Long-time traces of the initial condition in relaxation phenomena near criticality

The time evolution of systems relaxing towards thermal equilibrium is examined near the critical temperature Tc, with special attention paid to the role of the initial value mᵢ of the order parameter {φ}. To this end, the n-component model A [model A according to P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977)] for a cube of length L is investigated. The common belief that all memory of mᵢ is necessarily lost after a microscopic time span is shown to be unfounded. General arguments and the exact solution of the limit n{→}{∞} show that mᵢ leaves its traces in both the linear and nonlinear long-time relaxation of {φ} near or at Tc. Specifically, for linear relaxation near Tc or at Tc with L{∞}, the amplitude of the exponential decay depends on mᵢ and the short-time exponent {θ}'=(xᵢ-x_φ)/z, provided tᵢ{~}mᵢᵢ^-z/x is comparable to or larger than other time scales. Here, xᵢ is the scaling dimension of mᵢ, z is the dynamic bulk exponent, and x_φ is the usual equilibrium scaling dimension of {φ}.