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statistical mechanics | what molecule would have molar entropy $R \ln 2$ at $0K$? | https://physics.stackexchange.com/questions/74972/what-molecule-would-have-molar-entropy-r-ln-2-at-0k | <p>I was browsing my friends old notes and I came across the following problem that I am not sure if it's correct.</p>
<blockquote>
<p>Q. Prove that the molar entropy of <strong>CO</strong> of $0K$ would be $R \ln 2$.</p>
</blockquote>
<p>Here, it is considered that the thermodynamic probability of $N$ molecules of... | <p>CO is carbon monoxide. If you lay down these molecules in a lattice, each can have two orientations, almost equal in energy. Because of this, even at 0 K, carbon monoxide has <a href="http://en.wikipedia.org/wiki/Residual_entropy" rel="nofollow">residual entropy</a> because it's not a perfect crystal.</p>
<p>There ... | 900 |
statistical mechanics | Boltzmann distribution with interaction between particles? | https://physics.stackexchange.com/questions/78524/boltzmann-distribution-with-interaction-between-particles | <p>First of all, I would like to apologize in advance if I make stupid mistakes. I am a mathematician and I am trying to apply the Boltzmann distribution to places where I am not sure if it is applicable (albeit I have no choice). </p>
<p>The situation is: I have a system which consists in a discrete line of $M$ posit... | <p>Sure, it's no problem to do this. The thing that has to change is that $i$ should index over all possible configurations of the $N$ elements, and the energy in the Boltzmann distribution has to be the total energy of the system.</p>
<p>So if $M=10$, $N=3$ and $D=2$ then, for example,
$$
p([1,0,0,1,0,0,1,0,0,0]) = \... | 901 |
statistical mechanics | Logical understanding of the canonical probability distribution (canonical ensemble) | https://physics.stackexchange.com/questions/83325/logical-understanding-of-the-canonical-probability-distribution-canonical-ensem | <p>I am having problems in understanding the logic of this distribution:</p>
<p>$P(\Psi_{j})=\displaystyle\frac{e^{-E_{j}/kT}}{\displaystyle\sum_{j'}e^{-E_{j'}/kT}}$</p>
<p>The book I am studying use the case of a sample in contact with a reservoir at thermal equilibrium to derive this distribution. I understand the ... | <p>Maybe your intuition about energy and temperature need to be revisited. Your system can exchange energy with the reservoir at a given temperature. The system+reservoir will iterate through all microstates with equal probability (total energy being fixed), but you can show by using entropy arguments, that the probabi... | 902 |
statistical mechanics | Micro-canonical ensemble and classical reality | https://physics.stackexchange.com/questions/37740/micro-canonical-ensemble-and-classical-reality | <p>I seem to find a contradiction in the notion of probability density used by Landau and the notion of micro-canonical ensemble.</p>
<p>To see this, take an isolated classical system and we know experimentally that its energy lies between $E-\Delta$ and $E+\Delta$. So, we take a hypershell corresponding to these ener... | <p>No, you're not doing anything wrong, this is all correct. As an analogy, imagine I roll a die and hide it under a cup. Since you don't know which side of the die is facing upward, you represent it with a probability distribution, with an equal probability assigned to each of the six spaces. This probability distribu... | 903 |
statistical mechanics | Thermal equilibrium and non correlations | https://physics.stackexchange.com/questions/47709/thermal-equilibrium-and-non-correlations | <p>I read in a book on quantum fluctuations and quantum noise that, at thermal equilibrium the classical canonical variables are uncorrelated, ie: $$\langle xp\rangle=\langle x\rangle\langle p\rangle$$
But I am not sure to understand the sense of <em>at thermal equilibrium</em>, for me it just means $$\langle x^2\rangl... | 904 | |
statistical mechanics | Examples of systems with energy as an intensive variable | https://physics.stackexchange.com/questions/49789/examples-of-systems-with-energy-as-an-intensive-variable | <p>I need to consider a couple of examples of systems which have energies that are intensive variables - not extensive. I'be been thinking about this and I am not coming up with anything. My understanding is that extensive variables (at least wrt usual energies) scales with mass or length (system size). It also seems t... | <p>The total internal energy of a system is completely out of the question as an answer, of course. I would even go as far as saying that it is the quintessential and most important extensive variable of a system. Therefore, most things that have to do with 'energies' within thermodynamics will also be extensive variab... | 905 |
statistical mechanics | Statistical Mechanic | https://physics.stackexchange.com/questions/53381/statistical-mechanic | <p>One can define entropy as $$S=k\log{\omega(E)},$$ where $\omega(E)$ is the numbers of states with energy equal $E$; and the canonical partition function for a set of N particles is defined as$$Z_N=\sum_{\phi}e^{-\beta E[\phi]}=e^{-\beta F(\beta,N)},$$ where the sum run on states $\phi$ and the free energy is defined... | 906 | |
statistical mechanics | microcanonical distribution | https://physics.stackexchange.com/questions/58102/microcanonical-distribution | <p>We know that in an isolated system, the density matrix is the microcanonical distribution matrix. That this the possibility for all the states with energy in a certain interval is a constant?
But how can I deduce this from the postulate of equal probability? </p>
| <p>The assertion that the density matrix for an isolated system is that of the microcanonical ensemble <em>implies</em> the postulate of equal a priori probabilities since, as you indicate, it assigns equal probabilities to each of the energy eigenstates of the system.</p>
<p>I would then ask you the following questio... | 907 |
statistical mechanics | classical quantum particles in grand canonical ensemble | https://physics.stackexchange.com/questions/60256/classical-quantum-particles-in-grand-canonical-ensemble | <p>To derive Bose-Einstein and Fermi-Dirac distribution, we need to apply grand canonical ensemble:$Z(z,V,T)=\displaystyle\sum_{N=0}^{\infty}[z^N\sideset{}{'}\sum\limits_{\{n_j\}}e^{-\beta\sum\limits_{j}n_j\epsilon_j}]$. There is a constraint $\sideset{}{'}\sum\limits_{\{n_j\}}$ for quantum particles(bosons and fermion... | <p>For fermions, there is a constraint that each occupation number $n_i$ can only be either 0 or 1 because of the Pauli exclusion principle; no two fermions can occupy the same quantum state, but for bosons, there is no such constraint on the occupation numbers. For classical particles, namely those for which energy l... | 908 |
statistical mechanics | Number of particles in a microcanonical ensemble | https://physics.stackexchange.com/questions/62226/number-of-particles-in-a-microcanonical-ensemble | <p>Is it always assumed that, in a microcanonical ensemble, the number of particles is $N \gg 1$ ?</p>
<p>If no, are all the theorems related to the microcanonical description true even if the number of particles is small ?</p>
| <p>Numerical simulations are a good example to see with your own eyes that statistical mechanics results can be gotten without an infinite number of molecules in general and in particular in the microcanonical ensemble. However, one has to be aware of the finite size effects and see what is the difference with what you... | 909 |
statistical mechanics | Is this geometrical 'derivation' of Brownian motion legitimate? | https://physics.stackexchange.com/questions/12297/is-this-geometrical-derivation-of-brownian-motion-legitimate | <p>Here's a simple 'derivation' of the Brownian motion law that after N steps of unit distance 1, the total distance from the origin will be sqrt(N) on average. It's certainly not rigorous, but I'm wondering if people think it's reasonable, or possibly even a commonly known.</p>
<ol>
<li><p>An object takes one step fr... | <p>As a heuristic description, this is exactly right and correctly captures the essence of the subject. To turn it from a heuristic "derivation" into an actual derivation, you just need to make precise the notion that the two vectors are perpendicular on average. The precise meaning of "perpendicular on average" that's... | 910 |
statistical mechanics | How can I derive the analog of the susceptibility sum rule for the specific heat? | https://physics.stackexchange.com/questions/434199/how-can-i-derive-the-analog-of-the-susceptibility-sum-rule-for-the-specific-heat | <p>How can I derive the analog of the susceptibility sum rule for the specific heat? Does an infinite correlation length imply an infinite specific heat?
