Delta E$ . However, in the thermodynamical limit   $N \\to \\infty$ , such a dependence can be dropped and we can regard S as a function of E only.\n\nTo see this, we consider\n\n$$\n\\begin{array}{l} S (E, \\alpha \\Delta E) = k _ {B} \\ln (\\text { number   of   states   with   energy   between } E \\text { and } E + \\alpha \\Delta E) \\\\ = k _ {B} \\ln [ \\alpha \\times (\\text { number   of   states   with   energy   between } E \\text { and } E + \\alpha \\Delta E) ] \\\\ = S (E, \\Delta E) + k _ {B} \\ln \\alpha \\tag {1.2.17} \\\\ \\end{array}\n$$\n\nSince   $S(E, \\Delta E) \\sim N$ , as long as   $\\alpha = O(N^{n})$ ,   $k_{B} \\ln \\alpha$  term can be dropped.", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 4, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 5"], "page_start": 5, "page_end": 5, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0006", "text": "E₁\nE₂\n(a)\n\n2 \\overline{E}\n(b)\n\nE\n(c)\n\nΓ( E₁ ) Γ( E₂ )\nE₁\nΓ( E̅ ) Γ( E̅ )\n¯E\n\n# Additive property of entropy\n\nConsider two systems both with N spins. The first system has total energy   $E_{1}$  and the second   $E_{2}$ . The first system has   $\\Gamma_{1}=C_{N}^{E_{1}/\\epsilon_{0}}\\equiv\\Gamma(E_{1})$  possible states and the second   $\\Gamma_{2}=C_{N}^{E_{2}/\\epsilon_{0}}=\\Gamma(E_{2})$  possible states.\n\nIf we put the two systems together, but forbid any exchange of energy between them (see Fig. 1.5a), then the combined system will has  $\\Gamma = \\Gamma_{1}\\Gamma_{2}$ possible states. The entropy of the combined system  $S = k_{B}\\ln \\Gamma$ is the sum of the sub systems\n\n$$\nS = S _ {1} + S _ {2} \\tag {1.2.18}\n$$\n\nIf we allow the two system to exchange energy, the two systems will reach an equilibrium state. The subsystem will have the same average energy   $\\bar{E} = (E_{1} + E_{2})/2$  in the equilibrium state. The equilibrium state of the combined system will have a total energy   $2\\bar{E}$ . The number of possible states become   $\\bar{\\Gamma} = C_{2N}^{2\\bar{E}/\\epsilon_{0}}$ . Since   $\\bar{\\Gamma} = \\sum_{E_{1}} \\Gamma(E_{1}) \\Gamma(2\\bar{E} - E_{1})$ , it is clear that the   $\\bar{\\Gamma} > \\Gamma = \\Gamma(E_{1}) \\Gamma(E_{2})$  and the equilibrium state has a higher entropy (see Fig. 1.6). Thus reaching equilibrium always increase entropy (the second law of thermodynamics).\n\nAfter the two systems reach the equilibrium, we now forbid the energy exchange. The total number states is then reduced to   $\\bar{\\Gamma}^{\\prime} = \\Gamma(\\bar{E})\\Gamma(\\bar{E})$ . We like to show that   $\\ln\\Gamma(\\bar{E})\\Gamma(\\bar{E}) = \\ln\\bar{\\Gamma}$  in the thermodynamical limit, ie the system Fig. 1.5b and the system Fig. 1.5c have the same entropy. As the maximum of the   $\\Gamma(E_{1})\\Gamma_{2}(\\bar{E}-E_{1})$ , we find   $\\bar{\\Gamma}^{\\prime} > \\bar{\\Gamma}/2N$ , where 2N is the number of possible distinct values of   $E_{1}$ . We also have   $\\bar{\\Gamma}^{\\prime} < \\bar{\\Gamma}$ . Thus\n\n$$\n\\ln \\bar {\\Gamma} > \\ln \\bar {\\Gamma} ^ {\\prime} > \\ln (\\bar {\\Gamma} / 2 N) \\tag {1.2.19}\n$$\n\nor\n\n$$\n\\bar {S} > \\bar {S} ^ {\\prime} > \\bar {S} - k _ {B} \\ln (2 N) \\tag {1.2.20}\n$$", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 5, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 6"], "page_start": 6, "page_end": 6, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0007", "text": "E\n4 states\n7 states\n\n4 states\n4 states\n\nSince   $S\\bar{S}$  and   $\\bar{S}'$  is of order N. In large N limit, we can regard   $\\bar{S} = \\bar{S}'$ .