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Critical and Stochastic Dynamics

Near a continuous transition, long spatial correlations are accompanied by slow dynamics. Stochastic field theory explains how conservation laws, noise, and reversible couplings determine relaxation, universality, and aging.

Enter this chapter

Section titled “Enter this chapter”

Begin with Dynamic Scaling and Critical Slowing Down to relate a growing correlation length to a diverging relaxation time.

Helpful background. Langevin Field Equations and Noise and Stochastic Processes and Correlation Functions develop the definitions used by the chapter’s opening pages.

Topics

Section titled “Topics”
  1. Dynamic Scaling and Critical Slowing Down — Defines dynamic scaling and the exponent relating relaxation time to correlation length.
  2. Langevin Field Equations and Noise — Combines slow-variable drift with specified noise correlations.
  3. Fokker–Planck Evolution and Stationary Measures — Derives probability evolution and tests candidate stationary measures.
  4. The MSRJD Response Functional — Constructs a response-field functional for stochastic dynamics.
  5. Detailed Balance and Fluctuation–Dissipation — Distinguishes stationarity from equilibrium time-reversal relations.
  6. Dynamic Universality Classes — Classifies critical dynamics by conservation laws and slow couplings.
  7. Multiplicative, Colored, and Conserved Noise — Explains how nontrivial noise changes locality and stochastic prescriptions.
  8. Quenches, Coarsening, and Aging — Uses two-time correlations to describe domain growth and aging.

How the ideas fit together

Section titled “How the ideas fit together”

The same equilibrium free energy can generate different dynamics. A nonconserved order parameter can relax locally; a conserved one must redistribute through currents. Additional slow densities and reversible couplings can change the dynamic universality class without changing static critical exponents Hohenberg and Halperin 1977, §§ II–III.

Langevin, Fokker–Planck, and response-functional descriptions must represent the same noise and stochastic prescription. In particular, a stationary distribution need not imply detailed balance: circulating probability currents can persist. Following a quench, aging also prevents replacing two-time observables by functions of time difference alone.

Where to go next

Section titled “Where to go next”

Continue to Kinetic Theory and Transport Equations to identify when a phase-space distribution describes the excitations.

References

Section titled “References”
  • Hohenberg, P. C., and Halperin, B. I. (1977). “Theory of Dynamic Critical Phenomena.” Reviews of Modern Physics 49, 435–479. DOI.