Critical Langevin dynamics of the O(N) Ginzburg-Landau model with correlated noise

We use the perturbative renormalization group to study classical stochastic processes with memory. We focus on the generalized Langevin dynamics of the phi(4) Ginzburg-Landau model with additive noise, the correlations of which are local in space but decay as a power law with exponent a in time. These correlations are assumed to be due to the coupling to an equilibrium thermal bath. We study both the equilibrium dynamics at the critical point and quenches towards it, deriving the corresponding scaling forms and the associated equilibrium and non-equilibrium critical exponents eta, v, z and theta. We show that, while the first two retain their equilibrium values independently of a, the non-Markovian character of the dynamics affects z and theta for alpha < alpha(c)(D, N) where D is the spatial dimensionality, N the number of components of the order parameter, and alpha(c)( x, y) a function which we determine at second order in 4 - D. We analyze the dependence of the asymptotic fluctuation-dissipation ratio on various parameters, including a. We discuss the implications of our results for several physical situations. © 2012 IOP Publishing Ltd and SISSA.