Critical and Stochastic Dynamics
Stochastic field theory describes selected slow variables after faster degrees of freedom have been coarse grained into drift, dissipation, and noise. A model is defined only when its stochastic convention, noise covariance, regulator, probability current, response Jacobian, slow-variable content, and scaling regime are all specified.
From stochastic equations to critical dynamics
Section titled “From stochastic equations to critical dynamics”Langevin field equations fix deterministic relaxation, noise normalization, response, and stationary covariance. Fokker–Planck evolution converts that process into probability flow and tests normalizability and uniqueness. The MSRJD response functional makes response fields, determinants, ghosts, and discretization Jacobians explicit.
Detailed balance and fluctuation–dissipation distinguishes equilibrium from a merely stationary state. Dynamic scaling and critical slowing down derives the exponent and Kibble–Zurek freeze-out under a controlled ramp. Dynamic universality classes show why static exponents and order-parameter symmetry are insufficient without conserved densities and reversible couplings.
Generalized noise treats multiplicative conventions, colored memory, and conserved forcing. Quenches, coarsening, and aging then develops domain growth and two-time scaling when stationarity and equilibrium FDT fail.
| Question | Start with | Decisive check |
|---|---|---|
| Does the noise thermalize to the intended measure? | Langevin and Fokker–Planck pages | Derive the probability current and verify normalization and boundary conditions. |
| Is the response functional convention complete? | MSRJD page | Track the determinant, response contour, and value of the equal-time retarded function. |
| Is a stationary state in equilibrium? | Detailed-balance page | Test irreversible probability current and fluctuation–dissipation, including odd variables. |
| Which dynamic exponent applies? | Scaling and universality pages | List every conserved or reversibly coupled slow variable and test the scaling window. |
| Does a generalized-noise model change the drift? | Multiplicative and colored noise | Translate Itô/Stratonovich prescriptions and take the white-noise limit explicitly. |
| Is late-time evolution aging? | Quenches and coarsening | Test two-time scaling in both $t$ and waiting time rather than fitting equilibrium FDT. |
Scope boundary
Section titled “Scope boundary”Noise is an effective description, not evidence that microscopic dynamics is fundamentally random. White Gaussian noise, Markovian drift, and a small set of slow variables require scale separation. Driven, active, glassy, hydrodynamic, quantum, and non-Markovian systems may need additional fields or different symmetries.
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.
- Janssen, H.-K. (1976). “On a Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties.” Zeitschrift für Physik B 23, 377–380. DOI.