## Phase Transitions and Critical Phenomena

*28 Oct 2014 19:07*

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.)

- Recommended (big picture):
- 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 [Also in Sethna's excellent statistical mechanics textbook]
- Geoffrey Sewell, Quantum Mechanics and Its Emergent Macrophysics
- Julia Yeomans, The Statistical Mechanics of Phase Transitions

- Recommended (details):
- 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] - Iván Calvo, Juan C. Cuchí, José G. Esteve,
Fernando Falceto, "Generalized Central Limit Theorem and Renormalization
Group", Journal of
Statistical Physics
**141**(2010): 409--421, arxiv:1009.2899 - Giovanni Jona-Lasinio, "Renormalization Group and Probability Theory", Physics Reports
**352**(2001): 439--458 = arxiv:cond-mat/0009219

- To read:
- 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"]
- L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, "Lagrangian phase transitions in nonequilibrium thermodynamic systems", arxiv:1005.1489
- Binney, Dowrick, Fisher and Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group
- 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
- E. Edlund and Martin Nilsson Jacobi, "Renormalization of cellular
automata and
self-similarity", Journal
of Statistical Physics
**139**(2010): 972--984, arxiv:1108.3982 - Roberto Franzosi and Marco Pettini, "Topology and Phase
Transitions"
- and Lionel Spinelli, "Theorem on a necessary relation", math-ph/0505057
- "Entropy and Topology", math-ph/0505058

- A. Guionnet and B. Zegarlinski, Lectures on Logarithmic Sobolev Inequalities [120 pp. PDF]
- Leo Kadanoff
- "More is the Same; Phase Transitions and Mean Field Theories", arxiv:0906.0653
- "Theories of Matter: Infinities and Renormalization", arxiv:1002.2985

- Michael Kastner, "Phase transitions and configuration space topology", cond-mat/0703401
- Alon Manor and Nadav M. Shnerb, "Multiplicative Noise and Second Order Phase Transitions", Physical
Review Letters
**103**(2009): 030601 - 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
- Uwe C. Tauber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior
- 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
- Paolo Zanardi, Paolo Giorda, and Marco Cozzini,
"Information-Theoretic Differential Geometry of Quantum Phase
Transitions", Physical Review
Letters
**99**(2007): 100603