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I've only recently started studying statistical mechanics and I'm quite confused with the MaxEnt and anti-MaxEnt ideas. I'm looking for a concise answer, if it is possible, not really a description or debate of the critiques and arguments of MaxEnt, I just want to have a clear idea of where it stands nowadays.

Also, the debate got me thinking about the current status of Statistical physics. I'd like to know to what extent statistical mechanics is experimentally validated. It seems to me that most of e work done on statistical mechanics is through simulations, I tend to favor experimental validation over theoretical derivations or simulations, so hard evidence that, for example, entropy values calculated with statistical mechanical methods are consistent to a high degree of accuracy with experimental entropy values would be appreciated.

I hope that the question is concise and clear enough, I'm aware that this kind of questions can sometimes fuel subjective debate.

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  • $\begingroup$ What is clear is that you can't take the equal prior probability postulate literally, but doing so leads to results that do work. For classical systems you can work around this, but real systems are quantum mechanical. The question is then why does statistical mechanics work? The eigenstate thermalization hypothesis seems to be a good explanation. $\endgroup$ – Count Iblis Aug 29 '15 at 17:54
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    $\begingroup$ What is MaxEnt? $\endgroup$ – Kyle Kanos Aug 29 '15 at 17:55
  • $\begingroup$ Ok, probably a good idea I guess. $\endgroup$ – Ignacio Aug 30 '15 at 16:03
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The Maximum Entropy principle, principally popularized by Jaynes, is known by most people having studied statistical physics. The way I see it, although Jaynes considered it as crucial in the foundations of equilibrium statistical mechanics and other people in the field still do (like Roger Balian for instance), it is more taught and thought of as a useful way of retrieving the Gibbs' ensembles without thinking too much about what it is we are doing. So, I would say that the MaxEnt is viewed as an interesting and curious tool/phenomenon to which not so much importance is given in practice. Although I am quite partisan of the MaxEnt idea, I reckon it may not be enough on its own to explain the successes and solve the foundational problems of equilibrium statistical thermodynamics. In the recent years, I think the mainstream theoretical physics community has lost a bit of interest in the MaxEnt principle/idea in favour of an allegedly "more objective" way of getting the statistical physics framework based on the theory of large deviations.

As for the validity of statistical mechanics; it is validated in many ways by being able to find out the right constitutive relations between thermodynamics variables in various systems and by relating these thermodynamic observables to microscopic parameters. The whole of the theoretical understanding of gases, liquids and more generally condensed matter physics rely on it. It is fair to say that it is extremely successful. As for comparing actual simulations or theoretical numbers with experiments, there has been huge successes in the calculations of the specific heat of solids for instance. Melting temperatures of crystals are computed on a daily basis with usually very good agreement with experiments and effective interactions (like this one) between mesoscopic particles can only be apprehended with this framework etc... In fact, the list is so long and so broad that I don't know where to really start the list.

That being said, it is worth noting that predictions in statistical mechanics rely of course on the theoretical framework but also equally on the microscopic model used with the theory. A dramatic example is that of the liquid-gas phase transition is atomic and molecular systems. If the chosen range of the attractive potential used between the atoms/molecules is too short, then one never sees the transition as its corresponding critical point is then located below the fluid-solid transition.

Thus, dealing with statistical mechanics is a hard job and testing its predictions is also a very hard job. And when disagreement occur, considering the successes of the framework so far, it is often wiser to look at the model first before reconsidering the theory.

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    $\begingroup$ I don't see what you mean by "getting the statistical physics framework based on the theory of large deviations". Large deviations theory, by definition, requires a probabilistic framework, which is precisely what is difficult to derive in an objective way from the underlying mechanical theory. The only thing that large deviations help you with is moving from one description (say, microcanonical) to another one (say, canonical). But you'd still need to derive the microcanonical probability measure in some way, and this is the hard part. $\endgroup$ – Yvan Velenik Aug 30 '15 at 7:41
  • $\begingroup$ @YvanVelenik: you are right but that's not my point. MaxEnt does not avoid the problem of the prior probability to put in the Shannon entropy (that is one of its weaknesses). From the point of view of large deviations, starting from some prior models (micro canonical or canonical), one can get meaningful results for macroscopic observables involving known thermodynamic potentials; which in turn can "validate" the prior probability used. $\endgroup$ – gatsu Aug 30 '15 at 22:37
  • $\begingroup$ Sure, but a interpretation of probabilities in statistical mechanics is the core of any derivation of the latter. MaxEnt provides a subjective interpretation of the latter, the ergodicity/mixing approaches attempt (and fail) to provide a mechanical interpretation. I don't see how large deviations theory can be considered as an "alternative" approach to MaxEnt in this respect, as it has exactly nothing at all to say about this issue. $\endgroup$ – Yvan Velenik Aug 31 '15 at 6:08
  • $\begingroup$ The explanatory power of the MaxEnt is slightly fallacious in that most of its treatments in textbooks forget or avoid to say that proper Bayesian inference has to do with prior and posterior probabilities. Dismissing entirely the prior probability problem as if it didn't exist, like it is often done, doesn't make the method any better. Besides, stat. mech. is more that subjective, it may be relative to how we define quantities but that doesn't make it subjective. In that respect, large deviation chooses a prior and see what happens. $\endgroup$ – gatsu Aug 31 '15 at 6:33

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