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I have been reading papers by E.T. Jaynes recently about viewing all of statistical mechanics as just Bayesian inference applied to physics. (For an introduction: https://journals.aps.org/pr/abstract/10.1103/PhysRev.106.620)

I find this perspective very elegant and enlightening, especially compared to the conventional point of view when being introduced to statistical physics, which I believe creates a great deal of unnecessary confusion by conflating epistemology and ontology. It also has the added advantage that it doesn't rely on the ergodic hypothesis, or require things to reach "thermal equilibrium", since temperature is just a Lagrange multiplier for making a maximally unbiased probability distribution that accounts for an expected energy. When the system is too complicated to consider any further relevant quantities or effects, we ignore them, much like we often do when estimating the probabilities of "random" event's like a coin being tossed in regards to the initial conditions of it's orientation, velocity, and angular momentum. If we notice effects or quantities that are "non-ergodic" or that our system is "non-equilibrium", we can simply add in the additional relevant information with more Lagrange multipliers.

However, there is a situation in which I'm having a hard time wrapping my head around the Bayesian statistical physics point of view. I want to hear an explanation of phase transitions that explicitly acknowledge temperature as an epistemic quantity, and not ontological.

The Bayesian perspective seems to imply that different phases themselves are epistemic, and that we just categorize collections of micro-states as being in different phases. For example, it seems nonsensical to consider whether a single molecule of H2O is either in a "liquid" or "gas" state, so what are we really saying when $~10^{20}$ are? Is there a particular length scale or number of molecules I can start calling it a "liquid" or "gas" (Normally statistical physics considers the infinite limit)? Are these just arbitrary empirical categorizations?

There is another example where I believe thinking in terms of temperature obscures the physical ontology. The Hamiltonian of a ferromagnet has an $SO(3)$ rotational symmetry, but we are taught that below a critical temperature the spins all settle into alignment and the $SO(3)$ breaks into an $SO(2)$ symmetry.

However, if we consider a single spin while the ensemble is above the critical temperature, we find it is already breaking the $SO(3)$ symmetry of the Hamiltonian since the spin itself is pointing in a particular direction. Additionally if we have domain walls after the spins align below the critical temperature, then we still have an $SO(3)$ symmetry when considering length scales larger than the size of the domain walls. It appears that the "phase" depends entirely on the length scale considered, so it's not clear what role temperature plays in this.

I suppose my question can be abbreviated to this: What is happening during symmetry breaking or a phase transition from an epistemic, Bayesian perspective?

Edit: Grammar

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    $\begingroup$ I think the correct way to think about symmetry breaking is in terms of ergodicity breaking. It's true that if you measure any one spin above the transition, it will point in a random direction. But if you average your measurements across time, it will on average point nowhere. That's in contrast to below the phase transition, where on average it points in a specific direction. That requires not thinking in terms of microstates alone, but in terms of transitions between microstates. In one phase, you can go from one microstate to any other; in the other, you're restricted to a smaller set. $\endgroup$ – Jahan Claes Sep 7 '18 at 16:03
  • $\begingroup$ True, but then couldn't I then just consider time scales as well? Before the phase transition, I could notice that the symmetry is broken for short lengths and small timescales. Surely as I get closer to this transition, I will notice larger correlation lengths in addition to longer periods of correlations, so I can simply include time as another dimension of length. And in the Bayesian perspective, ergodicity is irrelevant anyway. Hard phase transitions seem to be an epistemic property unique only to systems whose size is taken to be infinity. $\endgroup$ – Connor Dolan Sep 7 '18 at 21:37
  • $\begingroup$ I agree, and you only get true ergodicity breaking in the thermodynamic limit anyway. But I think everyone will agree that, Bayesian perspective or no, phase transitions only happen for infinite systems. $\endgroup$ – Jahan Claes Sep 8 '18 at 2:54
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First, a macrostate in statistical mechanics does not correspond to a particular realization of the system (say, spins), but to a probability measure on the set of microscopic configurations (or microstates). So, when one says that the system is invariant under the action of a particular symmetry group, this means that the measure is invariant, not the particular realization. In particular, this holds even for a single spin in your system, since it is equally likely to point in each direction (and so its distribution is indeed isotropic).

In some situations, there may be several distinct macrostates for the same set of thermodynamical parameters. In such a case, one says that there is a first-order phase transition, and each of these probability measures (or rather, each of the extremal ones) corresponds to a phase of the system.

Concerning your second point, about domain walls. In simple systems (say, the nearest-neighbor classical ferromagnetic Heisenberg model, or the Ising model), there will be no domain walls at equilibrium (there will be small, localized excitations, but the system does not split into large regions with different orientations of the spins). If you consider more complicated systems in which such domain walls do appear and are stable, then I would not say that they undergo a symmetry breaking.

Finally, I wouldn't say that the temperature in Jaynes' approach is epistemic. For a large system, it is an unambiguously defined quantity that possesses all the properties ascribed to the thermodynamic temperature and, as such, possesses an objective meaning. What is nice in this approach is that you start with an a priori ambiguous, subjective point of view (describing your knowledge of the system rather than its actual state), but you end up with completely deterministic (and thus objective) predictions (for very large systems). In a sense, in macroscopic systems, the measure maximizing your ignorance already provides complete, deterministic information on macroscopic properties.

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  • $\begingroup$ My problem with that is that a probability measure isn't a real physical quantity. Observers with different information about a system would yield different probability distributions, so would they disagree on which phase a system is in? And couldn't you have a distribution that spans several of these measure, making the state undefined? Also, for the domain walls, see my comment about time-scales above. Even in the case of domain walls, couldn't you just say that the correlation length gets longer and the timescales get longer both in a smooth way? $\endgroup$ – Connor Dolan Sep 7 '18 at 21:43
  • $\begingroup$ Concerning your question about different observers using different distributions: no, they should all agree if they provide consistent information. That's what I tried to explain above: a first observer with maximal ignorance can already deduce deterministic values for all macroscopic observables (in the thermodynamic limit). So, if another observer used more macroscopic information on the system to build his measure, then this additional information could either already be deduced by the first observer or is inconsistent. $\endgroup$ – Yvan Velenik Sep 8 '18 at 6:54
  • $\begingroup$ Of course, the second observer might instead know more about the microscopic details of the system (say, be aware of additional degrees of freedom). Then their predictions might indeed become different, but that's because the first observer had an incomplete model. I think that Jaynes has an example of this type when discussing the second law. $\endgroup$ – Yvan Velenik Sep 8 '18 at 6:55
  • $\begingroup$ Concerning your second question about time scales, I prefer not to comment, because I am not sure what to say: non-equilibrium statistical mechanics is very poorly understood. I agree that in principle a Jaynes type approach should be applicable also in this case, it becomes so involved that I don't think one can say much. $\endgroup$ – Yvan Velenik Sep 8 '18 at 6:57

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