\n\nFrom Fig. 1.6 we also see that as system go from Fig. 1.5a to the equilibrium state Fig. 1.5b or Fig. 1.5c, the entropy of the system is maximized. Or equilibrium state has maximum entropy.\n\n# Reversible and irreversible processes\n\nThe system in Fig. 1.5b is evolved from that in Fig. 1.5a. Thus there are only  $\\Gamma(E_1)\\Gamma(E_2)$ possible initial states, and there will be only  $\\Gamma(E_1)\\Gamma(E_2)$ possible final states. Those the system Fig. 1.5b has  $\\bar{\\Gamma}$ states with energy  $2\\bar{E}$, it will only be in one of  $\\Gamma(E_1)\\Gamma(E_2)$ possible final states. But we have no clue about which are the  $\\Gamma(E_1)\\Gamma(E_2)$ possible final states. We lost the information. We only know the total energy of the system, and we only know the state can be in one of the  $\\bar{\\Gamma}$ states. This is how the entropy get increased.\n\nThe evolution from Fig. 1.5a to Fig. 1.5b is also presented in Fig. 1.7a. The Fig. 1.7b represent a reversible (or adiabatic) evolution, say, caused by a change in   $\\epsilon_{0}$ . We see that reversible (or adiabatic) processes do not change the entropy, since the number of possible states is not changed.\n\n# 1.3 Application to classical ideal gas\n\nEach degree of freedom is described by a point in phase space   $(q,p)$ . A particle has three degrees of freedom and its state is described by   $(x,p_{x},y,p_{y},z,p_{z})$ .\n\nConsider a N-particle system. How many states with total energy below E. The answer is infinity. We need quantum physics to get a sensible result. Each state in a degree freedom occupies a finite area   $\\Delta q\\Delta p = h$ . For the N particle system, the phase space is 6N dimensional. Each   $h^{3N}$  volume in the 6N dimensional phase space correspond to one state. Thus the number of states with total energy below E is given by\n\n$$\nN _ {<  } (E) = \\frac {1}{h ^ {3 N}} \\int_ {\\sum \\pmb {p} _ {i} ^ {2} / 2 m <   E} d ^ {3 N} \\pmb {q} d ^ {3 N} \\pmb {p} = \\frac {V ^ {N} S _ {3 N} (\\sqrt {2 m E}) ^ {3 N} / 3 N}{h ^ {3 N}} \\tag {1.3.1}\n$$", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 6, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 7"], "page_start": 7, "page_end": 7, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0008", "text": "where  $S_{n}$ is the solid angle in  $n$ dimension and  $\\int_0^R S_n r^{n-1} dr = S_n R^n / n$ is the volume of a  $n$-dimensional ball of radius  $R$. The number states between  $E$ and  $E + \\Delta E$ is\n\n$$\n\\Gamma (E) = N _ {<  } (E + \\Delta E) - N _ {<  } (E) = \\frac {V ^ {N} S _ {3 N} (\\sqrt {2 m E}) ^ {3 N - 2}}{2 h ^ {3 N}} \\Delta E \\tag {1.3.2}\n$$\n\nTo obtain  $S_{n}$, we note\n\n$$\n\\begin{array}{l} \\int d ^ {n} \\pmb {x} e ^ {- \\pmb {x} ^ {2}} = \\int S _ {n} r ^ {n - 1} d r e ^ {- r ^ {2}} \\\\ = \\frac {1}{2} S _ {n} \\int (r ^ {2}) ^ {(n - 2) / 2} d r ^ {2} e ^ {- r ^ {2}} \\\\ = \\frac {1}{2} S _ {n} \\Gamma (n / 2) = \\pi^ {n / 2} \\tag {1.3.3} \\\\ \\end{array}\n$$\n\nWe find that\n\n$$\nS _ {n} = \\frac {2 \\pi^ {n / 2}}{\\Gamma (n / 2)} \\tag {1.3.4}\n$$\n\nThe entropy can now be calculated as\n\n$$\n\\begin{array}{l} k _ {B} ^ {- 1} S (E) \\\\ = N \\ln N + N \\ln v + \\frac {3 N}{2} \\ln N + 3 N \\ln ((2 m \\epsilon) ^ {1 / 2} / h) + 3 N \\ln \\sqrt {\\pi} - \\frac {3 N}{2} \\ln (3 N / 2) + 3 N / 2 + \\ln \\Delta E \\\\ = N \\ln N + N \\ln \\frac {v (2 m \\epsilon) ^ {3 / 2}}{h ^ {3}} + N \\left(\\frac {3}{2} \\ln \\frac {2 \\pi}{3} + \\frac {3}{2}\\right) + \\ln \\Delta E \\tag {1.3.5} \\\\ \\end{array}\n$$\n\nwhere v = V/N is the volume per particle and   $\\epsilon = E/N$  is the average energy per particle.