<span class="math-container">$$
\chi = \frac{\partial M}{\partial H}
= \frac{1}{N}\sum_{i,j} \Gamma(i,j)
$$</span></p>
| <p>I'll restrict my answer to the nearest neighbour Ising model with <span class="math-container">$N$</span> spins <span class="math-container">$s_i=\pm1$</span>,
for which the energy and magnetization are given by
<span class="math-container">$$
E=-J\sum_{\langle j,k\rangle} s_j s_k, \qquad M=\sum_j s_j
$$</span>
and... | 911 |
statistical mechanics | What is the sign of chemical potential of a noninteracting classical ideal gas obeying MB distribution? | https://physics.stackexchange.com/questions/454415/what-is-the-sign-of-chemical-potential-of-a-noninteracting-classical-ideal-gas-o | <p>The chemical potential of a noninteracting Bose gas can never be negative while that of a noninteracting Fermi gas can be both positive or negative. What can be said about the chemical potential of noninteracting classical ideal gas obeying MB distribution? </p>
| <p>The simplest way to compute this is through the grand canonical ensemble. The partition function for a single gas molecule is <span class="math-container">$Z_1 = V/\lambda^3$</span>, where <span class="math-container">$\lambda$</span> is the thermal de Broglie wavelength. Then the grand partition function is
<span c... | 912 |
statistical mechanics | Lagrange multipliers in Maxwell-Boltzmann statistics | https://physics.stackexchange.com/questions/461145/lagrange-multipliers-in-maxwell-boltzmann-statistics | <p>I'm following Wikipedia's derivation of <a href="https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics#Derivation_from_microcanonical_ensemble" rel="noreferrer">Maxwell-Boltzmann statistics</a>.</p>
<p>After applying Lagrange multipliers, we arrive at this expression for energy:</p>
<p><span class="ma... | <p>Here we are not choosing some constant. We are arriving at the values of <span class="math-container">$\beta$</span> and <span class="math-container">$ \alpha$</span>. In the first equations <span class="math-container">$\ln(W)$</span> and <span class="math-container">$N $</span> are the variables which are arbitrar... | 913 |
statistical mechanics | Electrons residing in an orbit with energy lower than the ground state energy | https://physics.stackexchange.com/questions/483835/electrons-residing-in-an-orbit-with-energy-lower-than-the-ground-state-energy | <p>Is it possible for an electron to reside in an energy level lower than that of the ground state? What happens to the electrons when an atom is brought down to 0K , do they come closer? What happens to the left of the orbitals ? </p>
| <p>As pointed out in a comment by another user in a previous post by you, you cannot have an electron occupy a lower energy state than the ground state. The ground state is the lowest energy state by definition. </p>
<p>As for the electron “coming closer” that’s more ambiguous in QM since we’re working with probabilit... | 914 |
statistical mechanics | Modern uses of classical statistical mechanics? | https://physics.stackexchange.com/questions/484734/modern-uses-of-classical-statistical-mechanics | <p>Most of the cases when I see applications of statistical mechanics is when Fermi-Dirac or Bose-Einstein statistic are used in condensed matter or the equilibrium equation of neutron stars.</p>
<p>Besides the Poisson-Boltzmann equation for colloids and plasma screening, I would like to know what are the modern devel... | <p>If by "classical statistical mechanics" one means the equilibrium statistical mechanics, i.e. excluding applications of statistical mechanics to systems out of equilibrium, the last half century or so has witnessed many new developments/applications. It is difficult to make an exhaustive list, but certainly it shoul... | 915 |
statistical mechanics | Approximation of the total number of accessible microstates | https://physics.stackexchange.com/questions/547933/approximation-of-the-total-number-of-accessible-microstates | <p>So, here is a system having two subsystems <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> where the two subsystems can exchange energy between them, then the total number of accessible microstates of the whole system is given by, <span class="math-container">$$\Omega(E)=... | <p>The approximation is
<span class="math-container">$$
\Omega_\alpha(\tilde{E}_\alpha) \gg \sum_{E_\alpha \ne \tilde{E}_\alpha} \Omega(E_\alpha)
$$</span>
or in words: the number of microstates of the most occupied macrostate (which is also very close to the one having the mean energy) dominates not just some of the o... | 916 |
statistical mechanics | Probability of particle overcoming an energy barrier | https://physics.stackexchange.com/questions/619723/probability-of-particle-overcoming-an-energy-barrier | <p>I'm reading this article called:An experiment to demonstrate the canonical distribution(by M. D. Sturge and Song Bac Toha)
Department of Physics, Dartmouth College, Hanover, New Hampshire 03755.</p>
<p>They talked about the probability of a particle overcoming and energy barrier of height <span class="math-container... | <p>According to Boltzman distribution, the probability of finding the particle in a energy <span class="math-container">$E$</span> is propotion to <span class="math-container">$exp(-\frac{E}{KT})$</span>. Consider a degeneracy for energy <span class="math-container">$E$</span> is <span class="math-container">$g(E)$</sp... | 917 |
statistical mechanics | Show that these definitions are equivalent | https://physics.stackexchange.com/questions/645667/show-that-these-definitions-are-equivalent | <ol>
<li><strong>Consider the three definition of entropy namely
<span class="math-container">\begin{eqnarray}
S &\equiv& k\log\Gamma(E), \label{1.1}\\
S &\equiv& k\log\Sigma(E), \label{1.2}\\
S &\equiv& k\log\omega(E), \label{1.3}
\end{eqnarray}</span>
where <span class="math-container"... | <p>Let's show the definition 2 and 3 are equivalent. First,
<span class="math-container">$$
\frac{\partial}{\partial E} \log \Sigma = \frac{\omega}{\Sigma}.
$$</span>
Therefore,
<span class="math-container">$$
\log \omega = \log\Sigma + \log \left( \frac{\partial}{\partial E} \log \Sigma \right).
$$</span>
Since the se... | 918 |
statistical mechanics | Contrasting the microcanonical ensemble with the general approach to thermal equilibrium | https://physics.stackexchange.com/questions/711617/contrasting-the-microcanonical-ensemble-with-the-general-approach-to-thermal-equ | <p>For those wondering precisely what I am referencing throughout, I am contrasting the discussion in Reif Chapter 3.4 (general thermal equilibrium) with Chapter 6.2 (microcanonical ensemble).</p>
<p>In the most general approach to thermal equilibrium (let us consider subsystems A and A' in which no external parameters... | 919 | |
statistical mechanics | Sharpness of multiplicity function | https://physics.stackexchange.com/questions/222543/sharpness-of-multiplicity-function | <p>This is quoted from Daniel Schroeder's <em>An introduction to thermal Physics</em>:</p>
<blockquote>
<p><span class="math-container">$$\Omega= \left(\frac{e}{N}\right)^{2N} \; e^{N\ln (q/2)^2} e^{-N(2x/q)^2}\;=\; \Omega_\text{max} \cdot e^{-N(2x/q)^2}\;. $$</span></p>
<p>A function of this form is called <strong>Gau... | <p>The concept the paragraph is trying to drive home is that although the absolute size of the Gaussian peak is very large, it's small when compared to the length of the graph. Since $q$ is going to be of comparable size to $N$, the total scale of the Gaussian will be about $N$ while the peak will cover $\sqrt{N}$. </p... | 920 |
statistical mechanics | How does $\rho(\dot{q_1}\mathrm dt)(\mathrm dq_2, \ldots,\mathrm dp_f )$ represent the no. of systems that would enter the volume in $\mathrm d t\;?$ | https://physics.stackexchange.com/questions/237826/how-does-rho-dotq-1-mathrm-dt-mathrm-dq-2-ldots-mathrm-dp-f-represe | <p>I've been following Reif's <em>Fundamentals of Statistical and Thermal Physics</em>; there I came before the derivation of <a href="https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)" rel="nofollow noreferrer">Liouville's theorem</a>:</p>
<p><a href="https://i.sstatic.net/puKHUl.png" rel="nofollow no... | <p>In the figure the volume element is moving from right to left. So the shaded region on the left multiplied by the "coarse-grained" density at that region is the number of systems entering the volume element. </p>
<p>The width of the region on the left is equal to the change in $q_1$ in time $dt$ is equal to $\dot{q... | 921 |
statistical mechanics | Average value of Force in rotating bead using statistical physics | https://physics.stackexchange.com/questions/241561/average-value-of-force-in-rotating-bead-using-statistical-physics | <p>Consider a mass m fixed to the middle point of a string of length $L$ whose extremities are a distance $$l$$ apart, and pulled with a tension $$F$$. The system is in thermal equilibrium, and one supposes that the only effect of thermal fluctuations is to make the system rotate about the horizontal (dashed) axis. As ... | <p>You only need to take the average of the expression you have found for $F$. Since $r\sin\theta$ is a constant, you can simply take it out of the average. That means $<F> = {1 \over r\sin\theta} <mv^2>$. Write $r\sin\theta$ in terms of $l$ and $L$, put the average of $mv^2$ in its place, and you will be d... | 922 |
statistical mechanics | how to interprete that the random forces in Langevin Equation are assumed to be delta-correlated | https://physics.stackexchange.com/questions/271440/how-to-interprete-that-the-random-forces-in-langevin-equation-are-assumed-to-be | <p>I mean that, is there anything more fundamental to yields the result that the random force in Langevin Equation is delta-correlated?<br/>
