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Phase Transitions and Critical Phenomena

24 Oct 2024 11:07

One of the central areas of statistical mechanics for the last, oh, forty years, to the point where it has seriously shaped --- one might even say, warpped --- how those of us trained in that tradition look at the world in general. (See power laws and especially self-organized criticality.)

Things I want to understand better. Rigorously separated phases seem to only exist in infinite-system limits; what are the large-but-finite regimes like? Connections between phase transitions and changes in the topology of the phase space. Do there exist ways of deducing the order parameter from either microscopic Hamiltonians or from macroscopic observations? Is there a way of detecting phase transitions from macroscopic observables other than the order parameter and the thermodynamic potential?

Why are there so few fixed points to the renormalization group?

Connections between power law distributions and critical fluctuations. While I understand the physical arguments for why we see power-law-distributed fluctuations at the critical point, I find myself wanting a more probabilistic explanation as well. A crude sketch would go as follows. Far from the critical point, the microscopic dynamics are rapidly mixing in space and time --- and mixing in the technical, ergodic theory sense, so that the central limit theorem applies, and averages over spatio-temporal regions large compared to the mixing scales are approximately Gaussian. (Cf. Rosenblatt, 1956.) As one approaches the critical point, however, giant, correlated fluctuations begin to appear, i.e., the mixing scales diverge, and one is dealing with a process with long-range memory (in both space and time). Under these circumstances, averaging can deliver a non-Gaussian but still self-similar distribution, which is where the power-law tails come from. The stable distributions, including the Gaussian, emerge from the central limit theorem for independent variables because they are unchanged under convolution (averaging) with themselves --- there are ways, in renormalization group theory, of trading off infinite variance (as in the non-Gaussian stable limits) for infinite range-correlation. This, I should understand better. (The review paper by Jona-Lasinio is a start, but does not leave me with enough intuition that I feel entirely comfortable with what's going on.)

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