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

TL;DR

This paper investigates how the initial conditions influence the long-time relaxation behavior of systems near criticality, demonstrating that initial order parameter values leave persistent traces contrary to previous assumptions.

Contribution

It provides a theoretical analysis showing that initial conditions affect long-time relaxation near critical points, supported by exact solutions in the large-n limit.

Findings

Initial conditions influence long-time relaxation behavior.

Memory of initial order parameter persists beyond microscopic times.

Amplitude of exponential decay depends on initial conditions and scaling exponents.

Abstract

The time evolution of systems relaxing towards thermal equilibrium is examined near the critical temperature $T_c$, with special attention paid to the role of the initial value $m_i$ of the order parameter $\phi$. To this end, the $n$-component model A for a cube of length $L$ is investigated. The common belief that all memory of $m_i$ is necessarily lost after a microscopic time span is shown to be unfounded. General arguments and the exact solution of the limit $n\to\infty$ show that $m_i$ leaves its traces in both the linear and nonlinear long-time relaxation of $\phi$ near or at $T_c$. Specifically for linear relaxation near $T_c$, or at $T_c$ with $L<\infty$, the amplitude of the exponential decay depends on $m_i$ and the short-time exponent $\theta'=(x_i-x_{\phi})/z$, provided $t_i\sim m_i^{-z/x_i}$ is comparable to or larger than other time scales. Here $x_i$ is the scaling dimension of $m_i$, $z$ is the dynamic bulk exponent, and $x_{\phi}$ is the usual equilibrium scaling dimension of $\phi $.