\n\nA big problem, the entropy is NOT extensive due to the  $N \\ln N$ term. We need to use a concept from quantum physics - identical particle. For identical particles\n\n$$\nN _ {<  } (E) = \\frac {1}{h ^ {3 N} N !} \\int_ {\\sum \\boldsymbol {p} _ {i} ^ {2} / 2 m <   E} d ^ {3 N} \\boldsymbol {q} d ^ {3 N} \\boldsymbol {p} \\tag {1.3.6}\n$$\n\nUsing  $\\ln N! = N\\ln N - N$, we find\n\n$$\nk _ {B} ^ {- 1} S (E) = N \\ln \\frac {v (2 m \\epsilon) ^ {3 / 2}}{h ^ {3}} + N (\\frac {3}{2} \\ln \\frac {2 \\pi}{3} + \\frac {5}{2}) \\tag {1.3.7}\n$$\n\nFor identical particles, the entropy is extensive. The entropy per particle, s, is given by\n\n$$\n\\begin{array}{l} k _ {B} ^ {- 1} s = k _ {B} ^ {- 1} S / N = \\ln {\\frac {v (2 m \\epsilon) ^ {3 / 2}}{h ^ {3}}} + (\\frac {3}{2} \\ln {\\frac {2 \\pi}{3}} + \\frac {5}{2}) \\\\ \\approx \\ln {\\frac {v (2 m \\epsilon) ^ {3 / 2}}{h ^ {3}}} \\\\ = \\ln \\frac {v}{\\lambda^ {3}} \\tag {1.3.8} \\\\ \\end{array}\n$$\n\nMeaning:   $\\epsilon$  average energy per particle.   $(2m\\epsilon)^{1/2}$  the corresponding momentum.   $\\lambda = h/(2m\\epsilon)^{1/2}$  the corresponding wave length.   $v/\\lambda^{3}$  number wave packets that can be fitted into the volume per particle.\n\nClassical gas:  $v / \\lambda^3 \\gg 1$.\n\nQuantum gas:  $v / \\lambda^3 \\sim 1$.\n\n(Question: is air at room temperature a quantum gas or a classical gas?)", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 7, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 8"], "page_start": 8, "page_end": 8, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0012-statistical-ensembles-f5a73e3e-98ac552d:page-0009", "text": "# Thermodynamical function   $E(S, V, N)$\n\nFrom  $\\epsilon = \\frac{h^2}{2mv^{2/3}} e^{2s/3k_B}$ we get\n\n$$\nE (S, V, N) = N \\frac {h ^ {2} N ^ {2 / 3}}{2 m V ^ {2 / 3}} e ^ {2 S / 3 N k _ {B}} \\tag {1.3.9}\n$$\n\nThe equation of state: The temperature\n\n$$\nT = \\left. \\frac {\\partial E}{\\partial S} \\right| _ {V} = \\frac {2}{3 N k _ {B}} N \\frac {h ^ {2} N ^ {2 / 3}}{2 m V ^ {2 / 3}} e ^ {2 S / 3 N k _ {B}} \\tag {1.3.10}\n$$\n\nThe pressure\n\n$$\nP = - \\frac {\\partial E}{\\partial V} \\Big | _ {S} = \\frac {2}{3 V} N \\frac {h ^ {2} N ^ {2 / 3}}{2 m V ^ {2 / 3}} e ^ {2 S / 3 N k _ {B}} \\tag {1.3.11}\n$$\n\nWe obtain the equation of state\n\n$$\nP V = N k _ {B} T \\tag {1.3.12}\n$$", "source": "mit-ocw", "source_doc_id": "0012-statistical-ensembles-f5a73e3e-98ac552d", "source_title": "Chapter 1: Statistical Ensembles", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 8, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Chapter 1: Statistical Ensembles", "Page 9"], "page_start": 9, "page_end": 9, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0013-problem-set-1-1cd56d51-120aedfd:page-0001", "text": "# 8.08 Problem Set # 1\n\nFeb. 2, 2005\n\nDue Feb. 9, 2005\n\n# Problems:\n\n1. Problem 12.1 in K. Huang's book\n\n2. Problem 12.3 in K. Huang's book. But instead answer the following questions:\n\n- (a) How many states are there when the length of the chain is  $L$. (Assume the ceiling does not block the chain.)\n- (b) Find the entropy and the energy of the chain when the length of the chain is  $L$. (Assume  $N$ is large and  $L \\ll Na$.)\n- (c) Find the temperature  $T$ of the chain when the length of the chain is  $L$.\n- (d) Find the length of the chain in terms of  $T$. Show that the length of the chain is proportional to the force  $mq$ for a fixed  $T$ and a small force.