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The concept of entropy is notoriously difficult to grasp. Even the consummate mathematician and physicist Johnny von Neumann claimed that "nobody really knows what entropy is anyway." Although we have an exact and remarkably simple formula for the entropy of a macrostate in terms of the number of corresponding microsta... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.6 Qualitative Features of Entropy",
"token_count": 2040,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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is homogeneous at all temperatures. Thus the high-temperature and reentrant homogeneous phases are in fact the same phase. (Reentrant phases are also encountered in type-II superconductors.)
#### Rust
According to a common misconception, "The entropy law describes the tendency for all objects to rust, break, fall... | {
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"Header 2": "2.6 Qualitative Features of Entropy",
"token_count": 692,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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The four examples above should caution us about relying on qualitative arguments concerning entropy. Here is another situation<sup>10</sup> to challenge your intuition: The next page shows two configurations of 13<sup>2</sup> = 169 squares tossed down on an area that has 35 × 35 = 1225 empty spaces, each of which could... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.6 Qualitative Features of Entropy",
"Header 3": "2.6.2 Entropy and the lattice gas",
"token_count": 2019,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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In the lattice gas model there are many "orderly" configurations (such as the checkerboard pattern of figure 2.6) that are members of both classes. There are many other "orderly" configurations (such as the solid block pattern of figure 2.7) that are members only of the larger (higher entropy!) class.<sup>13</sup> The ... | {
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"Header 2": "2.6 Qualitative Features of Entropy",
"Header 3": "2.6.2 Entropy and the lattice gas",
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References: Schroeder sections 3.1, 3.4, and 3.5. Reif section 3.3. Kittel and Kroemer pages 30–41.
You will remember that one of the problems with our ensemble approach to statistical mechanics is that, while one readily sees how to calculate ensemble average values of microscopic (mechanical) quantities like the ki... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.7 Using Entropy to Find (Define) Temperature and Pressure",
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Consider two systems, A and B, enclosed in our perfectly reflecting no-skid walls. System A has energy EA, volume VA, and particle number NA; similarly for system B.
I have drawn the two systems adjacent, but they might as well be miles apart because they can't affect each other through... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.7 Using Entropy to Find (Define) Temperature and Pressure",
"Header 3": "2.7.2 Two isolated systems",
"token_count": 2046,
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\tag{2.57}$$
In fact, this result is too famous, because people often forget that it applies only to a classical monatomic ideal gas. It is a common misconception that temperature is defined through equation (2.57) rather than through equation (2.56), so that temperature is always proportional to the average energy p... | {
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"Header 2": "2.7 Using Entropy to Find (Define) Temperature and Pressure",
"Header 3": "2.7.2 Two isolated systems",
"token_count": 1545,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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The upshot of all our definitions is that
$$dS = \frac{1}{T} dE + \frac{p}{T} dV - \frac{\mu}{T} dN. \tag{2.70}$$
What do these definitions mean physically? The first term says that if two systems are brought together so that they can exchange energy but not volume or particle number, then in the system with high t... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.7 Using Entropy to Find (Define) Temperature and Pressure",
"Header 3": "2.7.6 The meaning of chemical potential",
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#### 2.25 Accessible configurations in another spin system
The Ising model for a ferromagnetic material such as nickel is different from the "ideal paramagnet" discussed in problem 2.9. In this model the spins reside at lattice sites and may point either up or down, but in contrast to the ideal paramagnet model, two ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "2.8 Additional Problems",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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We've seen how to calculate—at least in principle, although our techniques were primitive and difficult to use—the entropy S(E, V, N). We've also argued ("the booty argument", section 2.7.1) that for most systems entropy will increase monotonically with energy when V and N are fixed. Thus we (can)/(will always be able ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.1 Heat and Work",
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$$dE = T dS - p dV + \mu dN$$
$$= dQ + dW_{conf} + dW_{diss}$$
$$\geq dQ + dW_{conf}$$
but
$$dW_{\rm conf} = -p \, dV + \mu \, dN$$
so
$$T dS \geq dQ$$
where the equality holds for quasistatic changes.
The operational definition of entropy is that, for a quasistatic change,
$$dS = \frac{dQ}{T}$$
for quasi... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.2 Entropy",
"token_count": 2005,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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The emphasis of this course is on the properties of matter. This section is independent of the properties of matter! It is included because:
- 1. historically important
- 2. expected coverage (e.g. for GRE)
- 3. important from an applications and engineering standpoint (See, for example, "Hurricane heat engines" by H... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.3 Heat Engines",
"token_count": 1357,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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What is a section on multivariate calculus doing in a physics book. . . particularly a physics book which assumes that you already know multivariate calculus? The answer is that you went through your multivariate calculus course learning one topic after another, and there are some subtle topics that you covered early i... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.4 Multivariate Calculus",
"token_count": 1962,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Suppose
$$df = A(x,y) dx + B(x,y) dy. (3.28)$$
Then
$$A(x,y) = \frac{\partial f}{\partial x}\Big|_{y}$$
and $B(x,y) = \frac{\partial f}{\partial y}\Big|_{x}$ .
But because
$$\frac{\partial^2 f(x,y)}{\partial x \partial y} = \frac{\partial^2 f(x,y)}{\partial y \partial x}$$
it follows that
$$\left. \frac{... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.4 Multivariate Calculus",
"Header 3": "3.4.3 Maxwell relations",
"token_count": 963,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Suppose f(x, y) and g(x, y) are two function of the variables x and y. Then again we have
$$df = \frac{\partial f}{\partial x} \Big|_{y} dx + \frac{\partial f}{\partial y} \Big|_{x} dy \tag{3.35}$$
for any differential change dx and dy.
