Phase Transitions and Critical Phenomena

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 — in part, I think, because nobody entirely understands things.)

P. W. Anderson, Basic Notions of Condensed Matter Physics
L. D. Landau and E. M. Lifshitz, Statistical Physics
Joel L. Lebowitz, "Statistical mechanics: A selective Review of Two Central Issues", Reviews of Modern Physics 71 (1999): S346--S357 = math-ph/0010018 [One of the two issues is first-order phase transitions.]
James Sethna, "Order Parameters, Broken Symmetry, and Topology", pp. 243--265 in Lynn Nadel and Daniel L. Stein (eds.), 1990 Lectures in Complex Systems
Geoffrey Sewell, Quantum Mechanics and Its Emergent Macrophysics
Julia Yeomans, The Statistical Mechanics of Phase Transitions

Somendra M. Bhattacharjee and Flavio Seno, "A measure of data collapse for scaling", Journal of Physics A: Mathematical and General 35 (2001): 6375--6380 [thanks to Aaron Clauset for the pointer]
Giovanni Jona-Lasinio, "Renormalization Group and Probability Theory", Physics Reports 352 (2001): 439--458 = arxiv:cond-mat/0009219

N. G. Antoniou, F. K. Diakonos, E. N. Saridakis, and G. A. Tsolias, "An efficient algorithm simulating a macroscopic system at the critical point", physics/0607038 [Getting around critical-slowing down, using the fact that "dynamics in the order parameter space is simplified significantly ... due to the onset of self-similarity in the [fluctuations]. ... [T]he effective action at the critical point obtains a very simple form. ... [T]his simplified action can be used in order to simulate efficiently the statistical properties of a macroscopic system exactly at the critical point"]
Amir Dembo and Andrea Montanari, "Gibbs Measures and Phase Transitions on Sparse Random Graphs", arxiv:0910.5460
Cyril Domb, The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena
Roberto Franzosi and Marco Pettini, "Topology and Phase Transitions"
1. and Lionel Spinelli, "Theorem on a necessary relation", math-ph/0505057
2. "Entropy and Topology", math-ph/0505058
Leo Kadanoff, "More is the Same; Phase Transitions and Mean Field Theories", arxiv:0906.0653
Michael Kastner, "Phase transitions and configuration space topology", cond-mat/0703401
O. C. Martin, R. Monasson and R. Zecchina, "Statistical mechanics methods and phase transitions in optimization problems," cond-mat/0104428
Oliver Muelken, Heinrich Stamerjohanns, and Peter Borrmann, "The Origins of Phase Transitions in Small Systems," cond-mat/0104307
Marco Pettini, Roberto Franzosi and Lionel Spinelli, "Topology and Phase Transitions: towards a proper mathematical definition of finite N transitions," cond-mat/0104110
Javier Rodriguez-Laguna, "Real Space Renormalization Group Techniques and Applications," cond-mat/0207340
Martin Weigel and Wolfhard Janke, "Cross Correlations in Scaling Analyses of Phase Transitions", Physical Review Letters 102 (2009): 100601 = arxiv:0811.3097
Ji-Feng Yang, "Renormalization group equations as 'decoupling' theorems", hep-th/0507024

Notebooks

Phase Transitions and Critical Phenomena