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The ergodic hypothesis is not part of the foundations of statistical mechanics. In fact, it only becomes relevant when you want to use statistical mechanics to make statements about time averages. Without the ergodic hypothesis statistical mechanics makes statements about ensembles, not about one particular system. To understand this answer you have to ...

27

As for references to other approaches to the foundations of Statistical Physics, you can have a look at the classical paper by Jaynes; see also, e.g., this paper (in particular section 2.3) where he discusses the irrelevance of ergodic-type hypotheses as a foundation of equilibrium statistical mechanics. Of course, Jaynes' approach also suffers from a number ...

21

If you have only one species of particles then working with $(\mu,p,T)$ ensemble does not make sense, as its thermodynamic potential is $0$. $$U = TS -pV + \mu N,$$ so the Legendre transformation in all of its variables (i.e. $S-T$, $V-(-p)$ and $N-\mu$) $$U[T,p,\mu] = U - TS + pV - \mu N$$ is always zero. The fact is called Gibbs-Duhem relation, i.e. $$0 ... 21 Actually, temperature is defined as$$\frac{1}{T} = \frac{\partial S}{\partial E} = \frac{k_B}{\Omega}\frac{\partial\Omega}{\partial E} So in order to have zero temperature, you would need a system with either zero multiplicity, which you can't have by definition, or an infinite derivative $\partial\Omega/\partial E$ even though the multiplicity itself ...

18

UPDATE: Below I am answering yes to the first question in the post (are the two kinds of entropy the same up to a constant). This has led to some confusion as both Matt and John gave answers saying "the answer is no", however I believe they are referring to the title "Does entropy measure extractable work?". Although the author uses the two interchangeably, ...

17

Statistical Mechanics is the theory of the physical behaviour of macroscopic systems starting from a knowledge of the microscopic forces between the constituent particles. The theory of the relations between various macroscopic observables such as temperature, volume, pressure, magnetization and polarization of a system is called thermodynamics. first ...

17

Arnold Neumaier's comment about statistical mechanics is correct, but here's how you can prove it using just thermodynamics. Let's imagine two bodies at different temperatures in contact with one another. Let's say that body 1 transfers a small amount of heat $Q$ to body 2. Body 1's entropy changes by $-Q/T_1$, and body 2's entropy changes by $Q/T_2$, so ...

14

From a fundamental (i.e., statistical mechanics) point of view, the physically relevant parameter is coldness = inverse temperature $\beta=1/k_BT$. This changes continuously. If it passes from a positive value through zero to a negative value, the temperature changes from very large positive to infinite (with indefinite sign) to very large negative. ...

13

The realization of non-Abelian statistics in condensed matter systems was first proposed in the following two papers. G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991) X.-G. Wen, Phys. Rev. Lett. 66, 802 (1991) Zhenghan Wang and I wrote a review article to explain FQH state (include non-Abelian FQH state) to mathematicians, which include the explanations ...

13

If you were to surround the atmosphere by an adiabatic envelope and allow it to come to equilibrium, it probably would settle into such a state. However, the atmosphere is not a static place. It is actively mixed due to heating of the ground by the sun, and by cooling of the upper atmosphere by radiation into space. This makes the surface air less dense than ...

13

A convenient operational definition of temperature is that it is a measure of the average translational kinetic energy associated with the disordered microscopic motion of atoms and molecules. The underlying framework of all matter is quantum mechanical. This means that the Heisenberg Uncertainty principle holds. Even for a single particle the HUP ...

12

The Ising model is a model, originally developed to describe ferromagnetism, but subsequently extended to more problems. Basically, it is an interaction model for spins. Imagine you have a system which is a collection of $N$ spins. Each spin $S_i$ has two possible states $+1$ or $-1$. Here you can imagine already a possible extension to more states. You can ...

12

Another use of the SLE approach seems to be (I haven't read the papers below much beyond their abstracts) as a tool to probe for the presence of conformal invariance in various systems, when a direct (numerical or experimental) verification is difficult. In this approach, (i) one extracts suitable non self-crossing paths, (ii) one determines (empirically) ...

11

SLEs can be used in order to define a certain kind of QFT. You can check M. Douglas' talk, from page 28 forward: Foundations of Quantum Field Theory (PDF). There's also another great article, Conformal invariance and 2D statistical physics. You may also like SLE and the free field: Partition functions and couplings. Finally, there's an approach to try and ...

11

Hannesh, you are correct that the second law of thermodynamics only describes what is most likely to happen in macroscopic systems, rather than what has to happen. It is true that a system may spontaneously decrease its entropy over some time period, with a small but non-zero probability. However, the probability of this happening over and over again tends ...

10

I guess so - I mean, as far as I know, there's no law of physics that strictly prohibits those "exotic" states from being realized. As long as the state exists and can be reached by some path from the "center" of the state space where the likely states are, there should be a nonzero (not even infinitesimal, really) probability of accessing it. But for a ...

10

First of all we must distinguish between two things that are called entropies. There's a microscopic entropy, also called Shannon Entropy, that is a functional over the possible probability distributions you can assign for a given system: $\displaystyle H[p] = -\sum_{x \in \mathcal{X}}\; p(x) \log(p(x))$ where $\mathcal{X}$ is the set where your variable ...

10

First and foremost, the BEC systems studied in detail today do not involve the formation of any bonds between atoms. Bose-Einstein Condensation is a quantum statistical phenomenon, and would happen even with noninteracting particles (though as a technical matter, that's impossible to arrange, but you can make a condensate and then manipulate the interactions ...

10

A somewhat similar (yet inverted) question was posed, infamously, by Einstein, Podolsky, and Rosen as an argument against the same phenomenon you are challenging. The basic argument was that such quantum wave collapses are indistinguishable from dice falling under couches, but under opposite grounds as yours -- both must be "determined" by what were later ...

10

From a fluid dynamics standpoint, as a body moves through a fluid, a small region of fluid is dragged along with it. This is what forms the boundary layer. In the near-body region, odor will be dragged along with the body. Likewise, behind a moving person is a turbulent wake and a low pressure region. The low pressure reason will "suck" the odor along with ...

10

Here's a self-contained (hopefully clear) derivation. Step 1. Setup and definition of differential scattering cross-section Let $\mathcal L$ denote the incident luminosity (number of incident particles per unit area, per unit time) of a beam to be scattered. We assume that we have a spherical detector at infinity with which to measure scattered particles, ...

10

I think gatsu's right: it's because of an entropic attraction resulting from the fact that two spheres whose centers are less than $2a$ apart leave more room for other spheres. To see why this happens, it may help to draw a picture: Here, the blue spheres all have radius $\frac12a$. The gray dashed circle around each sphere has radius $a$, and shows the ...

9

Generalities on Conformal Invariance In two dimensions, a lot is known / conjectured about statistical models at criticality. For instance, at $T_c$, the spin configuration that you see will not only be self-similar (what others here have been calling "fractal") but actually fully conformally invariant (in the continum limit); that is, the probability ...

9

It is important to digest appropriately Anderson's comments about scaling. In Physics, when one talks of "scaling phenomena", what's really being talked about are these two things: Renormalization Group; & Effective Field Theory. And, as i mentioned above, conformal symmetry plays a leading role in all of this discussion. Roughly, the bottom line is ...

9

There's no temperature. If we use the following definition "temperature is the average kinetic energy of the particles". Then no particles - no temperature. As the first sight this answer doesn't seem to be good enough, but if you want to calculate "average spin" or "average charge" those parameters will have no sense if there's no particles to calculate ...

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