What if we are interested, not in any change, but in a change along a contour... | {
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"Header 2": "3.4 Multivariate Calculus",
"Header 3": "3.4.5 Multivariate chain rule",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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By a thermodynamic "quantity" I mean either a variable or a function. We have already seen that for fluids we may regard the entropy as a function of energy, volume, and number, S(E, V, N), or the energy as a function of entropy, volume, and number, E(S, V, N): using the term "quantity" avoids prejudging the issue of w... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.5 Thermodynamic Quantities",
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$$S(E, V, N) = \left(\frac{k_B^2}{v_0 T_0}\right)^{1/3} (EVN)^{1/3}$$
(3.55)
$$S(E, V, N) = \left(\frac{k_B v_0^2}{T_0^2}\right)^{1/3} \left(\frac{EN}{V}\right)^{2/3}$$
(3.56)
$$S(E, V, N) = \left(\frac{k_B v_0^2}{T_0^2}\right)^{1/3} \frac{EN}{V}$$
(3.57)
$$S(E, V, N) = \left(\frac{k_B}{T_0}\right)^{1/2} \left(... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.5 Thermodynamic Quantities",
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We know that in many circumstances (e.g. pure, non-magnetic fluids) the thermodynamic state of a system is uniquely specified by giving the entropy S, the volume V , and the particle number N. The master function for this description is the energy
$$E(S, V, N) \tag{3.63}$$
whose total differential is
$$dE = T dS ... | {
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"Header 2": "3.6 The Thermodynamic Dance",
"Header 3": "3.6.1 Description in terms of variables (S, V, N)",
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In this description, the master function
$$G(T, p, N) = F + pV \tag{3.76}$$
<sup>5</sup> If a material existed for which these two experiments did not give identical results, then we could use that substance to build a perfect heat engine. (See page 59.)
is called the "Gibbs potential" (or the "Gibbs free energy"... | {
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"Header 2": "3.6 The Thermodynamic Dance",
"Header 3": "3.6.4 Description in terms of variables (T, p, N)",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Here the master function is
$$\Pi(T, V, \mu) = F - \mu N \tag{3.80}$$
with master equation
$$d\Pi = -S dT - p dV - N d\mu. \tag{3.81}$$
You might wonder why Π doesn't have a name. This is because
$$\Pi = F - \mu N = F - G = -pV \tag{3.82}$$
or, to put it more formally,
$$\Pi(T, V, \mu) = -p(T, \mu)V, \tag... | {
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"Header 2": "3.6 The Thermodynamic Dance",
"Header 3": "3.6.5 Description in terms of variables (T, V, µ)",
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Perform the thermodynamic dance down to the function F(T, V, NA, NB) to show that
$$\mu_A(T, V, N_A, N_B) = \frac{\partial F}{\partial N_A} \Big|_{T, V, N_B}$$
(3.98)
and similarly for B.
b. Argue that, because µ<sup>A</sup> and µ<sup>B</sup> are intensive, their functional dependence on V , NA, and NB, must be t... | {
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"Header 2": "3.6 The Thermodynamic Dance",
"Header 3": "3.6.5 Description in terms of variables (T, V, µ)",
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All the results of the previous section relied on the underlying assumption that the system was a pure fluid and thus could be described by specifying, for example, the temperature T, the volume V , and the particle number N. Recall that the latter two were simply examples of many possible mechanical parameters that co... | {
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"Header 2": "3.7 Non-fluid Systems",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Is there any relation between
$$C_p(T,p) = T \left. \frac{\partial S}{\partial T} \right)_p,$$
(3.105)
the heat capacity at constant pressure, and
$$C_V(T, V) = T \left( \frac{\partial S}{\partial T} \right)_V,$$
(3.106)
the heat capacity at constant volume? Remembering that entirely different experiments are u... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.8 Thermodynamics Applied to Fluids",
"Header 3": "3.8.1 Heat capacities",
"token_count": 2025,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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It is possible to consider the energy as a function of T and V , and in this subsection we will find the total differential of E(T, V ).
We begin by finding the total differential of S(E, V ) and finish off by substituting that expression for dS into the master equation (3.127). The mathematical expression for that t... | {
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"Header 2": "3.8 Thermodynamics Applied to Fluids",
"Header 3": "3.8.1 Heat capacities",
"token_count": 1974,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
#### 3.34 Heat capacities in a magnetic system
For a magnetic system (see equation (3.102)), show that
$$C_H = T \frac{\partial S}{\partial T}\Big)_H$$
, $C_M = T \frac{\partial S}{\partial T}\Big)_M$ , $\beta = \frac{\partial M}{\partial T}\Big)_H$ , and $\chi_T = \frac{\partial M}{\partial H}\Big)_T$ (3.148... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.8 Thermodynamics Applied to Fluids",
"Header 3": "3.8.1 Heat capacities",
"token_count": 1292,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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A pure system is in two-phase coexistence, say at the vaporization temperature, and the system variables are pressure and temperature. One way to realize this experimentally is through the chamber sketched here:
The lid on the system keeps it pure (so air doesn't mix into the vapor) but... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.9 Thermodynamics Applied to Phase Transitions",
"token_count": 2037,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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So far in this book we have considered mostly pure substances. What happens if we have a mixture of several chemical species, say the four substances A, B, C, and D? In this case the mechanical parameters will include the numbers NA, NB, N<sup>C</sup> , and ND, and the entropy
$$S(E, V, N_A, N_B, N_C, N_D)$$
(3.170) ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.10 Thermodynamics Applied to Chemical Reactions",
"Header 3": "3.10.1 Thermodynamics of mixtures",
"token_count": 1247,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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I emphasized on page 73 that the equation of state did not contain full thermodynamic information. Thus, for example, knowledge of the equation of state V (T, p, N) is not sufficient to uncover the master function G(T, p, N). On the other hand, that knowledge is sufficient to restrict the functional form of the Gibbs p... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.10 Thermodynamics Applied to Chemical Reactions",
"Header 3": "3.10.3 Chemical potential of an ideal gas",
"token_count": 1985,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Show that
$$\mu_A(T, p, f_A, f_B) = k_B T \ln \left[ \left( \frac{h_0^2}{2\pi m_A (k_B T)^{5/3}} \right)^{3/2} p f_A \right],$$
(3.192)
and relate this expression to the form (3.187). Verify relation (3.176).
