# Phase Transitions from a Bayesian Perspective of Statistical Mechanics

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

• 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. – Jahan Claes Sep 7 '18 at 16:03
• 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. – Connor Dolan Sep 7 '18 at 21:37
• 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. – Jahan Claes Sep 8 '18 at 2:54