As is shown in the picture of a textbook below, its formula (3.4) is given by the assumption that "impacts are independent".However, it is still daunted for me to derive delta-cor... | <p>Delta-correlation is just an approximation. The actual forces that they represent are <em>not</em> truly delta-correlated. However, typical atomic-scale force autocorrelations last ~ 1 picosecond, so it's a pretty good approximation.</p>
<p>EDIT. To clarify, imagine dividing time up into tiny slices (~ 1 ps). The s... | 923 |
statistical mechanics | How to understand the map for the statistical mechanics? | https://physics.stackexchange.com/questions/309878/how-to-understand-the-map-for-the-statistical-mechanics | <p>Recently I found <a href="https://ocw.mit.edu/courses/physics/8-044-statistical-physics-i-spring-2013/readings-notes-slides/MIT8_044S13_L1.pdf" rel="nofollow noreferrer">a very interesting map</a> which seems to contain all these elements one will meet in the statistical mechanics.</p>
<p>So does anybody want to sha... | <p>From reading the PDF, issued by MIT OpenCourseWare, (that map is the first page of the PDF), it seems a straightforward outline of the key points of an SM - TD course, stressing the importance of probability, (the language reference) and then showing how it will be structured.</p>
<p>I can't see any more significa... | 924 |
statistical mechanics | What is the concept of Energy level in Maxwell-Boltzmann statistics? | https://physics.stackexchange.com/questions/587942/what-is-the-concept-of-energy-level-in-maxwell-boltzmann-statistics | <p>In Statistical thermodynamics Maxwell-Boltzmann statistics is considered a pre-quantum statistics. However in the mathematical treatment in all textbooks, and also in <a href="https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics" rel="nofollow noreferrer">Wikipedia article</a>, there is the concept of ... | <p>Yes, Maxwell and Boltzmann produced their theories well before quantum ideas were dreamed of, and they were developed by <a href="https://en.wikipedia.org/wiki/Elementary_Principles_in_Statistical_Mechanics" rel="nofollow noreferrer">Gibbs</a> into the form we know today using purely classical ideas. This involves ... | 925 |
statistical mechanics | Confusion about the Second Law of Thermodynamics and statistical mechanics | https://physics.stackexchange.com/questions/592443/confusion-about-the-second-law-of-thermodynamics-and-statistical-mechanics | <p>Suppose you have a container of volume V containing some gas with energy E and N particles. Let's assume the container to be isolated for now.</p>
<p>The microcanonical ensemble tells us that all microstates are equally likely. So a specific state in which all the molecules are at the top is as likely as a specific... | <p>Your question touches on some subtleties about which people very often get puzzled, and indeed there may not be universal agreement on the best way to describe the situation. I think the main point is that a macrostate should be defined in terms of the things which are contrained---so in your example, <span class="m... | 926 |
statistical mechanics | Mean values of position in Van der Waals coupled systems | https://physics.stackexchange.com/questions/593611/mean-values-of-position-in-van-der-waals-coupled-systems | <p>I'm trying to solve a problem regarding two systems interacting through a coupling hamiltonian:
<span class="math-container">$$ H(x,y) = H_1(x) + H_2(y) + kxy $$</span>
I am supposed to express the mean values <span class="math-container">$\langle x \rangle$</span> and <span class="math-container">$\langle y \rangle... | 927 | |
statistical mechanics | Is it possible for a particle to have all the energy of the Isolated System of particles? | https://physics.stackexchange.com/questions/596165/is-it-possible-for-a-particle-to-have-all-the-energy-of-the-isolated-system-of-p | <p>We have read the Fundamental postulate of statistical mechanics which says that :</p>
<blockquote>
<p>In a state of thermal equilibrium, All the accessible microstates of the system are equally probable.</p>
</blockquote>
<p>Suppose a system in thermal equilibrium with total energy to be <span class="math-container"... | <p>Because the macrostate is what you observe. Each macrostate is associated to a set o microstates. The probability of observing a macrostate is the sum of the probabilities of the equally likely microstates. Having all the particles at a single energy level has only one possibility. But having the particles spread ov... | 928 |
statistical mechanics | Apparent contradiction in Feynman's treatment of the velocity distribution of a gas | https://physics.stackexchange.com/questions/522369/apparent-contradiction-in-feynmans-treatment-of-the-velocity-distribution-of-a | <p>In Feynman's treatment of <a href="https://www.feynmanlectures.caltech.edu/I_40.html" rel="nofollow noreferrer">statistical mechanics</a>, <em>40–4 The distribution of molecular speeds</em>, Feynman found that the number of gas molecules per unit area per second who have an upper velocity component <span class="math... | 929 | |
statistical mechanics | Why when $-(\frac{\partial p}{\partial V})_T\geq 0$ we can say $-\frac{1}{V}(\frac{\partial V}{\partial p})_T\geq 0$? | https://physics.stackexchange.com/questions/535936/why-when-frac-partial-p-partial-v-t-geq-0-we-can-say-frac1v-fr | <p>Why when <span class="math-container">$-(\frac{\partial p}{\partial V})_T\geq 0$</span> we can say <span class="math-container">$-\frac{1}{V}(\frac{\partial V}{\partial p})_T\geq 0$</span> where V is the volume, p is the mean pressure of the system under consideration and T is the temperature which is kept fixed? Or... | <p>Assume a function <span class="math-container">$f{\left(x\right)}$</span> is invertible, e.g. by assuming that it is monotonic. Let its inverse be called, suggestively, <span class="math-container">$X{\left(F\right)}$</span>. We have</p>
<p><span class="math-container">$$X{\left(f{\left(x\right)}\right)} = x,$$</s... | 930 |
statistical mechanics | QHO in Microcanonical Ensemble: Problem with alternate derivation | https://physics.stackexchange.com/questions/101406/qho-in-microcanonical-ensemble-problem-with-alternate-derivation | <p>I am working through Franz Schwabl's book on Statistical Mechanics, and he has a number of derivations of thermodynamic quantities that are different than those I have seen before. I am also having difficulty finding them repeated elsewhere.</p>
<p>In particular, he has a method for calculating <span class="math-con... | <p>I) If we expect $\Omega(E)$ to depend analytically on the variable $\hbar\omega>0$ extended to (parts of) the complex plane, then we may regularize by introducing an $i\epsilon$ prescription, and substitute </p>
<p>$$\tag{1} \hbar\omega ~\longrightarrow ~ \hbar\omega (1-i\epsilon). $$</p>
<p>The variable </p>
... | 931 |
statistical mechanics | Calculating $C_v$ of a canonical ensemble | https://physics.stackexchange.com/questions/565081/calculating-c-v-of-a-canonical-ensemble | <p>I am writing code to find the heat capacity <span class="math-container">$C_v$</span> of a canonical <span class="math-container">$NVT$</span> ensemble. We know that,
<span class="math-container">$$C_v = \frac{\langle U^2 \rangle - \langle U \rangle ^2}{k_B T^2}$$</span></p>
<p>I have written a Metropolis algorithm ... | <p>The <span class="math-container">$P_m$</span> factors are unnecessary. Why? Because a state <span class="math-container">$i$</span> will be returned <span class="math-container">$\approx Np_i$</span> times in <span class="math-container">$N$</span> samples. Since you have a record of samples (indexed by <span class=... | 932 |
statistical mechanics | Microcanonical Ensemle | https://physics.stackexchange.com/questions/244657/microcanonical-ensemle | <p>The energy of a particle is given by E=|p|+|q|, where p and q are generalized momentum and coordinate respectively. All the states with E less than equal to E0 are equally probable and states with E greater than equal to E0 are inaccessible. What is the probability density of finding the particle at coordinate q wit... | 933 | |
statistical mechanics | How can we show that the BBGKY hierarchy is time symmetric? | https://physics.stackexchange.com/questions/249676/how-can-we-show-that-the-bbgky-hierarchy-is-time-symmetric | <p>I am trying to mathematically show that the BBGKY hierarchy for s particles is time symmetric by setting $t\rightarrow -t$. Using the Wikipedia notation for the s-particle we have</p>
<p>$\frac{\partial f_s}{\partial t} + \sum_{i=1}^s \dot{\mathbf{q}}_i \frac{\partial f_s}{\partial \mathbf{q}_i} + \sum_{i=1}^s \lef... | 934 | |
statistical mechanics | Probability distribution of two particle types system | https://physics.stackexchange.com/questions/108566/probability-distribution-of-two-particle-types-system | <p>Suppose that particles of two different species, A and B, can be chosen with
probability $p_A$ and $p_B$, respectively. </p>
<p>What would be the probability (and distribution) $p(N_A;N)$ that $N_A$ out of $N$ particles are of type A? </p>
<p>I'm trying to apply the Binomial distribution here but am bothered by th... | <p>These sorts of problems are easiest to think about if you build up from simpler problems.</p>
<p>Probability that n particles are all of type A: $p_A^{n}$.</p>
<p>Probability that, with two particles chosen, the first is of type A, and the second of type B: $p_A p_B$.</p>
<p>Probability that, with two particles c... | 935 |
statistical mechanics | state occupation rate $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}+{[1/-1/0]}}$ & density matrix $\rho _{m}=\frac{e^{-\frac{E_{m}}{kT}}}{Z(T)}$ | https://physics.stackexchange.com/questions/114718/state-occupation-rate-n-i-frac1e-beta-varepsilon-i-mu-1-1-0 | <p>Three kinds of distributions.