#### 3.43 Relations between equilibrium constants
You measure the equilibrium constants for these three... | {
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"Header 2": "3.10 Thermodynamics Applied to Chemical Reactions",
"Header 3": "3.10.3 Chemical potential of an ideal gas",
"token_count": 322,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Another non-fluid system (see section 3.7).
See Robert E. Kelly, "Thermodynamics of blackbody radiation," Am. J. Phys. 49 (1981) 714–719, and Max Planck, The Theory of Heat Radiation, part II.
How do you get electromagetic radiation — light — into thermal equilibrium? The light streaming from, say, a red neon tube ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.11 Thermodynamics Applied to Light",
"token_count": 417,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The fundamental thermodynamic equation for this system involves the master function E(S, V ). It is
$$dE = T dS - p dV. (3.193)$$
This equation differs from previously encountered master equations in that there is no term for µ dN. From the classical perspective, this is because radiation is made up of fields, not ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.11 Thermodynamics Applied to Light",
"Header 3": "3.11.1 Fundamentals",
"token_count": 2011,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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During a quasistatic adiabatic expansion, dE = −p dV , so
$$(V\Delta\lambda) d\bar{u} + (\bar{u}\Delta\lambda) dV + (\bar{u}V) d[\Delta\lambda] = -\frac{1}{3}\bar{u}\Delta\lambda dV.$$
(3.215)
During the expansion the volume and wavelengths are changing through (see equation 3.211)
$$\lambda = cV^{1/3}$$
$$d\la... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.11 Thermodynamics Applied to Light",
"Header 3": "3.11.1 Fundamentals",
"token_count": 549,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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#### 3.45 Cool mountain air
Model the earth's atmosphere as an ideal gas (nitrogen) in a uniform gravitational field. Ignore all winds. Let m denote the mass of a gas molecule, g the acceleration of gravity, and z the height above sea level.
a. Use ideas from Newtonian mechanics to show that the change of atmospher... | {
"Header 1": "Statistical Mechanics",
"Header 2": "3.12 Additional Problems",
"token_count": 2021,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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#### The canonical ensemble in general:
The probability that the system is in the microstate x is proportional to the "Boltzmann factor"
$$e^{-H(\mathbf{x})/k_BT}. (4.1)$$
The normalization factor is called the "partition function" or "sum over all states" (German "Zustandsumme"):
$$Z(T, V, N) = \sum_{\text{mic... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.1 The Canonical Ensemble",
"Header 3": "Summary",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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I've told you that it's easier to do calculations in the canonical ensemble than in the microcanonical ensemble. Today I'm going to demonstrate the truth of this assertion.
Remember how we found the entropy S(E, V, N) for a classical monatomic ideal gas? The hard part involved finding the volume of a shell in 3N-dime... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.3 Classical Monatomic Ideal Gas",
"token_count": 1964,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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\tag{4.21}$$
Use of the substitution u = p β/2m p in the momentum integral gives
$$\int_{-\infty}^{+\infty} dp \, e^{-\beta(p^2/2m)} = \sqrt{2m/\beta} \int_{-\infty}^{+\infty} e^{-u^2} \, du = \sqrt{2\pi m k_B T}. \tag{4.22}$$
Putting all this together gives us
$$Z(T, V, N) = \frac{V^N}{N! h_0^{3N}} \left(\sqrt... | {
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"Header 2": "4.3 Classical Monatomic Ideal Gas",
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The systems in the canonical ensemble are not restricted to having just one particular energy or falling within a given range of energies. Instead, systems with any energy from the ground state energy to infinity are present in the ensemble, but systems with higher energies are less probable. In this circumstance, it i... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.4 Energy Dispersion in the Canonical Ensemble",
"token_count": 2035,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Note distinction!
P(x) is the probability of falling in a particular microstate x. This happens to be a function only of the energy of the microstate, whence this function is often called P(H(x)). (For a continuous system, P(x) corresponds to p(Γ)dΓ. . . the probability of the system falling in some phase space point... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.5 Temperature as a Control Variable for Energy (Canonical Ensemble)",
"token_count": 1080,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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In a microcanonical ensemble the individual systems are all restricted, by definition, to have a given energy. In a canonical ensemble the individual systems are *allowed* to have any energy, from that of the ground state to that of an ionized plasma, but we have seen that (for large systems) they tend not to use this ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.5 Temperature as a Control Variable for Energy (Canonical Ensemble)",
"Header 3": "4.6 The Equivalence of Canonical and Microcanonical Ensembles",
"token_count": 597,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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This section uncovers results that are of interest in their own right, but it also serves as an example of a mathematically rigorous argument in statistical mechanics.