\n\n3. Problem 12.3 in K. Huang's book. But assume  $m = 0$ and answer the following questions:\n\n- (a) Find the partition function  $Q(L)$ when the length of the chain is  $L$ and the chain is in contact with a heat bath of temperature  $T$.\n- (b) Find the free energy of the chain  $A(T, L)$. (Assume  $N$ is large and  $L \\ll Na$.)\n- (c) Find the tension  $\\tau$ of the chain:  $\\tau = \\frac{\\partial A}{\\partial L}$. Show that the length of the chain is proportional to the tension  $\\tau$ for a small tension.\n- (d) When we pull the chain, we do work. but the internal energy of the chain is always zero:  $U = 0$. Where does the energy go? Do we still have energy conservation?\n\n4. Problem 12.4 in K. Huang's book", "source": "mit-ocw", "source_doc_id": "0013-problem-set-1-1cd56d51-120aedfd", "source_title": "8.08 Problem Set # 1: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 1: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0014-problem-set-2-9995c4e7-a2730b6f:page-0001", "text": "# 8.08 Problem Set # 2\n\nFeb. 9, 2005\n\nDue Feb. 16, 2005\n\n# Problems:\n\n# 1. Curie's Law from canonical ensemble:\n\nConsider a spin-1/2 in a magnetic field B. The   $S_{z} = 1/2$  state has an energy   $g\\mu_{B}B/2$  and the   $S_{z} = -1/2$  state has an energy   $-g\\mu_{B}B/2$ . Assume the spin is in contact with a heat bath of temperature T.\n\n- (a) Find the probability  $P(1/2)$ for the spin to be in the  $S_z = 1/2$ state and the probability  $P(-1/2)$ for the spin to be in the  $S_z = -1/2$ state.\n- (b) Find the average spin  $\\langle S_z\\rangle$.\n- (c) Find the spin susceptibility  $\\chi = \\frac{\\langle S_z\\rangle}{B}\\bigg|_{B\\to 0}$.\n\n2. Problem 12.9 in K. Huang's book\n\n(c) Show the equipartition of the energy. That is the averages of the potential energy and the kinetic energy of a particle are given by   $\\left\\langle\\frac{p_{i}^{2}}{2m}\\right\\rangle=\\left\\langle\\frac{1}{2}m\\omega^{2}q_{i}^{2}\\right\\rangle=\\frac{1}{2}k_{B}T$ .\n\n# 3. Cooling by adiabatic demagnetization:\n\n- (a) Consider  $N$ spin-1/2 spins in a magnetic field  $B$. Initially, the system has a temperature  $T$. If we slowly reduce the magnetic field to zero, what becomes the temperature of the system? (Hint: the entropy remains unchanged in the above adiabatic process.)\n- (b) Consider  $N$ spin-1/2 spins in a magnetic field  $B$. The spin system is in thermal contact with an ideal gas of  $N$ particles in a volume  $V$. Initially, the two systems have a temperature  $T$. Assume  $g\\mu_B B \\gg k_B T$. If we slowly reduce the magnetic field to zero, what becomes the temperature of the gas?\n\n- 4. Problem 6.3 in K. Huang's book\n- 5. Problem 6.4 in K. Huang's book\n\nAssume the air is an ideal gas. You may want to do (b) first.  $\\frac{\\gamma - 1}{\\gamma}$ is just a constant. Find the value of the constant. (Hint: the entropy per particle does not depend on height  $z$.)", "source": "mit-ocw", "source_doc_id": "0014-problem-set-2-9995c4e7-a2730b6f", "source_title": "8.08 Problem Set # 2: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 2: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0015-problem-set-3-117bceed-6e2640c9:page-0001", "text": "# 8.08 Problem Set # 3\n\nFeb. 16, 2005\n\nDue Feb. 23, 2005\n\n# Problems:\n\n1.  $S$ of a spin-1 spin can take three different values  $S = 0$ and  $S = \\pm 1$.\n\n(a) Consider one spin-1 spin in a magnetic field. The energy is given by  $E = -hS$. Find the average spin  $\\langle S \\rangle$ at temperature  $T$.\n\n(b) Consider a 1D Ising model formed by spin-1 spins\n\n$$\nE = - J \\sum_ {i = 1} ^ {N} S _ {i} S _ {i + 1}\n$$\n\nUsing the mean-field theory to find the critical temperature  $T_{c}$, below which the system spontaneously generate magnetization.