Critical Langevin dynamics of theO(N) Ginzburg–Landau model with correlated noise

Abstract

We use the perturbative renormalization group to study classical stochasticprocesses with memory. We focus on the generalized Langevin dynamics of theϕ4 Ginzburg–Landau model with additive noise, the correlations ofwhich are local in space but decay as a power law with exponentα in time. These correlations are assumed to be due to the coupling to anequilibrium thermal bath. We study both the equilibrium dynamics at thecritical point and quenches towards it, deriving the corresponding scalingforms and the associated equilibrium and non-equilibrium critical exponentsη,ν,z andθ. We show that, while the first two retain their equilibrium values independently ofα, the non-Markovian character of the dynamics affectsz andθ forα < αc(D, N) whereD is the spatialdimensionality, N the number of components of the order parameter, andαc(x, y) a function which we determine at second order in4 − D. Weanalyze the dependence of the asymptotic fluctuation-dissipation ratio on various parameters, includingα. We discuss the implications of our results for several physical situations.