Definitions. The grand partition function is
$$\Xi(T, V, \alpha) = \sum_{N=0}^{\infty} e^{-\alpha N} Z(T, V, N), \tag{4.45}$$
the Helmholtz free e... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.8 The Grand Canonical Ensemble in the Thermodynamic Limit",
"token_count": 1638,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
| | boundary | variables | probability of microstate | p.f. | master function |
|-----------------|-----------------------|-----------|---------------------------------------------------------------------------------|--... | {
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"Header 2": "4.9 Summary of Major Ensembles",
"token_count": 324,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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How shall we define a partition function for quantal systems? A reasonable first guess is that
$$Z(T, \text{parameters}) = \sum_{\text{all quantal states}} e^{-\beta E}.$$
[first guess]
This guess runs into problems immediately. Most quantal states are not energy eigenstates, so it's not clear how to interpret "E" ... | {
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"Header 2": "4.10 Quantal Statistical Mechanics",
"token_count": 1646,
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#### 4.5 Classical monatomic ideal gas in the canonical ensemble
In section 5.1 we will show that the canonical partition function of a classical monatomic ideal gas is
$$Z(T, V, N) = \frac{1}{N!} \left[ \frac{V}{\lambda^3(T)} \right]^N, \tag{4.67}$$
where
$$\lambda(T) \equiv \frac{h_0}{\sqrt{2\pi m k_B T}}. (4... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.11 Ensemble Problems I",
"token_count": 2004,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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\tag{4.85}$$
Show how this integral—as well as a host of other useful matrix elements—can be obtained easily from the well known result
$$\int_0^{\pi} \sin(ax)\sin(bx) \, dx = \frac{1}{2} \left[ \frac{\sin[(a-b)\pi]}{a-b} - \frac{\sin[(a+b)\pi]}{a+b} \right] \quad a \neq \pm b. \tag{4.86}$$
#### 4.14 Polymers
A... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.11 Ensemble Problems I",
"token_count": 1973,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Finally conclude that
$$K = k_B T \gamma N \langle \omega \rangle + F \tag{4.104}$$
or, in light of relationship (4.100),
$$K = F + B\langle \omega \rangle, \tag{4.105}$$
which should be compared to equation (4.93).
We have shown only that this ensemble scheme is "not inconsistent". It is not obviously wrong,... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.11 Ensemble Problems I",
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The remaining problems in this chapter deal with the same "principles" issues that the others do, but they assume some familiarity with physical and mathematical topics that we have not yet treated. I place them here because of their character, but I do not expect you to do them at this moment. Instead I list their pre... | {
"Header 1": "Statistical Mechanics",
"Header 2": "4.12 Ensemble Problems II",
"token_count": 840,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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We have already found the Helmholtz free energy of the classical monatomic ideal gas (section 4.3). I think you will agree that the canonical calculation is considerably easier than the corresponding microcanonical calculation. Here we will review the calculation, then go back and investigate just what caused the solut... | {
"Header 1": "Statistical Mechanics",
"Header 2": "5.1 Classical Monatomic Ideal Gases",
"token_count": 1255,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Indeed there is. Suppose a Hamiltonian is a sum of two pieces
$$H(\Gamma) = H_1(\Gamma_1) + H_2(\Gamma_2), \tag{5.7}$$
where Γ<sup>1</sup> and Γ<sup>2</sup> are exclusive. (That is, the phase space variables in the list Γ<sup>1</sup> and the phase space variables in the list Γ<sup>2</sup> together make up the whole... | {
"Header 1": "Statistical Mechanics",
"Header 2": "5.1 Classical Monatomic Ideal Gases",
"Header 3": "5.1.2 Theorem on decoupling Hamiltonians",
"token_count": 1019,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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#### 5.3.1 The equipartition theorem
Equipartition theorem. (For classical systems.) Suppose the Hamiltonian H(Γ) decouples into one piece involving a single phase space variable—call it H1, plus another piece which involves all the other phase space variables—call it H2(Γ2). Suppose further that the energy depends q... | {
"Header 1": "Statistical Mechanics",
"Header 2": "5.3 Heat Capacity of an Ideal Gas",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Estimate the crossover temperature between the non-relativistic and ultra-relativistic regimes.
#### 5.3 Another generalization of equipartition
Consider the same situation as the equipartition theorem in the text, but now suppose the single phase space variable takes on values from 0 to +∞. What is the correspondi... | {
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"Header 2": "5.3 Heat Capacity of an Ideal Gas",
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The classical partition function is
$$Z = \frac{1}{N!} \left[ \frac{V}{\lambda^3(T)} \frac{1}{h^2} \int_0^{\pi} d\theta \int_0^{2\pi} d\varphi \int_{-\infty}^{+\infty} d\ell_{\theta} \int_{-\infty}^{+\infty} d\ell_{\varphi} \, e^{-\beta(\ell_{\theta}^2 + \ell_{\varphi}^2)/2I} \right]^N.$$
(5.53)
Let's make sure we ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "5.4 Specific Heat of a Hetero-nuclear Diatomic Ideal Gas",
"token_count": 1936,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Now it is easy to use equations (5.62) and (5.63) to find
$$e^{\text{rot}} = -k_B \theta \left[ -3e^{-\theta/T} - 9e^{-2\theta/T} - 42e^{-3\theta/T} + \mathcal{O}(e^{-4\theta/T}) \right]$$
(5.69)
and
$$c_V^{\text{rot}} = k_B \left(\frac{\theta}{T}\right)^2 \left[3e^{-\theta/T} - 18e^{-2\theta/T} + 126e^{-3\thet... | {
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"Header 2": "5.4 Specific Heat of a Hetero-nuclear Diatomic Ideal Gas",
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#### 5.5 Decoupling quantal Hamiltonians
Prove the "decoupling Hamilton implies factoring partition function" theorem of section 5.1.2 for quantal systems.