The <strong>states occupation rates</strong>:</p>
<h2><strong>F.D.</strong> $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}+1}$ <strong>B.E.</strong> $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}-1}$ <strong>Boltzmann</strong> $n_{i}=e^{-\beta (\varepsilon _{i}-\mu )}$</h2>
<p>$... | 936 | |
statistical mechanics | ensembles and lagrange multipliers | https://physics.stackexchange.com/questions/137433/ensembles-and-lagrange-multipliers | <p>In the derivation of maxwell-boltzmann distributions, the method of Lagrange multiplier is </p>
<p>$\sum n_i = N$</p>
<p>$\sum n_i E_i = E$</p>
<p>where $N$ is the total number of particles, and $E$ is the total energy. And we try to find the macrostate with the most microstates, I think the derivation is familia... | 937 | |
statistical mechanics | Exponentially increasing $\Omega(E)$ | https://physics.stackexchange.com/questions/137501/exponentially-increasing-omegae | <p>If I choose the number of microstates for energy $E$ to be $\Omega(E) = e^{aE}$ ($a>0$), its temperature is constant:
$$
kT = \left( {d\ln \Omega \over dE} \right)^{-1} = 1/a
$$
If I choose $\Omega(E) = e^{aE^2}$ ($a>0$), its temperature decreases as the energy goes up:
$$
kT = \left( {d\ln \Omega \over dE} \r... | <p>In string theory appears the so-called <em>Hagedorn behavior</em>, which is an exponential behavior of the density of states. As you point out, in this case, temperature does not vary with energy (so it means that the system has an infinite heat capacity!)</p>
<p>This behavior appears in Little String Theory, e.g. ... | 938 |
statistical mechanics | The number of states for fermions, bosons, and Boltzman in statistical mechanics | https://physics.stackexchange.com/questions/142723/the-number-of-states-for-fermions-bosons-and-boltzman-in-statistical-mechanics | <p>This is related with Equation 8.58 in Kerson Huang's 2nd edition of Statistical Mechanics. </p>
<p>The partition functions for the ideal gases are given as
$ Q_N (V,T) =\sum_{\{ n_p \}} g\{n_p \}e^{-\beta E\{n_p \}} $ where $E\{n_p \} =\sum_p \epsilon_p n_p$ and the occupation numbers are subject to the $\sum_p n_p... | <p>Bose particles <strong>cannot</strong> be identified as different in a given state, whereas boltzmann particles can (even though both types can occupy a given energy state with more than one particle). Thus boltzmann statistics need to take into account the <strong>permutations</strong> ($n!$) of the $n$ particles i... | 939 |
statistical mechanics | What does Born Green equation signify physically? | https://physics.stackexchange.com/questions/149561/what-does-born-green-equation-signify-physically | <p>What does Born Green equation obtained from YBG hierarchy for the equilibrium particle densities signify? I mean how can you model the equation into a physical problem?I understood the steps involved in the derivation of the expression, but am still unsure if I understood it physically, am unable to explain it prope... | <p>I'm not sure if there is much to physically understand in the equation itself, the derivation is where all the physical insight takes place (Kirkwood's superposition approximation). </p>
<p>What of the equation, then? Why is it important and why should anyone care? It's because the BBGKY hierarchy, while exact, can... | 940 |
statistical mechanics | Statistical mechanics: What is a "microscopic realization" of a system? | https://physics.stackexchange.com/questions/133720/statistical-mechanics-what-is-a-microscopic-realization-of-a-system | <p>What is a "microscopic realization" of a system?</p>
<p>The context is statistical mechanics. The microscopic system consists of many atoms (too many to track individually) with an assigned probability density function <code>f(x,y,z,Vx,Vy,Vz,t)</code>. </p>
<p>The macroscopic system consists of the atoms taken tog... | <p>Statistical mechanics relies on a probabilistic understanding of the world and as such one needs to define a probability space. In classical statistical mechanics the probability space consists of a domain which is the set of all possible <em>microstates</em> (that is the position and velocity vectors of all the par... | 941 |
statistical mechanics | The BBGKY Hierarchy | https://physics.stackexchange.com/questions/133787/the-bbgky-hierarchy | <p>The collision term in the Boltzmann equation can be derived from the BBGKY hierarchy. </p>
<p><a href="http://en.wikipedia.org/wiki/BBGKY_hierarchy" rel="nofollow">Wikipedia</a> says:</p>
<blockquote>
<p>In statistical physics, the BBGKY hierarchy [...] is a set of equations describing the dynamics of a system o... | <p>No, this is talking about correlations between s random particles. The s-particle distribution function is a 2*d*s (so 6s in 3 dimensional space) dimensional PDF that statistically describes s particles. For s=1, this is just the normal density in phase space. For s=2, this might show, for example, that more often t... | 942 |
statistical mechanics | How to derive the Bhatnagar-Gross-Krook collision integral from Boltzmann one? | https://physics.stackexchange.com/questions/154371/how-to-derive-the-bhatnagar-gross-krook-collision-integral-from-boltzmann-one | <p>Let's have Boltzmann collision integral:
$$
I_{coll} =\int d \sigma d^{3}\mathbf p_{1}(ff_{1} - f{'}f{'}_{1})|\mathbf v_{rel}|.\tag{1}\label{1}
$$
How to transform $\eqref{1}$ to BGK collision integral,
$$
I_{coll} = \frac{1}{\tau}(f - f_{0})?\tag{2}\label{2}
$$
Here $\tau = \frac{|v_{rel}|}{l} \sim v_{rel}N\sigma $... | 943 | |
statistical mechanics | Number of states of a simple system | https://physics.stackexchange.com/questions/166369/number-of-states-of-a-simple-system | <p>I am trying working on a problem in which there are two energy states $E_{1}<E_{2}$, and three different (i.e. distinguishable) particles. </p>
<p>I cannot decide if the order of the particles matters. If it doesn't, then there are 8 states. If order does matter, there are 24. My problem is not knowing the logic... | <p>first you have to specify what ensemble you are working in. in case of microcanonical ensemble (total energy fixed). no. of microstates will be decided by no. of different ways total energy can be E.then as only two energies are possible . let n1 particle are in state e1 n2 in state e2 with constraint that n1+n2=3 a... | 944 |
statistical mechanics | how will the distribution of the no. of particles be in a system ,(N,V,E) if N tends to infinity? | https://physics.stackexchange.com/questions/171499/how-will-the-distribution-of-the-no-of-particles-be-in-a-system-n-v-e-if-n-t | <p>MB distribution is followed if there are N no. of non interacting and distinguishable particles. But if N tends to infinity why does the no. of micro states reduces?
Is there any peak in the graph?</p>
| 945 | |
statistical mechanics | Statistical Mechanics deals with the same systems that Thermodynamics does? | https://physics.stackexchange.com/questions/174627/statistical-mechanics-deals-with-the-same-systems-that-thermodynamics-does | <p>Thermodynamics deals with "equilibrium states of macroscopic matter", that is, considering macroscopic systems there are states which can be characterized fully by a few number of measured degrees of freedom and on such states we are not able, through macroscopic measurements to see the fact that the molecules and a... | <p>Indeed, statistical mechanics in principle deals with completely general systems.</p>
<p><a href="https://en.wikisource.org/wiki/Elementary_Principles_in_Statistical_Mechanics" rel="nofollow">From the man himself who coined the term "statistical mechanics"</a>:</p>
<blockquote>
<p>The laws of thermodynamics, as ... | 946 |
statistical mechanics | Strange vector matrix operation (in "A Modern Course in Statistical Physics" by Reichl) | https://physics.stackexchange.com/questions/182773/strange-vector-matrix-operation-in-a-modern-course-in-statistical-physics-by | <p>I am reading "A Modern Course in Statistical Mechanics" by Linda E. Reichl. Where i encountered this notation:</p>
<p>$$\Delta S = \bar g : \vec \alpha \vec \alpha$$</p>
<p>Here $\bar g$ is $$ g_{i,j}=-{ \partial^2 S \over \partial A_i \partial A_j}\bigg|_{A_i = A_i^0, A_j = A_j^0 } $$
a matrix of second derivat... | <p>I think what they could mean is that $\vec{\alpha}\vec{\alpha}$ is a second rank tensor that is contracted with $\overline{g}$. I saw this notation being used in the context of electrodynamics before. It is used to get a simple notation for multi-dimensional Taylor series. So we get</p>
<p>$$\Delta S=\overline{g}:\... | 947 |
statistical mechanics | Bending moment and Shear force | https://physics.stackexchange.com/questions/187147/bending-moment-and-shear-force | <p>Do bending moment and shear force of a beam depend on it's cross sectional dimentions??
