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(I am assuming that there is no external potential, apart from the box confining the particles; see Alexander's answer for a discussion of the case when this is not true.) The density is uniform as long as the free energy is strictly convex in the density. Namely assume that the overall density $\rho = N/V$ belongs to an interval $I$ such that $$f_\beta(\... 3 Consider the Green’s function for a system at a nonzero temperature. It depends on time. One can look at its analytic continuation to imaginary time, even though time isn’t actually imaginary. In imaginary time, the function turns out to be periodic with a period inversely proportional to the temperature. This analytic behavior of finite-temperature Green’s ... 2 So, you want to know about entropy. Well, in thermodynamics it is defined as the measure of how much energy or heat is 'spread'. Then there is the second law: Entropy always increases a.k.a. energy tends to spread out over time. - Second Law of thermodynamics But, in a modern sense, we don't think of entropy just as a thermodynamic quantity measuring the ... 2 It should first be noted that Off-Diagonal Long-Range Order (ODLRO) and superfluidity do no necessarily go hand in hand. ODLRO is associated with a Bose-Einstein condensed phase (BEC), which usually also behaves as a superfluid and the latter thus inherits the ODLRO property. However, you can have systems that are superfluid but where BEC is not possible (... 2 This is a great question and both you and @valerio have given good answers. I will just add that, as suggested in the Jaynes paper you cite, the Gibbs paradox is really confusion about macrovariables and what constitutes a "subsystem". Entropy is only ever defined with respect to some set of macrovariables, so if you want to compare the total ... 2 In equilibrium state of non-interacting particles, number density is uniform only for box potential with some boundary conditions, e.g.$$V(x)=\cases{0,x\in V\\ \infty,\text{othersize}}$$For general potential the distribution is not uniform. Consider non-interacting particles in harmonic potential, that distribute as Gaussian. This is exactly n(x) \sim \... 2 Ultimately, the idea boils down to the fact that in quantum mechanics, you evolve states forward in time by applying the time evolution operator$$U(t) =\exp\left[-it\hat H\right] where $\hat H$ is the Hamiltonian (energy) operator. At the same time, the state of a quantum system in contact with a heat bath at temperature $T$ is encoded in the thermal ...

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To familiarize your self with the subject you may consult the aforementioned Arnold's book, or Nakahara "Geometry, Topology and Physics" https://www.academia.edu/29696440/GEOMETRY_TOPOLOGY_AND_PHYSICS_SECOND_EDITION_Nakahara, which is a very pedagogical introduction for physicists with the mathematical machinery. In simple words, the symplectic ...

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If OP is already familiar with Poisson brackets then it seems that the central piece of information relevant to OP's question is the following theorem. Theorem: Let there be given a $2n$-dimensional manifold $M$. There is a canonical bijective correspondence between symplectic structures $\omega\in\Omega^2(M)$ and non-degenerate Poisson structures $\{\cdot,\... 1 The idea that microcanonical ensemble would be "more fundamental" than other ensembles is strongly related to a mechanics-based approach to statistical mechanics, where the clean starting point would be a Hamiltonian isolated system of N particles. Therefore, a system at constant energy. However, that this is not the only possible point of view was ... 1 I know it's late but maybe someone will find it useful someday. I agree, this can be a bit confusing. But try to think it that way. You have 10 coins, and imagine that you toss them in a way that they make a line. Then a microstate HHHHHHHHHH is as probable as microstate HTTTHHTHTT - think of an order, what is the chance that you get exactly HTTTHHTHTT ... 1 There are many good introductory books on thermodynamics and you can find this topic everywhere. I personally read Heat and Thermodynamics by Zemansky which gives a good account of thermal physics and some intro to statistical physics. Then there is Hill, T., An Introduction to Statistical Thermodynamics which is pretty famous introductory book. Also you can ... 1 I googled a bit, but did not see anything that specifically talks about$N_2$and$O_2$separating. Separation of lighter components does occur at altitudes about 100 miles because the mean free path becomes longer than the scale of motion that mixes gases. At this altitude, sunlight creates reactive species. Atomic$O\$ becomes a predominant component. See ...

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I've always thought this was due to Brownian motion at the molecular level and mixing due to thermal convection. In other words if the molecules were not jiggling and moving around then they would indeed stratify themselves in terms of density (not mass). A nice experiment would be to allow a very tall cooled sealed cylinder of air to rest/settle for some ...

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If I have a single hydrogen atom with an electron in the ground state, can light of the correct energy excite this electron? The answer to this question is yes. To bring in your discussion of equilibrium statistical mechanics phenomena, it is best to consider an ensemble of hydrogen atoms at zero temperature. Then your question may be recast as: can light ...

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