#### 5.6 Schottky anomaly
A molecule can be accurately modeled by a quantal two-state system with ground state energy 0 and excited state energy . Show that t... | {
"Header 1": "Statistical Mechanics",
"Header 2": "5.6 Problems",
"token_count": 1670,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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In the previous chapter, we found that at high temperatures, an ideal gas of diatomic molecules with spring interactions has a heat capacity of <sup>7</sup> 2 k<sup>B</sup> per molecule: <sup>3</sup> 2 k<sup>B</sup> from the translational degrees of freedom, k<sup>B</sup> from the rotational degrees of freedom, and k<s... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.1 Introduction",
"token_count": 265,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Before turning to statistical mechanics, let us review the quantal "interchange rule". The wavefunction for a system of three particles is a function of three variables: ψ(xA, xB, x<sup>C</sup> ). [The symbol x represents whatever is needed to specify the state: For a spinless particle in one dimension x represents the... | {
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"Header 2": "6.2 The Interchange Rule",
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This chapter considers collections of independent (i.e. non-interacting) identical monatomic particles. It does not treat mixtures or diatomic molecules. Notice that "independent" means only that the particles do not interact with each other. In contrast, each particle individually may interact with some background pot... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.3 Quantum Mechanics of Independent Identical Particles",
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It is of course possible that the resulting function vanishes, but this does not invalidate the procedure, because zero functions are both symmetric and antisymmetric!
When applied to a quantal wavefunction ψ(xA, xB, x<sup>C</sup> ), these processes result in the symmetric wavefunction
$$\hat{S}\psi(\mathsf{x}_{A},... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.3 Quantum Mechanics of Independent Identical Particles",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Thus, if |r, s, ti is an energy eigenstate, then
$$\hat{S}|r,s,t\rangle \equiv A_s[|r,s,t\rangle + |r,t,s\rangle + |t,r,s\rangle + |t,s,r\rangle + |s,t,r\rangle + |s,r,t\rangle]$$
(6.12)
is a symmetric state with the same energy, while
$$\hat{A}|r,s,t\rangle \equiv A_a[|r,s,t\rangle - |r,t,s\rangle + |t,r,s\rangl... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.3 Quantum Mechanics of Independent Identical Particles",
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To summarize the occupation number representation: an element of the symmetric basis is specified by the list
$$n_r$$
, for $r = 1, 2, ... M$ , where $n_r$ is $0, 1, 2, ...$ , (6.22)
and an element of the antisymmetric basis is specified by the list
$$n_r$$
, for $r = 1, 2, ... M$ , where $n_r$ is 0 or 1... | {
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"Header 2": "6.3 Quantum Mechanics of Independent Identical Particles",
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#### 6.3 Bases in quantum mechanics
We have just produced an energy eigenbasis for independent non-identical particles, one for independent bosons, and one for independent fermions. In each case did we produce the only possible energy eigenbasis or just one of several possible energy eigenbases? If the particles inte... | {
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"Header 2": "6.3 Quantum Mechanics of Independent Identical Particles",
"Header 3": "6.3.4 Problems",
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Now that we have an energy eigenbasis, the obvious thing to do is to calculate the canonical partition function
$$Z(\beta) = \sum_{\text{states}} e^{-\beta E}, \tag{6.26}$$
where for fermions and bosons, respectively, the term "state" implies the occupation number lists:
| fermions | (n1, n2, · · · , nM), nr<br>=... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.4 Statistical Mechanics of Independent Identical Particles",
"Header 3": "6.4.1 Partition function",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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In practice, these results from the grand canonical ensemble are used as follows: One uses these results to find quantities of interest as functions of temperature, volume, and chemical potential, such as the pressure p(T, V, µ). But most experiments are done with a fixed number of particles N, so at the very end of ... | {
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"Header 2": "6.4 Statistical Mechanics of Independent Identical Particles",
"Header 3": "6.4.1 Partition function",
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#### 6.10 Evaluation of the grand canonical partition function
Can you find a simple expression for Ξ(β, µ) for non-interacting particles in a one-dimensional harmonic well? For non-interacting particles in a one-dimensional infinite square well? For any other potential? Can you do anything valuable with such an expr... | {
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"Header 2": "6.4 Statistical Mechanics of Independent Identical Particles",
"Header 3": "6.4.4 Problems",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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"Particle in a box." Periodic boundary conditions. k-space. In the thermodynamic limit, the dots in k-space become densely packed, and it seems appropriate to replace sums over levels with integrals over k-space volumes. (In fact, there is at least one situation (see equation 6.83) in which this replacement is not corr... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.5 Quantum Mechanics of Free Particles",
"token_count": 433,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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In three dimensions, the chemical potential µ decreases with temperature. Why? It is clear that at T = 0, µ = E<sup>F</sup> > 0. But as the temperature rises the gas approaches the classical limit, for which µ < 0 (see problem 2.23). This is not proof, but it makes sense that µ should decrease with increasing temperatu... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.6 Fermi-Dirac Statistics",
"token_count": 2025,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
In most cases it is a good model to assume that the electrons are free and independent non-relativistic fermions at zero temperature. Consider a white dwarf of mass M and radius R, containing N electrons.