Since all the diagrams which I have draw so far don't involve any cross section details. So I think they do not depend on them and don't do any influences on the shear force diagrams and the bending moment diagram.</p>
| <blockquote>
<p>Do bending moment and shear force of a beam depend on it's cross sectional
dimentions??</p>
</blockquote>
<p>I never underestimate the power of experiment, and that indeed, physics and all of science is rooted deeply in experiment. Is it easier for me to snap a foot-long twig or the one foot high ... | 948 |
statistical mechanics | How Statistical Physics? | https://physics.stackexchange.com/questions/187970/how-statistical-physics | <p>It's a common fact that in physics, we use statistics (or maybe probabilities ) to describe the behaviour of a system. It was from the statistical analysis of a system where quantum statistics arose and then the theory of quantum mechanics began.</p>
<p>How is possible to make such descriptions and to construct the... | <p>We can use statistics by being willing to ask different questions.</p>
<p>No individual particle has a pressure or a temperature. But we can ask about subsystems with particular pressures or temperatures.</p>
<p>So we group collections into subsystems that are large enough and regular enough to have collective pro... | 949 |
statistical mechanics | Boltzmann distribution for angles? | https://physics.stackexchange.com/questions/189522/boltzmann-distribution-for-angles | <p>Consider a system whose sole degree of freedom is an angle $\theta$ that goes from $0$ to $2\pi$. Let $E(\theta)$ be its energy function. Obviously, $E(\theta)$ is $2\pi$-periodic. What's the general form for the Boltzmann distribution for $\theta$? Is it just: $P(\theta)\propto e^{-E(\theta)/kT}$? Or is there some ... | <p>There's no issue with the energy having an angular dependency. This is similar to the case of a spin in a magnetic field, in which the energy is</p>
<p>$$E = -\mathbf{\mu \bullet B}$$</p>
<p>or </p>
<p>$$E (\theta) = -\mu B cos(\theta)$$</p>
<p>This poses no problem. As you say, the Boltzmann factor is $e^{\mu B... | 950 |
statistical mechanics | The different in wear test when using Aluminum and Steel disc in pin on disc apparatus | https://physics.stackexchange.com/questions/203401/the-different-in-wear-test-when-using-aluminum-and-steel-disc-in-pin-on-disc-app | <p>In wear test of pin on disc apparatus i found that mass loss of pin when i used Aluminum disc is higher than when i used Steel disc under the same conditions ,pressure, velocity and contact time can anyone explain this behavior to me and give me the reason ?</p>
| <p>Sometimes, when you have a soft material like aluminum, and a brittle material like asbestos, in a pin/disk configuration, particles of asbestos break off and become embedded in the aluminum. And now you have created a very abrasive disk (a bit like diamond particles in phosphor bronze), and you will wear the pin mu... | 951 |
statistical mechanics | Energy transfer in form of work or heat? | https://physics.stackexchange.com/questions/205394/energy-transfer-in-form-of-work-or-heat | <p>Suppose a system A which is a vessel of water with two electrodes, connected by a resistor, placed in the water. </p>
<p>If you apply voltage to the electrodes, energy is transferred from the battery (not included in system A) to system A.</p>
<p>I read in a book that the form of energy transferred is work, and no... | <p>Sadly, as @Brionius comments, you must have misunderstood the book.</p>
<p>The 1st law of thermodynamics, which sets up the energy balance, says that:</p>
<p>$$\Delta U=Q-W$$</p>
<p>$U$ is internal energy, so $\Delta U$ covers the change in the total <em>contained</em> energy in the system. Both <strong>work</str... | 952 |
statistical mechanics | equilibrium state of a system in statistical mechanics | https://physics.stackexchange.com/questions/211202/equilibrium-state-of-a-system-in-statistical-mechanics | <p>Why we consider the maximum number of micro states or complexions as equilibrium state of a macro state or a system in statistical physics?</p>
| <p>In statistical mechanics all micro-states are considered to be equally likely. This means that the most likely macro-state is the one that contains the most micro-states. This macro-state is considered at equilibrium because one the system is there, it is unlikely to move away from that macro state, as the vast majo... | 953 |
statistical mechanics | Difference between macroscopic variable, macroscopic observable, parameter and generalized force in Thermodynamics | https://physics.stackexchange.com/questions/228668/difference-between-macroscopic-variable-macroscopic-observable-parameter-and-g | <p>When I read Books about statistical physics, then often names like "macroscopic variable / observable", parameter of the macroscopic state and generalized force are used, and I want to know, what is the difference, and wether there are definitions for that. Plus, I want to know of what type are the commonly quantiti... | <p>The general principle is that macroscopic variables and macrostates are not "real" from the microscopic, Hamiltonian perspective. They're things that we, human beings on the scale of $10^{23}$ atoms, make up based on what we can observe. </p>
<p>For example, let's take pressure. Given a microstate $\Gamma$, you can... | 954 |
statistical mechanics | How is $ \left(1-\frac{p^2}{2mE}\right)^{3N/2-2} =\; \exp\left(-\frac{3N}{2}\frac{p^2}{2mE}\right)\;?$ | https://physics.stackexchange.com/questions/227654/how-is-left1-fracp22me-right3n-2-2-exp-left-frac3n2-fra | <p>How is
$$ \left(1-\frac{p^2}{2mE}\right)^{3N/2-2} = \exp\left(-\frac{3N}{2}\frac{p^2}{2mE}\right)$$
(Karder, Statistical Physics of Particles, Page 107)</p>
<p>in the large $E$ limit. Here $N$ is particle, of the order of $10^{23}$, $E$ is the total energy.
I roughly guess that it should be $\exp(-\frac{p^2}{2m})$... | <p>Add a comment needs 50 reputation, and I got only 46 now. So I write my opinion here.
I have read the textbook, the original formula is
$$p(\vec{p_1})=(1-{{\vec{p_1}^2}\over {2mE}})^{3N/2-2}\cdots\cdots$$</p>
<p>So $\vec{p_1}$ is the momentum of only one particle in the ensemble. Considering the system has very lar... | 955 |
statistical mechanics | Partition function of primon bosonic gas | https://physics.stackexchange.com/questions/229111/partition-function-of-primon-bosonic-gas | <p>Can we interpret the <strong>Euler product formula</strong> " $\sum\frac{1}{n^s} = \prod_{p\;\mathrm{prime}} \frac{1}{1-p^{-s}} $ " in a stat. physical sense, as a product of single-particle system <em>partition functions</em>, considering them <em>statistically independent</em> ?</p>
| <p>Umm...OK well lets see what happens.</p>
<p>Lets let $s = \beta\varepsilon$, where $\varepsilon$ is some fixed energy and
$$
Z\left(\frac{s}{\varepsilon}\right) = \zeta(s)
$$.</p>
<p>To get some kind of idea for what kind of system $Z$ describes we need to find the energy levels of the system and to do that we ne... | 956 |
statistical mechanics | Does fluctuation really occur in equilibrium as its microstates are allowed to occur by Fundamental Postulate in equilibrium? | https://physics.stackexchange.com/questions/234396/does-fluctuation-really-occur-in-equilibrium-as-its-microstates-are-allowed-to-o | <p>The Fundamental Postulate says:</p>
<blockquote>
<p>In <em>equilibrium</em>, all accessible microstates are equally likely.</p>
</blockquote>
<p>Accessible means having same energy.(right?)</p>
<p>Let a container is taken full of gas having number of particles $N_,$ volume $V$ and energy $E\:_;$ the system is i... | <blockquote>
<p>At equilibrium, the system would be in that macrostate which would have the maximum multiplicity or the largest number of microstates; that would correspond to gas totally dispersed over the whole volume $V\;.$</p>
</blockquote>
<p>This is wrong, and based on a misunderstanding of terminology. You ha... | 957 |
statistical mechanics | Negative amount of particles in a grand canonical ensemble | https://physics.stackexchange.com/questions/210152/negative-amount-of-particles-in-a-grand-canonical-ensemble | <p>Knowing that, in the $\mu$-canonical (or micro-canonical) and canonical ensembles, the number of particles is held constant and usually reflects the actual number of particles, in which case $N_{MC} \simeq N_{actual}$, and also $N_C \simeq N_{actual}$. Or otherwise </p>
<p>$\text{sgn}(N_{MC}) = \text{sgn}(N_{actual... | <p>No, this cannot happen, simply because in general $N_G \ge 0$, which can easily be proven in a quantum-statistical setting.</p>
<p>$N_G$ is defined as $N_G = \frac 1 {Z_G} \mathrm{Tr}(\rho_G N)$, now consider the following expansion of the trace in terms of a complete set of states $\left|n\alpha\right>$ which a... | 958 |
statistical mechanics | Derivation of Fermi-Dirac Distribution | https://physics.stackexchange.com/questions/240623/derivation-of-fermi-dirac-distribution | <p>How can I derive the Fermi-Dirac distribution function using simple mathematics?