\n\n2. Water-vapor transition can be described by the following Ginzberg-Landau free energy (or more precisely Gibbs potential) near the critical point\n\n$$\nG = (h _ {1} T - h _ {2} P) \\delta n + (- a _ {1} + a _ {2} T + a _ {3} P) \\delta n ^ {2} + b \\delta n ^ {4}\n$$\n\nAll the constant coefficients  $h_{1,2}$,  $a_{1,2,3}$, and  $b$ are positive. Here  $\\delta n = n - n_c$.  $n$ is the density of water molecules and  $n_c$ is the density at the critical point  $(T_c, P_c)$.\n\n(a) Determine  $P_{c}$ and  $T_{c}$ of the critical point from the constants  $h_{1,2}$,  $a_{1,2}$ and  $b$.\n\n(b) Determine first order phase transition line between water and vapor the  $T - P$ space. Sketch such a line in the  $T - P$ space.\n\n(c) Calculate the change of the density  $\\Delta n$ across the first order transition line. (ie  $\\Delta n = n_{water} - n_{vapor}$ on the two sides of the first phase transition.)\n\n3. Problem 14.6 in K. Huang's book. (20 pts)\n\n(It may be less confusing to rename the order parameter from  $S$ to  $m$ so not to be confused with entropy.)", "source": "mit-ocw", "source_doc_id": "0015-problem-set-3-117bceed-6e2640c9", "source_title": "8.08 Problem Set # 3: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 3: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0016-problem-set-4-d4b41591-d3278c93:page-0001", "text": "# 8.08 Problem Set # 4\n\nFeb. 23, 2005\n\nDue March 2, 2005\n\n# Problems:\n\n1. Problem 13.1 in K. Huang's book.\n\n2. A surface has  $N_{s}$ sites which can adsorb 1 or 2 atoms. It costs no energy to adsorb 1 atom or 2 atoms. The surface is in contact with a gas of the atoms.\n\n(a) Assume the gas has a chemical potential  $\\mu$ and a temperature  $T$.\n\n(i) Find the probabilities for a site to be empty, occupied by one atom, and occupied by two atoms.\n\n(ii) Find the average number of atoms absorbed on the surface.\n\n(b) Assume the gas is describe by the van der Waals model. Its free energy is given by\n\n$$\nA = N k _ {B} T [ \\ln (\\frac {N \\lambda^ {3}}{V - N v _ {0}}) - 1 ] + \\frac {N ^ {2} \\bar {v}}{V}\n$$\n\nwhere   $v_{0}$  and   $\\bar{v}$  are two constants, N is the total number of atoms in the gas, and   $\\lambda = \\sqrt{2\\pi\\hbar^{2}/mk_{B}T}$ . Find the chemical potential of the van der Waals gas as a function of T and n = N/V.", "source": "mit-ocw", "source_doc_id": "0016-problem-set-4-d4b41591-d3278c93", "source_title": "8.08 Problem Set # 4: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 4: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0017-problem-set-5-bdf0a26e-b655152f:page-0001", "text": "# 8.08 Problem Set # 5\n\nMarch 2, 2005\n\nDue March 9, 2005\n\n# Problems:\n\n1. At finite temperature, a semiconductor contains electrons and holes. An electron and a hole can annihilate and release an energy  $\\Delta$:\n\n$$\ne + h \\leftrightarrow \\Delta\n$$\n\n(You may assume each electron and each hole have an internal energy   $\\Delta/2$ .) Here we assume the electrons and the holes have the same mass m and the temperature is T.\n\n- (a) Find the densities of the electrons  $n_e$ and the holes  $n_h$ in a undoped semiconductor. (In a undoped semiconductor  $n_e = n_h$.)\n- (b) Find the densities of the electrons  $n_e$ and the holes  $n_h$ in a doped semiconductor. (In a doped semiconductor  $n_e - n_h = n_d$ where  $n_d$ is the density of doping which is fixed.)\n\n2. If we roll two dices, we get a pair of random numbers  $(n_1, n_2)$.\n\n(a) Consider two random numbers\n\n$$\nk _ {+} = n _ {1} + n _ {2}, \\quad k _ {-} = n _ {1} - n _ {2}.