- a. Show that to a very good approximation, N = M/mp, where m<sup>p</sup> is the mass of a proton.
- b. Show tha... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.6 Fermi-Dirac Statistics",
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For T < T0(ρ) it is always true that µ = 0, but with sufficient cleverness you can again derive all equilibrium properties. Because these are two dramatically different procedures, we should expect that each part will behave quite differently, i.e. we expect a sudden change of behavior... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.6 Fermi-Dirac Statistics",
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#### 6.23 Character of the Bose function
What are the limits of the Bose function b(E) (equation 6.77) as E → ±∞? Is the curvature of the function greater when the temperature is high or when it is low?
#### 6.24 Thermodynamics of the Bose condensate
For temperatures less than the Bose condensation temperature T0... | {
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"Header 2": "6.6 Fermi-Dirac Statistics",
"Header 3": "6.7.3 Problems",
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There is a "set of the pants" heuristic argument showing that at low temperatures,
$$C_V(T) \approx k_B G(\mathcal{E}_F)(k_B T).$$
(6.91)
This argument is not convincing in detail as far as the magnitude is concerned and, indeed, we will soon find that it is wrong by a factor of $\pi^2/3$ . On the other hand it is... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.8 Specific Heat of the Ideal Fermion Gas",
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$$A_{\alpha}(x_0) = \int_0^{\infty} x^{\alpha - 1} g(x) \, dx$$
$$= \left[\frac{x^{\alpha}}{\alpha}g(x)\right]_{0}^{\infty} - \int_{0}^{\infty} \frac{x^{\alpha}}{\alpha}g'(x) dx$$
$$= \frac{1}{4\alpha} \int_{0}^{\infty} \frac{x^{\alpha}}{\cosh^{2}((x-x_{0})/2)} dx \qquad (6.108)$$
It's now easy to plot the piec... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.8 Specific Heat of the Ideal Fermion Gas",
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\tag{6.113}$$
This is the desired series in powers of $1/x_0$ ... and surprisingly, all the odd powers vanish! This result is known as "the Sommerfeld expansion".
Now we apply this formula to equation (6.104) to obtain the function $\mu(T)$ :
$$1 \approx \frac{\mu^{3/2}}{\mathcal{E}_F^{3/2}} \left[ 1 + \frac{\p... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.8 Specific Heat of the Ideal Fermion Gas",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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\tag{6.121}$$
Wow! At long last we're able to say
$$C_V = \frac{\partial E}{\partial T}\Big|_{NV} = \frac{\partial E}{\partial (k_B T/\mathcal{E}_F)}\Big|_{NV} \left(\frac{k_B}{\mathcal{E}_F}\right)$$
(6.122)
whence
$$C_V \approx \frac{3}{5}Nk_B \left[ \frac{5\pi^2}{6} \left( \frac{k_B T}{\mathcal{E}_F} \right)... | {
"Header 1": "Statistical Mechanics",
"Header 2": "6.8 Specific Heat of the Ideal Fermion Gas",
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#### 6.27 What if there were no interchange rule?
Suppose that the interchange rule did not apply, so that the basis on page 153 were the correct one for three identical particles. (Alternatively, consider a gas of N non-identical particles.) Find and sketch the heat capacity as a function of temperature.
#### 6.28... | {
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"Header 2": "6.9 Additional Problems",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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#### 7.1 The Problem
#### 7.1 The harmonic Hamiltonian
The Hamiltonian for lattice vibrations, in the harmonic approximation, is
$$\mathcal{H} = \frac{1}{2} \sum_{i=1}^{3N} m_i \dot{x}_i^2 + \frac{1}{2} \sum_{i=1}^{3N} \sum_{j=1}^{3N} x_i A_{ij} x_j.$$
(7.1)
Notice that this Hamiltonian allows the possibility t... | {
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"Header 2": "Harmonic Lattice Vibrations",
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Model a one-dimensional crystal as a chain of N atoms (each of mass m) connected by springs (each of spring constant K). The nth atom has a deviation from equilibrium position un(t). To find the motion of the nth atom, we need the force on that atom. (This could be found using the spring law F = −Kx, but it's easier to... | {
"Header 1": "Statistical Mechanics",
"Header 2": "7.3 Normal Modes for a One-dimensional Chain",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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If
$$G(\omega) d\omega = \text{number of normal modes with frequencies from } \omega \text{ to } \omega + d\omega$$
(7.9)
then
$$E^{\text{crystal}} = \int_0^\infty G(\omega) e^{\text{SHO}}(\omega) \, d\omega \quad \text{and} \quad C_V^{\text{crystal}} = \int_0^\infty G(\omega) c_V^{\text{SHO}}(\omega) \, d\omega ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "7.5 Low-temperature Heat Capacity",
"token_count": 1980,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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#### 7.6 Spin waves
For harmonic lattice vibrations at low frequencies, ω = csk. There are analogous excitations of ferromagnets called "spin waves" which, at low frequencies, satisfy ω = Ak<sup>2</sup> . Find the temperature dependence of the heat capacity of a ferromagnet at low temperatures. (Do not bother to eval... | {