I am now tired of looking for the derivation on the net.So please help me to understand how actually electrons are distributed between various energy levels?</p>
| <p>Okay, so do you understand derivation of thermodynamics of an ideal gas and what partition function is? This derivation can be done through use of principle of maximization of entropy and Lagrange multipliers. Do you know how to derive partition function of an ideal gas with fixed number of particles (canonical ens... | 959 |
statistical mechanics | Langevin Paramagnetism for dipoles rotating in 2D | https://physics.stackexchange.com/questions/677796/langevin-paramagnetism-for-dipoles-rotating-in-2d | <p>The energy for dipoles in a magnetic field can be described by <span class="math-container">$$H = - \mathbf{m} \cdot \mathbf{B}.$$</span> What I did for an exercise was integrate the partition function for the case where I allow rotation in three dimensions (reducing it to the integral <span class="math-container">$... | 960 | |
statistical mechanics | Question regarding expectation value of energy and gibbs factor | https://physics.stackexchange.com/questions/679293/question-regarding-expectation-value-of-energy-and-gibbs-factor | <p>In lecture we introduced the average energy for a single particle as <span class="math-container">$$E = \langle \varepsilon \rangle = \frac{\int d \varepsilon g(\varepsilon) e^{- \beta \varepsilon} \varepsilon}{Z_1}$$</span>
where <span class="math-container">$Z_1 = \int d \varepsilon e^{- \beta \varepsilon}$</span>... | <p>To start with your second question</p>
<blockquote>
<p>How does the definition of the expectation value here account for indistinguishable particles?</p>
</blockquote>
<p>It doesn't. What you have calculated here is not the expected energy of <span class="math-container">$N$</span> indistinguishable particles, it is... | 961 |
statistical mechanics | MCE nr. of microstates & proof that entropy is extensive (ideal gas in a box) | https://physics.stackexchange.com/questions/682081/mce-nr-of-microstates-proof-that-entropy-is-extensive-ideal-gas-in-a-box | <p>I am trying to find the nr. of microstates inside a box of dimensions <span class="math-container">$L_1,L_2,L_3$</span> of a hypersphere in the phase space.We have a gas of N particles in 3D. While we have a 6N dimensional phase space because we assumed that the gas is ideal, we do not have potential energy in the H... | <ol>
<li><p><span class="math-container">$R$</span> is not <span class="math-container">$E$</span>, since you are thinking about the <span class="math-container">$3N$</span>-dimensional hyper-space of momenta, and <span class="math-container">$\sum_i \mathbf{p}_i^2=2mE$</span>, so the radius is really <span class="math... | 962 |
statistical mechanics | In statistical mechanics, is the partition function a mathematical cumulative distribution function as in probability mathematics? | https://physics.stackexchange.com/questions/685153/in-statistical-mechanics-is-the-partition-function-a-mathematical-cumulative-di | <p>In statistical mechanics, is the partition function a mathematical cumulative distribution function as in probability mathematics ?</p>
<p>Why is the partition function the fundamental objects for statistical mechanisms ?</p>
<p>Why isn't it the probability <em>density</em> function ?</p>
| 963 | |
statistical mechanics | Microstates and Combinations | https://physics.stackexchange.com/questions/693650/microstates-and-combinations | <p>There are two boxes (which I will call 1 and 2) that are initially thermally isolated and have a sliding door in between them. We can write the probability of configuration <span class="math-container">$A$</span> in box 1 as,</p>
<p><span class="math-container">$$P_1(A)=\frac{1}{\Omega_1}$$</span></p>
<p>Similarly, ... | <p>I believe it does actually work out using Stirling's approximation (it's a lot of annoying algebra):</p>
<p><span class="math-container">$$\ln a!\approx a\ln a - a$$</span></p>
<p>First expression:</p>
<p><span class="math-container">$$\ln \left( \frac{n!}{r!(n-r)!} \right)^2 = 2(\ln n! - \ln r! - \ln(n-r)!)$$</span... | 964 |
statistical mechanics | bridging the connection from the Helmholtz free energy in classical thermo to stat mech | https://physics.stackexchange.com/questions/182774/bridging-the-connection-from-the-helmholtz-free-energy-in-classical-thermo-to-st | <p>The Helmholtz-free energy from classical thermo is defined as
$$\text{F=u-TS}$$</p>
<p>taking the differential and algebraic manipulation, we arrive at</p>
<p>$$\text{dF=-pdv-sdT}$$</p>
<p>Observe that:</p>
<p>$$\text{p=-(}\frac{\text{$\delta $F}\backslash }{\text{$\delta $v}})_T$$ and $$\text{S=-(}\frac{\text{... | <p>There are multiple ways to justify this identification.</p>
<ul>
<li>The first one is to recognize that in thermodynamics the Helmholtz free energy is the right thermodynamic potential to figure out spontaneous evolution of your thermodynamic system (in virtue of the second principle of thermodynamics) at fixed (N,... | 965 |
statistical mechanics | Occupation number of a bosonic gas for high temperatures | https://physics.stackexchange.com/questions/694716/occupation-number-of-a-bosonic-gas-for-high-temperatures | <p>If we consider the average occupancy for a bose gas, we know:</p>
<p><span class="math-container">$$\langle n \rangle_B=\frac{1}{e^{\beta(\epsilon-\mu)} -1}$$</span></p>
<p>I also know that for high temperatures this transforms itself to the occupancy for a classical Boltzmann gas:</p>
<p><span class="math-container... | 966 | |
statistical mechanics | Can susceptibility be infinite even for small systems? | https://physics.stackexchange.com/questions/694727/can-susceptibility-be-infinite-even-for-small-systems | <p>Susceptibility can be espressed in terms of Gibbs free energy as:</p>
<p><span class="math-container">$$\chi^{-1}= \frac{\partial^2g}{\partial m^2}$$</span></p>
<p>Where <span class="math-container">$g$</span> is the intensive Gibbs free energy. So if the second derivative of <span class="math-container">$g$</span> ... | <p>I don't think so. Typically, the nonanalyticity of the free energy comes from the thermodynamic limit. In finite systems, the free energy is (almost) always a finite sum of analytic functions so there is no possibility of divergence of derivatives.</p>
<p>The divergence of the susceptibility is typically linked to a... | 967 |
statistical mechanics | Boltzmann distribution: derivation from canonical distribution | https://physics.stackexchange.com/questions/106288/boltzmann-distribution-derivation-from-canonical-distribution | <p>I'm trying to understand the Maxwell-Boltzman Distribution, and in particular the derivation from the boltzman distribution for energy. I have successfully created an incorrect derivation, but I'm not sure what's wrong with it :). Any guidance would be much appreciated!</p>
<p>I believe that the probability density... | 968 | |
statistical mechanics | Phase-space diagrams of microstates-infinite-dimensional? | https://physics.stackexchange.com/questions/695242/phase-space-diagrams-of-microstates-infinite-dimensional | <p>I've just begun learning Statistical Mechanics and my question concerns my professor's statement and I quote:</p>
<blockquote>
<p>Consider a gas of <span class="math-container">$N$</span> atoms which can have position <span class="math-container">$q_i$</span> and momentum <span class="math-container">$p_i$</span>. I... | <p>Sometimes in a lecture, it may happen to make careless mistakes. I would advise everybody to check in the textbook, and if this doesn't provide an answer, ask the lecturer directly.</p>
<p>In the present case, it is clearly false that the space of microstates for a finite system is infinite-dimensional. However, it ... | 969 |
statistical mechanics | How do we know there are no local maxima in the number of microstates with respect to energy? | https://physics.stackexchange.com/questions/705610/how-do-we-know-there-are-no-local-maxima-in-the-number-of-microstates-with-respe | <p>In my textbook, the definition of temperature begins by determining the maximum number of microstates for two systems in thermal equilibrium of energies <span class="math-container">$E_1$</span> and <span class="math-container">$E_2$</span> and microstates <span class="math-container">$\Omega_1(E_1)$</span> and <spa... | 970 | |
statistical mechanics | What is the particle density for a Gaussian distribution in position and velocity? | https://physics.stackexchange.com/questions/711853/what-is-the-particle-density-for-a-gaussian-distribution-in-position-and-velocit | <p>Consider that the position and momentum of my particles have a Gaussian distribution.