\n$$\n\nAre  $k_{+}$ and  $k_{-}$ independent random numbers?\n\n(b) Consider two random numbers\n\n$$\nm _ {+} = \\left(n _ {1} + n _ {2}\\right) \\bmod 6, \\quad m _ {-} = n _ {1} \\bmod 6.\n$$\n\nAre  $m_{+}$ and  $m_{-}$ independent random numbers?\n\n- 3. (a) A pendulum is formed by a mass M and string of length L. Calculate the thermal fluctuations of the position of the mass:   $\\Delta x = \\sqrt{\\langle(x - \\bar{x})^{2}\\rangle}$ . Assume the air temperature is T.\n- (b) Calculate the value of  $\\Delta x$ assuming  $M = 1\\mathrm{g}$,  $L = 10\\mathrm{cm}$, and  $T = 300\\mathrm{K}$.\n\n4. Problem 12.11 in K. Huang's book.", "source": "mit-ocw", "source_doc_id": "0017-problem-set-5-bdf0a26e-b655152f", "source_title": "8.08 Problem Set # 5: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 5: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0018-problem-set-6-664f17bb-d5091dc4:page-0001", "text": "# 8.08 Problem Set # 6\n\nMarch 9, 2005\n\nDue March 16, 2005\n\n# Problems:\n\n1. A chamber of volume   $L \\times L \\times 2L$  is divided by a wall into two chambers, each with a volume   $L \\times L \\times L$ . The wall can move freely in the horizontal direction. Assume the two chambers is filled with ideal gas. The number of the particles in each chamber is N. Also the chambers are in contact with a heat bath which keep the temperature of the chambers at a constant T. Find the fluctuation of the position of the wall:   $\\Delta x = \\sqrt{\\langle(x - \\bar{x})^{2}\\rangle}$ , where   $\\bar{x} = \\langle x \\rangle$  is the average position of the wall. Does your result depend on the mass of wall?\n\nWall\nx\n\n- 2. Problem 10.3 in K. Huang's book.\n- 3. Problem 10.5 in K. Huang's book.\n- 4. Problem 10.6 in K. Huang's book.\n\n(c) Also find out by what factor does the total number of photon in the universe increases (or decreases) after the expansion.", "source": "mit-ocw", "source_doc_id": "0018-problem-set-6-664f17bb-d5091dc4", "source_title": "8.08 Problem Set # 6: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 6: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0019-problem-set-7-a495c460-d328f3d8:page-0001", "text": "# 8.08 Problem Set # 7\n\nMarch 16, 2005\n\nDue March 30, 2005\n\n# Problems:\n\n- 1. Problem 10.7 in K. Huang's book.\n- 2. Problem 10.8 in K. Huang's book.\n- 3. Problem 11.9 in K. Huang's book.\n- 4. Problem 11.10 in K. Huang's book.", "source": "mit-ocw", "source_doc_id": "0019-problem-set-7-a495c460-d328f3d8", "source_title": "8.08 Problem Set # 7: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 7: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0020-problem-set-8-5b3b1186-e25401f2:page-0001", "text": "# 8.08 Problem Set # 8\n\nMarch 30, 2005\n\nDue April 6, 2005\n\n# Problems:\n\n1. Consider a gas of bosonic sodium atoms confined in a quadratic potential well  $U(\\pmb{r}) = \\frac{1}{2} m\\omega_0^2 |\\pmb {r}|^2$ where  $m$ is the mass of the sodium atom. The characteristic length of the oscillator potential is  $r_0 = \\sqrt{\\hbar / m\\omega_0} = 5\\times 10^{-3}\\mathrm{cm}$.\n\n- (a) Ignore the interaction between the sodium atoms, fined the size of the condensed sodium atoms at  $T = 0$. How does the size of the condensation depends on the number of particles?