"Header 1": "Statistical Mechanics",
"Header 2": "7.8 Additional Problems",
"token_count": 231,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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The subject of this chapter is also called "real gases", or "dense gases", or "non-ideal gases", or "imperfect gases", or "liquids and dense gases". The many names are a clue that the same problem has been approached by many different scientists from many different points of view, which in turn is a hint that the probl... | {
"Header 1": "Statistical Mechanics",
"Header 2": "8.1 Introduction",
"token_count": 1060,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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#### 8.2.1 Fluids in the canonical ensemble
The partition function for a fluid is
$$Z(T, V, N) = \frac{1}{h^{3N} N!} \int d\Gamma e^{-\beta H(\Gamma)}$$
(8.3)
where
$$H(\Gamma) = \frac{1}{2m} \sum_{i=1}^{N} \mathbf{p}_i^2 + U_N(\mathbf{r}_1, \dots, \mathbf{r}_N).$$
(8.4)
The momentum integrals can be performe... | {
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"Header 2": "8.1 Introduction",
"Header 3": "8.2 Perturbation Theory",
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The grand canonical partition function is
$$\Xi(T, V, \mu) = \sum_{N=0}^{\infty} e^{\beta \mu N} Z(T, V, N)$$
(8.9)
$$= \sum_{N=0}^{\infty} \left(\frac{e^{\beta\mu}}{\lambda^3(T)}\right)^N Q_N(T, V) \tag{8.10}$$
$$= \sum_{N=0}^{\infty} z^N Q_N(T, V) \tag{8.11}$$
$$= 1 + Q_1(T, V)z + Q_2(T, V)z^2 + Q_3(T, V)z^3 ... | {
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"Header 2": "8.1 Introduction",
"Header 3": "8.2.3 Fluids in the grand canonical ensemble",
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Begin with
$$\rho(z) = z + 2b_2 z^2 + 3b_3 z^3 + \mathcal{O}(z^4), \tag{8.33}$$
then use
$$z = \rho + \mathcal{O}(z^2)$$
$$z^2 = (\rho + \mathcal{O}(z^2))(\rho + \mathcal{O}(z^2))$$
$$z^3 = (\rho^2 + \mathcal{O}(z^3))(\rho + \mathcal{O}(z^2))$$
$$= \rho^2 + 2\rho\mathcal{O}(z^2) + \mathcal{O}(z^4)$$
$$= \... | {
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"Header 2": "8.1 Introduction",
"Header 3": "8.2.7 Appendix: Inverting ρ(z) to find z(ρ)",
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#### 8.4.1 One-particle distribution functions
What is the mean number of particles in the box of volume d 3 r<sup>A</sup> about rA?
The probability that particle 1 is in d 3 r<sup>A</sup> about r<sup>A</sup> is
$$\frac{d^3r_A \int d^3r_2 \int d^3r_3 \cdots \int d^3r_N \int d^3p_1 ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "8.4 Distribution Functions",
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"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Thus the mean number of particles in d 3 r<sup>A</sup> about r<sup>A</sup> is
$$n_{1}(\mathbf{r}_{A}) d^{3}r_{A}$$
$$= N \frac{d^{3}r_{A} \int d^{3}r_{2} \int d^{3}r_{3} \cdots \int d^{3}r_{N} \int d^{3}p_{1} \cdots \int d^{3}p_{N} e^{-\beta H(\mathbf{r}_{A}, \mathbf{r}_{2}, \mathbf{r}_{3}, \dots, \mathbf{r}_{N},... | {
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"Header 2": "8.4 Distribution Functions",
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Thus the mean number of pairs with one particle in $d^3r_A$ about $\mathbf{r}_A$ and the other in $d^3r_B$ about $r_B$ is
$$n_{2}(\mathbf{r}_{A}, \mathbf{r}_{B}) d^{3}r_{A}d^{3}r_{B}$$
$$= N(N-1) \frac{d^{3}r_{A} d^{3}r_{B} \int d^{3}r_{3} \cdots \int d^{3}r_{N} \int d^{3}p_{1} \cdots \int d^{3}p_{N} e^{-... | {
"Header 1": "Statistical Mechanics",
"Header 2": "8.4 Distribution Functions",
"token_count": 1170,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Consider a fluid in which each atom is modeled by a "hard sphere" of volume v0. In this system the potential energy vanishes unless two spheres overlap, while if they do overlap it is infinite. The criterion for overlap is that the centers of the two spheres are separated by a distance of 2r<sup>0</sup> or less, where ... | {
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"Header 2": "8.6 The Hard Sphere Fluid",
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The N-spin one-dimensional Ising model consists of a horizontal chain of spins, s1, s2, . . . , s<sup>N</sup> , where s<sup>i</sup> = ±1.
A vertical magnetic field H is applied, and only nearest neighbor spins interact, so the Hamiltonian is
$$\mathcal{H}_N = -J \sum_{i=1}^{N-1} s_i s... | {
"Header 1": "Statistical Mechanics",
"Header 2": "9.2 Free Energy of the One-Dimensional Ising Model",
"token_count": 2010,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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\tag{9.22}$$
Finally, using equation (9.17), we find the free energy per spin
$$f(T,H) = -J - k_B T \ln \left[ \cosh \frac{mH}{k_B T} + \sqrt{\sinh^2 \frac{mH}{k_B T} + e^{-4J/k_B T}} \right]. \tag{9.23}$$
#### Results
Knowing the free energy, we can take derivatives to find any thermodynamic quantity (see prob... | {
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"Header 2": "9.2 Free Energy of the One-Dimensional Ising Model",
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#### 9.4.1 Motivation and definition
How does a spin at one site influence a spin at another site? This is not a question of thermodynamics, but it's an interesting and useful question in statistical mechanics. The answer is given through correlation functions.
