Dynamical Relaxation and Universal Short-Time Behavior in Finite Systems: The Renormalization Group Approach

TL;DR

This paper investigates the universal relaxation dynamics of finite-sized systems near criticality using a renormalization-group approach, revealing universal initial-condition effects and confirming results with simulations.

Contribution

It develops a renormalization-group improved perturbation theory for finite systems and derives a stochastic equation capturing universal short-time behavior.

Findings

Universal dependence of magnetization on initial conditions.

Recovery of anomalous short-time power-law increase.

Agreement with Monte Carlo simulations for 3D Ising model.

Abstract

We study how the finite-sized n-component model A with periodic boundary conditions relaxes near its bulk critical point from an initial nonequilibrium state with short-range correlations. Particular attention is paid to the universal long-time traces that the initial condition leaves. An approach based on renormalization-group improved perturbation theory in 4-epsilon space dimensions and a nonperturbative treatment of the q=0 mode of the fluctuating order-parameter field is developed. This leads to a renormalized effective stochastic equation for this mode in the background of the other q=0 modes; we explicitly derive it to one-loop order, show that it takes the expected finite-size scaling form at the fixed point, and solve it numerically. Our results confirm for general n that the amplitude of the magnetization density m(t) in the linear relaxation-time regime depends on the initial magnetization in the universal fashion originally found in our large-$n$ analysis [J.\ Stat. Phys. 73 (1993) 1]. The anomalous short-time power-law increase of m(t) also is recovered. For n=1, our results are in fair agreement with recent Monte Carlo simulations by Li, Ritschel, and Zheng [J. Phys. A 27 (1994) L837] for the three-dimensional Ising model.