\n- (b) For interacting bosons, the shape of condensation at  $T = 0$ is determined by\n\n$$\n\\Big [ - \\frac {\\hbar^ {2}}{2 m} \\partial_ {\\pmb {r}} ^ {2} + (U (\\pmb {r}) - \\mu) + g | \\psi (\\pmb {r}) | ^ {2} \\Big ] \\psi (\\pmb {r}) = 0\n$$\n\nIn Thomas-Fermi approximation, we assume the wave function   $\\psi$  is smooth and drop the   $\\partial_{r}^{2}$  term. In this case the shape of condensation is determined by\n\n$$\n[ (U (\\pmb {r}) - \\mu) + g | \\psi (\\pmb {r}) | ^ {2} ] \\psi (\\pmb {r}) = 0\n$$\n\nNow, how does the size of the condensation depends on the number of particles?\n\n(c) Fig. 15.2 of Huang's book shows measured shapes of condensation. The maximum density is  $10^{11}\\mathrm{cm}^{-3}$ for the shape near  $T = 0$. Using the data provided by the curve, find the scattering length  $a$ of the sodium atom. (Note  $a$ and  $g$ is related through Eq. (15.3) in Huang's book.)\n\n2. Problem 15.9 in K. Huang's book.", "source": "mit-ocw", "source_doc_id": "0020-problem-set-8-5b3b1186-e25401f2", "source_title": "8.08 Problem Set # 8: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 8: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0021-problem-set-9-d597eebe-b2bdb646:page-0001", "text": "# 8.08 Problem Set # 9\n\nApril 6, 2005\n\nDue April 13, 2005\n\n# Problems:\n\n1. Consider an interacting bosonic gas in 1D. The Ginzburg-Landau free energy is given by\n\n$$\nA = \\int_ {- \\infty} ^ {+ \\infty} d x \\left(\\frac {1}{2 m} | \\partial_ {x} \\psi | ^ {2} + \\left(\\frac {a (T)}{2} + U (x)\\right) | \\psi | ^ {2} + \\frac {b}{4} | \\psi | ^ {4}\\right)\n$$\n\nWhere   $\\psi$  is the amplitude of condensed bosons (the order parameter) and   $a(T)=a_{0}(\\frac{T}{T_{c}}-1)$  for T near   $T_{c}$ . Here   $a_{0}$ , b and m are constants. The external potential   $U(x)$  has the following form\n\n$$\nU (x) | _ {x <   0} = + \\infty , \\qquad U (x) | _ {x > 0} = 0\n$$\n\n- (a) Show that there is a boson condensation for  $T < T_{c}$ and find the amplitude of condensed bosons  $\\psi(x)$ for  $x \\to +\\infty$.\n- (b) Near  $x = 0$, the amplitude of condensed bosons is suppressed by the potential  $U(x)$. To gain a more quantitative understanding of the suppression, we assume  $\\psi(x)$ to have a form\n\n$$\n\\psi (x) | _ {x <   0} = 0, \\qquad \\psi (x) | _ {0 <   x <   \\xi} = \\frac {x}{\\xi} \\psi (+ \\infty), \\qquad \\psi (x) | _ {\\xi <   x} = \\psi (+ \\infty)\n$$\n\nWe want to adjust   $\\xi$  to minimize the total free energy for the above form of boson condensation. Calculate the   $\\xi$  dependence of the free energy. Find the value of   $\\xi$  that minimizes the free energy.\n\n(c) Show that near  $T_{c}$,  $\\xi$ diverges as  $\\xi \\propto |T_c - T|^{\\nu}$. Find the critical exponent  $\\nu$.\n\n(The length scale  $\\xi$ is called the coherent length. It is a very important length scale in superfluid. For example, the size of the vortex core is given by  $\\xi$.)\n\n- 2. Problem 8.2 in K. Huang's book.\n- 3. Problem 8.3 in K. Huang's book.\n- 4. Problem 9.4 in K. Huang's book.", "source": "mit-ocw", "source_doc_id": "0021-problem-set-9-d597eebe-b2bdb646", "source_title": "8.08 Problem Set # 9: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 9: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0022-problem-set-10-cfad4d05-d61ab082:page-0001", "text": "# 8.08 Problem Set # 10\n\nApril 13, 2005\n\nDue April 20, 2005\n\n# Problems:\n\n- 1. Problem 10.10 in K. Huang's book.\n- 2. (a - c) in Problem 9.3 in K. Huang's book.\n- 3. Problem 9.5 in K. Huang's book.\n- 4. (a - d) and (f) in Problem 9.8 in K. Huang's book.", "source": "mit-ocw", "source_doc_id": "0022-problem-set-10-cfad4d05-d61ab082", "source_title": "8.08 Problem Set # 10: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 10: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0023-problem-set-11-d5342fb9-fe658c35:page-0001", "text": "# 8.08 Problem Set # 11\n\nApril 20, 2005\n\nDue April 27, 2005\n\n# Problems:\n\n- 1. Problem 7.1 in K. Huang's book.