If I now calculate the number of particles in
<span class="math-container">$dx$</span> with momentum <span class="math-container">$dp$</span> then which one of the following would it be:</p>
<p><span class="math-container">$$d^2N=C... | <p>These are two different distributions, which one is correct depends on the system of particles you are trying to describe.</p>
<p>In the first one, the particles are uniformly distributed in space, with a Gaussian momentum distribution of zero mean and standard deviation <span class="math-container">$1$</span>. In t... | 971 |
statistical mechanics | Boltzmann distribution - why does distinguishability increase likelihood? | https://physics.stackexchange.com/questions/712644/boltzmann-distribution-why-does-distinguishability-increase-likelihood | <p>I am looking through derivations of the Boltzmann distribution. The method I've seen uses an argument that involves counting distinguishable microstates of a system with fixed energy, and then assuming that these distinguishable microstates are equally likely to occur.</p>
<p>A first assumption is that from an exper... | <p>I think that there are some basic concepts to be reviewed in your question.</p>
<p>First, a macrostate is defined by macroscopic variables, in your example of the four particles, fixing the macrostate is equal to fixing the total energy of the system. A microstate is defined by a configuration compatible with a macr... | 972 |
statistical mechanics | Confusion about fundamental assumption of statistical mechanics | https://physics.stackexchange.com/questions/713939/confusion-about-fundamental-assumption-of-statistical-mechanics | <p>I am confused about the fundamental assumption of statistical mechanics. It says, over a long time scale, that all microstates are equally accessible. I get it so far.</p>
<p>But for microstate, there are arrangements such that no particles(thinking of Einstein solid) have energy (Daniel schroeder), how is that phys... | 973 | |
statistical mechanics | Fermions and bosons weakly degenerate gases | https://physics.stackexchange.com/questions/729892/fermions-and-bosons-weakly-degenerate-gases | <p>I've tried to derive the pressure, <span class="math-container">$P$</span>, for a weakly degenerate gas of fermions (and analogously for bosons). The strange thing is that the expression I calculated is correct except for the sign of a term. I checked the calculations and i don't see any error, so the error might be... | <p>Your <span class="math-container">$\mu$</span> expression is correct in the classical limit only. There is a correction term which is the same order as the correction you have kept. That is you should substitute for
<span class="math-container">$\Omega^{MB}(T,V,\mu) = -\frac{VT}{\Lambda^3} e^{\beta \mu}$</span> to g... | 974 |
statistical mechanics | What does the chromatic polynomial have to do with the Potts model? | https://physics.stackexchange.com/questions/482/what-does-the-chromatic-polynomial-have-to-do-with-the-potts-model | <p><a href="http://en.wikipedia.org/wiki/Potts_model">Wikipedia</a> writes:</p>
<blockquote>
<p>In statistical mechanics, the Potts
model, a generalization of the Ising
model, is a model of interacting spins
on a crystalline lattice.</p>
</blockquote>
<p>From combinatorics conferences and seminars, I know tha... | <p>The relationship between the chromatic polynomial and the Potts model is a special case of the relationship between the Tutte polynomial and the random cluster model of Fortuin and Kastelyn. There's a very tiny bit about this in <a href="http://en.wikipedia.org/wiki/Tutte_polynomial#Definitions">the wikipedia page ... | 975 |
statistical mechanics | Applicability of Baxter's method for IRF models | https://physics.stackexchange.com/questions/6899/applicability-of-baxters-method-for-irf-models | <p>In a interaction-round-a-face model of $n^2$ particles in a lattice, a weight $W(a,b,c,d)$ is assigned to each face in the lattice based on the spins $a,b,c,d$ (listed say from the bottom-left corner in counter-clockwise fashion) of the particles at the corners of the face. Based on this, a partition function is fo... | 976 | |
statistical mechanics | Is there any physics behind flocking? | https://physics.stackexchange.com/questions/10578/is-there-any-physics-behind-flocking | <p>There are many articles published in physics journals about <a href="http://en.wikipedia.org/wiki/Flocking_%28behavior%29" rel="nofollow">flocking</a>. Is there a physical reason for these phenomena or is it just because physics methods are being used to study collective motion?</p>
<p>It seems there is no true mec... | <p>For your first question: Yes, there is a physical reason for this phenomenon as is (or should be) for every observable phenomenon.</p>
<p>For your second part: Depends on who is debating the validity or applicability of that theorem. Can you please add a reference, then perhaps someone may be able to answer it.</p>
... | 977 |
statistical mechanics | Can somebody provide some sort of crash course on random walk and its problems at the level of a beginning undergraduate student in physics? | https://physics.stackexchange.com/questions/12733/can-somebody-provide-some-sort-of-crash-course-on-random-walk-and-its-problems-a | <p>I really need some very simple discussions of random walk (probability). Couldn't get anything from class, more so from Reif. Thanks!</p>
| <p>Random walk is intimately connected with diffusion, heat equation, Laplacian, harmonic functions, quadratic forms and Gaussian distribution. Here's a sketch of the relationship in the discrete case (continuous case is conceptually similar but requires much more familiarity with probability theory). I'll try to be as... | 978 |
statistical mechanics | Cross-field diffusion from Smoluchowski approximation | https://physics.stackexchange.com/questions/13804/cross-field-diffusion-from-smoluchowski-approximation | <p>I'm reading <em>An Introduction to Stochastic Processes in Physics</em> by Don S Lemons. Problem 10.2 leads to a pair of equations:</p>
<p>$dV_x = -\gamma V_xdt+V_y\Omega dt-V_y\sqrt{2\gamma dt}N_t(0,1)$</p>
<p>$dV_y = -\gamma V_ydt-V_x\Omega dt+V_x\sqrt{2\gamma dt}N_t(0,1)$</p>
<p>($\gamma$, $\Omega$ constants,... | <h3>Original wrong stuff</h3>
<p>Well, the OP asked for a solution, but I still can't figure out how this is physics. Add the two equations with an i to get the equation for $V=V_x + i V_y$ (dividing by dt--- I hate finance/mathematics conventions, and calling the white noise $\eta$)</p>
<p>${dV\over dt} = (-\gamma+i... | 979 |
statistical mechanics | How "to take" this integral? | https://physics.stackexchange.com/questions/28529/how-to-take-this-integral | <p>When I learned anharmonic model of crystal, I read that considering anharmonic oscillations and Boltzmann distribution for the "atoms" of crystal we can get the dependence of distance between the "atoms" from a temperature as</p>
<p>$$
\langle r \rangle = r_{0} + \alpha T.
$$</p>
<p>As I understood the words be... | <p>You expand the top and bottom integral in a power series in b, and keep the lowest order term in $b$:</p>
<p>For the top integral:</p>
<p>$$ \int x e^{-ax^2} (1 + bx^3) = b\int x^4 e^{-ax^2} = b {d^2\over da^2} \int e^{-ax^2} = b{d^2\over da^2} \sqrt{\pi\over a} = -\sqrt{\pi\over a} {3b\over 4a^2} $$</p>
<p>For t... | 980 |
statistical mechanics | Spin 3/2 Statistical Mechanics Problem | https://physics.stackexchange.com/questions/43625/spin-3-2-statistical-mechanics-problem | <p>I am trying to solve a problem from the book 'Introductory Statistical Mechanics' (Bowley, Sanchez). The question reads:</p>
<p>Calculate the free energy of a system of N particles, each with spin 3/2 with one particle per site, given that the levels associated with the four spin states have energies e, 2e, -e, -2e... | <p>If there are no term in the Hamiltonian lifting the degeneracy on the spin, then you'll have a degree of degeneracy equal to $$g_s=(2s+1)$$ where s is the spin ($=\frac32$ so $g_s=4$ in your case).</p>
<p>Each correspond to a projection of the spin along some axis (the z usually) $$s_z = \frac32,\frac12,-\frac12,-\... | 981 |
statistical mechanics | Why do we need different ensembles in statistical mechanics? | https://physics.stackexchange.com/questions/54758/why-do-we-need-different-ensembles-in-statistical-mechanics | <p>Why do we study these different ensembles, microcanonical, canonical, grand canonical ensemble ? Are they used for studying different physical system or scenarios?(e.g. in some system you can only treat it as mirocanonical, and in other cases you can only apply canonical ensemble) Do they have the same result at the... | <p>If somebody tells you what the entropy is as a function of energy, volume, and number of particles, you have all the information you need (for a standard plain vanilla system). It is not <em>necessary</em> to define any other ensemble, but it is <em>convenient</em>. If your system for instance is in contact with a b... | 982 |
statistical mechanics | Partition function for multidimensional scaling energy | https://physics.stackexchange.com/questions/55525/partition-function-for-multidimensional-scaling-energy | <p>Let $D_{ij}$ a random matrix with i.i.d positive coefficients. One can take for instance $D_{ij}$ uniformly distributed in [0,1]. We consider the following energy function $H(x)$ defined for $x=(x_i)_1^n$, with each $x_i\in \mathbb{R}^k$, where $n>k$ are two positive integers:</p>
<p>$$H(x) = \sum_{i,j} \left(\|... | 983 | |
statistical mechanics | Why the chemical potential of massless boson is zero? | https://physics.stackexchange.com/questions/60499/why-the-chemical-potential-of-massless-boson-is-zero | <p>In Bose-Einstein condensation, the chemical potential is less than the ground state energy of the system($\mu<\epsilon_g$). But why does the massless boson such as photon have zero chemichal potential($\mu=0$)?</p>
| <p>The chemical potential is a complementary variable to $N$, the number of particles (of a certain kind), and they get combined in the same sense as $-\beta,H$ and similar pairs. The chemical potential "punishes" too high or too low number of particles in grand canonical and similar distributions such as
$$\exp(-\beta... | 984 |
statistical mechanics | What is the minimum non-integer dimension for which the XY model shows a phase transition? (if well-defined) | https://physics.stackexchange.com/questions/64552/what-is-the-minimum-non-integer-dimension-for-which-the-xy-model-shows-a-phase-t | <p>I know that <a href="http://en.wikipedia.org/wiki/Classical_XY_model" rel="nofollow">XY statistical model</a> for $d=2$ doesn't show a regular phase transition , while the $3d$ has, I was wondering what is the behaviour for $2< d < 3$.</p>
<p>If it is simpler one could consider another model in its class of u... | 985 | |