Consider an Ising model with spins s<sup>i</sup> = ±1 ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "9.4 Correlation Functions in the Ising Model",
"token_count": 1883,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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$$\sum_{i} G_{i} = \sum_{i} \langle s_{0} s_{i} \rangle - \langle s_{0} \rangle^{2}$$
$$(9.36)$$
$$= \sum_{i} \left[ \frac{\sum_{\mathbf{S}} s_0 s_i e^{-\beta \mathcal{H}}}{Z} - \frac{\left(\sum_{\mathbf{S}} s_0 e^{-\beta \mathcal{H}}\right)^2}{Z^2} \right]$$
(9.37)
$$= \frac{\sum_{\mathbf{S}} s_0(\mathcal{M}/m... | {
"Header 1": "Statistical Mechanics",
"Header 2": "9.4 Correlation Functions in the Ising Model",
"token_count": 1080,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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The previous subsection shows us how to find one thermodynamic quantity, χ(T, H) at some particular values for temperature and field, by knowing the correlation function Gi(T, H) for all sites i, at those same fixed values for temperature and field. In this subsection we will find how to calculate any thermodynamic qua... | {
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"Header 3": "9.4.3 All of thermodynamics from the susceptibility",
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I cannot resist telling you how the results of this section change when applied to fluid rather than magnetic systems. In that case the correlation function g2(r; T, V, N) depends upon the position r rather than the site index i. Just as the sum of G<sup>i</sup> over all sites is related to the susceptibility χ, so the... | {
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"Header 3": "9.4.4 Parallel results for fluids",
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#### 9.5.1 Basic strategy
If we're going to use a computer to solve problems in statistical mechanics, there are three basic strategies that we could take. I illustrate them here by showing how they would be applied to the Ising model.
- 1. Exhaustive enumeration. Just do it! List all the states (microstates, confi... | {
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"Header 2": "9.5 Computer Simulation",
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select a spin to flip]
Compute wa→b for transition to candidate
With probability wa→b, make the transition
Gather data concerning configuration [e.g. find M:=n↑ − n↓; Msum:=Msum + M]
END DO
Summarize and print data [e.g. Mave:=Msum/Nconfigs]
```
I'll make three comments concerning this algorithm. First of all, note t... | {
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| ❛ | ❛ | ❛ | ❛ |
|---|---|----|----|
| 8 | 9 | 10 | 11 |
| ❛ | ❛ | ❛ | ❛ |
| 4 | 5 | 6 | 7 |
| ❛ | ❛ | ❛ | ❛ |
| 0 | 1 | 2 | 3 |
It is not hard to show that, with skew-periodic boundary conditions, the four neighbors of site number l (l for location) are
```
(l + 1) mod NSites, (l − 1) mod NSites, (l +... | {
"Header 1": "Statistical Mechanics",
"Header 2": "9.5 Computer Simulation",
"token_count": 1979,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
#### 9.1 Denaturation of DNA
(This problem is modified from one posed by M.E. Fisher. It deals with a research question born in the 1960s that is still of interest today: Douglas Poland and Harold A. Scheraga, "Occurrence of a phase transition in nucleic acid models," J. Chem. Phys. 45 (1966) 1464–1469; M.E. Fisher, ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "9.6 Additional Problems",
"token_count": 2014,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
(Hint: tanh x = x − 1 3 x <sup>3</sup> + <sup>15</sup>x <sup>5</sup> + · · ·.) (Remark: Experiment shows that the magnetization is not precisely of the character predicted by mean field theory: While it does approach zero like (T<sup>c</sup> − T) β , the exponent β is not 1/2 — for two-dimensional systems β = 1/8, whil... | {
"Header 1": "Statistical Mechanics",
"Header 2": "9.6 Additional Problems",
"token_count": 1900,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \cdots \quad \text{for } |x| < 1 \quad \text{(the "geometric series")}$$
(A.1)
$$e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \frac{1}{4!}x^4 + \cdots$$
(A.2)
$$\ln(1+x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + \dots \quad \text{for } |x| < 1$$
(A.3) ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Series and Integrals",
"token_count": 251,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The integral
$$\int_{-\infty}^{+\infty} e^{-x^2} dx,\tag{B.1}$$
called the Gaussian integral, does not fall to any of the methods of attack that you learned in elementary calculus. But it can be evaluated quite simply using the following trick.
Define the value of the integral to be A. Then
$$A^{2} = \int_{-\in... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Evaluating the Gaussian Integral",
"token_count": 548,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The gamma function $\Gamma(s)$ is defined, for s > 0, by
$$\Gamma(s) = \int_0^\infty x^{s-1} e^{-x} dx. \tag{C.1}$$
Upon seeing any integral, your first thought is to evaluate it. Stay calm... first make sure that the integral exists. A quick check shows that the integral above converges when s > 0.
There is no... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Clinic on the Gamma Function",
"token_count": 719,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
I will call the volume of a d-dimensional sphere, as a function of radius, Vd(r). You know, of course, that
$$V_2(r) = \pi r^2 \tag{D.1}$$
(two-dimensional volume is commonly called "area") and that
$$V_3(r) = \frac{4}{3}\pi r^3. (D.2)$$
But what is the formula for arbitrary d? There are a number of ways to fin... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Volume of a Sphere in d Dimensions",
"token_count": 1742,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
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