\n- 2. Problem 7.2 in K. Huang's book.\n- 3. Problem 5.9 in K. Huang's book. There are errors in this problem. There should be a  $n!$ in the numerator in part (b).\n- 4. Problem 5.10 in K. Huang's book. You need to estimate the mean-free path first.\n- 5. Problem 7.5 in K. Huang's book.", "source": "mit-ocw", "source_doc_id": "0023-problem-set-11-d5342fb9-fe658c35", "source_title": "8.08 Problem Set # 11: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 11: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0024-problem-set-12-08058f08-96219369:page-0001", "text": "# 8.08 Problem Set # 12\n\nApril 27, 2005\n\nDue May 6, 2005\n\n# Problems:\n\n1. Problem 7.7 in K. Huang's book.\n\n2. Silicon   $S_{i}$  solid has a density   $2.3g/cm^{3}$ . The velocity of the sound is 2200m/sec. The thermal conductivity at T = 300K is   $1.5\\frac{W}{cm\\cdot K}$ . Assuming the thermal conductivity is due to the phonons, estimate the mean free path of the phonons. (Hint: you may need to estimate the phonon density first.)\n\n3. Copper   $C_{u}$  solid has a density   $8.9g/cm^{3}$  and an electric resistivity   $1.7 \\times 10^{-6}\\Omega cm$ . Assume the density of the conduction electrons is the same as the density of the   $C_{u}$  atom and the electron mass is the same as electron mass in the vacuum. Estimate relaxation time   $\\tau$  and the mean free path   $\\lambda$  of the electrons in   $C_{u}$  solid from the Drude model.\n\n4. (a) Estimate the mean free path of air molecules at 300K and 1 atm. Estimate the viscosity of the air.\n\n(b) Consider a ball of radius  $0.1\\mathrm{mm}$ moving in the air. Estimate the friction coefficient of the ball.\n\n(c) Assume the ball is floating in the air and is doing the Brownian motion. Estimate how long does it take for the ball drift  $1\\mathrm{mm}$.\n\n5. Problem 18.2 in K. Huang's book with the following changes:\n\n(a)  $\\varphi(t)$ has a form  $\\varphi(t) = Ae^{-t^2/\\tau^2}$.\n\n(b) Calculate the power spectrum for all frequencies.", "source": "mit-ocw", "source_doc_id": "0024-problem-set-12-08058f08-96219369", "source_title": "8.08 Problem Set # 12: Problems:", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["8.08 Problem Set # 12: Problems:", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}
{"unit_id": "mit-ocw:0025-practice-problem-for-quizzes-and-final-exam-1cc13336-6872ed29:page-0001", "text": "# Practice Problem for Quizzes and Final Exam\n\nPractice problems for quiz 1:\n\n4.2, 4.3, 4.5, 6.6, 6.14, 6.15, 12.5, 12.7, 14.1, 14.5\n\nPractice problems for quiz 2:\n\nHomework problems plus 6.1, 6.2, 6.7, 6.14, 10.11, 11.4, 11.5, 11.6, 11.7, 13.2, 13.3\n\nPractice problems for final:\n\nAll homework problems\n\nAdditional practice problems:\n\n5.1, 5.4, 5.5, 6.5, 6.6, 6.9, 6.11, 6.12, 6.13, 7.3, 7.9, 8.4, 8.5, 8.6, 8.7, 8.8, 9.2, 9.6, 10.9,\n\n11.3, 11.8, 13.8, 13.10, 13.11, 14.2, 14.7, 16.4, 16.5", "source": "mit-ocw", "source_doc_id": "0025-practice-problem-for-quizzes-and-final-exam-1cc13336-6872ed29", "source_title": "Practice Problem for Quizzes and Final Exam", "domain": "physics", "subdomain": "statistical_physics", "level": "undergraduate", "order_index": 0, "metadata": {"module_id": null, "source_format": "pdf", "extraction_method": "mineru_vlm_auto_engine_v1", "license": "CC-BY-NC-SA-4.0", "language": "en", "hierarchy_path": ["Practice Problem for Quizzes and Final Exam", "Page 1"], "page_start": 1, "page_end": 1, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null}}