statistical mechanics | Why is velocity normally distributed in a gas, but not energy? | https://physics.stackexchange.com/questions/74660/why-is-velocity-normally-distributed-in-a-gas-but-not-energy | <p>If one looks at a cubic box of gaseous atoms all initially flying in the same direction at the same speed (but flying at an angle to the walls, so as not to reflect up-and-down against the box walls forever), they will collide with the walls and each other, their previously uniform velocities becoming messed up rand... | <p>The more natural relationship between the two distributions is the opposite one. The Boltzmann distribution
$$\exp(-E/kT)$$
is the more general one (connected with the microscopic definition of the temperature $T$ in any system in physics) and one may simply substitute the kinetic energy $mv^2/2$ for $E$ to get the ... | 986 |
statistical mechanics | Practical difference between canonical and grand canonical ensembles | https://physics.stackexchange.com/questions/80050/practical-difference-between-canonical-and-grand-canonical-ensembles | <p>I'm currently doing some calculations which require evaluating various standard thermal expectation values in the canonical ensemble (both bosons and fermions). Now, in order to make my theoretical machinations easier, I am actually using the grand canonical ensemble, where the chemical potential acts as a Lagrange ... | <p>As you know, the thermodynamic limit requires the system to grow to infinite size while keeping the same density, which lets you neglect surface effects. It also requires the lack of long-range interactions so that distant parts can act independently. So, you need to neglect gravitational interactions, allow the sys... | 987 |
statistical mechanics | Gap exponents and homogeneous functions | https://physics.stackexchange.com/questions/89637/gap-exponents-and-homogeneous-functions | <p>Looking at <a href="http://www.tcm.phy.cam.ac.uk/~bds10/phase/scaling.pdf" rel="nofollow">this paper on page 1</a> how is the first limit obtained? That is, if I have some homogeneous function $g_f(h/t^{\Delta})$, how does setting the gap exponent $\Delta$ to $3/2$ ensure that $$\lim_{x \to 0} g_f(x) = -1/u?$$</p>
| <p>Setting $\Delta=3/2$ is useful only to ensure the correct behavior of the second limit.</p>
<p>The first limit is given by the condition $f(t,0)\propto t^2$. Because $f(t,h)=t^2 g_f(h/t^\Delta)$, you directly get that $g_f(x)\to {\rm const}$ for $x\to 0$.</p>
<p>For the sake of completeness, let's do the other cas... | 988 |
statistical mechanics | What are the key properties of and differences between classical and quantum statistical mechanics? | https://physics.stackexchange.com/questions/89850/what-are-the-key-properties-of-and-differences-between-classical-and-quantum-sta | <p>I'm studying different ensembles and different statistics (<a href="http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution" rel="nofollow">M-B</a>, <a href="http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics" rel="nofollow">B-E</a>, <a href="http://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_st... | <p>Let's recall basics of classical and quantum mechanics for non-statistical systems.</p>
<p>In classical Hamiltonian mechanics, one models the non-statistical state of a system as a point in phase space $\mathcal P$. If the configuration space (space of <em>spatial</em> positions) of the system is $N$-dimensional, ... | 989 |
statistical mechanics | Where does the Maxwell-Boltzmann distribution come from? | https://physics.stackexchange.com/questions/91708/where-does-the-maxwell-boltzmann-distribution-come-from | <p>I understand that <a href="http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution" rel="nofollow">Maxwell-Boltzmann distributions</a> arise for distributions of weakly interacting particles at equilibrium. But I'd like to know if there's a deeper reason behind why they are specifically Maxwellian. </p>
... | <p>Maxwell derived it from simple assumptions about collisions of the molecules. If the interaction is weak (decreases fast enough with distance), use of the Boltzmann-Gibbs probability distribution of states of the molecule</p>
<p>$$
\rho(\mathbf r,\mathbf p) = \frac{e^{-\frac{E(\mathbf r,\mathbf p)}{k_B T}}}{Z},
$$
... | 990 |
statistical mechanics | 'Fermi-Dirac'-like occupation probability at high temperature | https://physics.stackexchange.com/questions/93075/fermi-dirac-like-occupation-probability-at-high-temperature | <p>Consider an ensemble of $N\to\infty$ free particles, each of which can assume energy states $E_i\in\{0,E\}$. Using the canonical ensemble one can compute the occupation probability for a single of those particles to be in the excited state $E_i=E$ (or equivalently the expectation value for what fraction of all parti... | <p>Perhaps the following (which basically involves investigating what happens for a general system with discrete energy spectrum) will help.</p>
<p>The canonical partition function for a quantum system with discrete spectrum $\{E_n\}$ is
\begin{align}
Z = \sum_ne^{- E_n/(kT)}
\end{align}
and the population fraction ... | 991 |
statistical mechanics | Classical regime for Fermi-Dirac and Bose-Einstein gases | https://physics.stackexchange.com/questions/93638/classical-regime-for-fermi-dirac-and-bose-einstein-gases | <p>I'm studying statistical mechanics, in particular classical regime for Fermi Dirac and Bose Einstein gases. Time average value for occupation numbers in FDBE statistics:
$$ \langle n_\epsilon\rangle_{FB} = \frac{1}{e^{(\epsilon-\mu)\beta}\pm1} $$
For Boltzmann Statistics:
$$ \langle n_\epsilon \rangle_B = e^{(\mu-\e... | 992 | |
statistical mechanics | Boltzmann–Gibbs-distribution as resulting from a limiting density of states? | https://physics.stackexchange.com/questions/95174/boltzmann-gibbs-distribution-as-resulting-from-a-limiting-density-of-states | <p>I'm interested in the relation between the probability distribution $p_i$ over states of a system on the one side and the density of states $\rho(\eta)$ of its environment. (Meaning, $\int_{\eta_a}^{\eta_b} \rho(\eta) ~ \mathrm{d} \eta$ is the number of environment states with energies in the interval $[\eta_a, \eta... | <p>When one proves that a small part of a greater system is described canonical ensemble, even though the greater system is described by a microcanonical ensemble, the key point is that the density of states of the greater system has the exponential form you mention, <em>over a certain interval of energy</em>.</p>
<p>... | 993 |
statistical mechanics | The Maxwell and the Boltzmann distributions | https://physics.stackexchange.com/questions/103453/the-maxwell-and-the-boltzmann-distributions | <p>I am trying to understand where the Boltzmann distribution comes from. I recently learned some interesting things of which my interpretation follows below. Did I interpret correctly? If so, is this all there is to it, or is this only part of the story?</p>
<p>When sampling coordinates in a high-dimensional Euclidea... | 994 | |
statistical mechanics | Statistical mechanics of a coin toss | https://physics.stackexchange.com/questions/108156/statistical-mechanics-of-a-coin-toss | <p>I'd like to ask some questions about flipping two coins related to statistical mechanics, e.g. microcanonical distribution, phase space distribution function etc... after I rephrase the coin flipping problem into the language of statistical mechanics.</p>
<p>In probability theory, given the following problem</p>
<... | 995 | |
statistical mechanics | What is an intuitive explantion for the fact that the Maxwell-Boltzmann distribution of energies is independent of mass? | https://physics.stackexchange.com/questions/119739/what-is-an-intuitive-explantion-for-the-fact-that-the-maxwell-boltzmann-distribu | <p>If you take the Maxwell-Boltzmann distribution of velocities (which depends on the mass) and substitute $v=\sqrt{\frac{2E}{m}}$ you get the distribution for the energies, which turns out to be independent of mass. What physical reality does this reflect? Why is the velocity distribution mass dependent, whereas the e... | <p>The simplest way to think about this is that, when considering the Maxwell-Boltzmann distribution of velocities, the sole energy scale in the system (as is mostly the case when considering thermal equilibrium systems) is set by the temperature as $k_BT$. So the probabilities ($p(E)\mathrm{d}E$) being dimensionless c... | 996 |
statistical mechanics | Derivation of Landau diamagnetism | https://physics.stackexchange.com/questions/122081/derivation-of-landau-diamagnetism | <p>In deriving the magnetic susceptibility of free electrons, we need to calculate</p>
<p>$$\chi = \left( \frac{\partial M}{\partial H} \right)_N = - \left( \frac{\partial^2 F}{\partial H^2} \right)_N.$$</p>
<p>Here, $F = E- TS $ is the free energy. It should be emphasized that the particle number $N$ is held fixed. ... | <p>$$-\left(\frac{\partial^2\Omega}{\partial H^2}\right)_\mu=\left(\frac{\partial M}{\partial H}\right)_\mu=\left(\frac{\partial M}{\partial H}\right)_N+\left(\frac{\partial M}{\partial N}\right)_H\left(\frac{\partial N}{\partial H}\right)_\mu$$</p>
<p>In principle, there is a difference between these quantities, at l... | 997 |
statistical mechanics | Statistical mechanics: Meaning of "accessible" in "accessible microstates" | https://physics.stackexchange.com/questions/133608/statistical-mechanics-meaning-of-accessible-in-accessible-microstates | <p>What does "accessibility" mean in statistical mechanics?</p>
<p>Is it an equivalent concept to accessibility in mathematical control theory?</p>
<p>I'll provide an example:
When two systems A and B interact on a subspace of their respective spaces, i.e. where they overlap in space, does accessibility of states of... | <p>When people say "accessible microstates" it means "microstates consistent with a set of <em>constraints</em> or <em>conditions</em> which you are supposed to keep in the back of your mind".</p>
<p>The most common example of a constraint is a fixed amount of energy. A closed physical system with exactly one constrai... | 998 |
statistical mechanics | Physical meaning of coefficient of variation | https://physics.stackexchange.com/questions/139866/physical-meaning-of-coefficient-of-variation | <p>While doing a course in statistical physics I came across a term called coefficient of variation. Now according to <a href="http://en.wikipedia.org/wiki/Coefficient_of_variation" rel="nofollow">Wikipedia</a>, coefficient of variation</p>
<blockquote>
<p>shows the extent of variability in relation to mean of the p... | <p>Suppose you have some radioactive material with a half life $\tau_{1/2}$. What that term "half life" means is that the amount of material $m(t)$ you have left after a time $t$ is</p>
<p>$$m(t) = m(0) \exp[ -t / \tau_{1/2}] . \qquad (*)$$</p>
<p>However, the material is made up of discrete atoms and each one decays... | 999 |
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