Dynamical relaxation and universal short-time behavior in finite systems The renormalization-group approach

We study how the finite-size 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 not equal 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.