Bounds on the decay of the autocorrelation in phase ordering dynamics

Abstract

We investigate the decay of temporal correlations in phase ordering dynamics by obtaining bounds on the decay exponent \ensuremath{\lambda} of the autocorrelation function [defined by ${\mathrm{lim}}_{{\mathit{t}}_{2}\mathrm{\ensuremath{\gg}}{\mathit{t}}_{1}}$〈\ensuremath{\varphi}(r,${\mathit{t}}_{1}$)\ensuremath{\varphi}(r,${\mathit{t}}_{2}$)〉\ensuremath{\sim}L (${\mathit{t}}_{2}$${)}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$]. For a nonconserved order parameter, we recover the Fisher and Huse inequality, \ensuremath{\lambda}\ensuremath{\ge}d/2. For a conserved order parameter, we find \ensuremath{\lambda}\ensuremath{\ge}d/2 only if ${\mathit{t}}_{1}$ = 0. If ${\mathit{t}}_{1}$ is in the scaling regime, then \ensuremath{\lambda}\ensuremath{\ge}d/2+2 for d\ensuremath{\ge}2 and \ensuremath{\lambda}\ensuremath{\ge}3/2 for d=1. For the one-dimensional scalar case, this, in conjunction with previous results, implies that the value of \ensuremath{\lambda} depends on whether ${\mathit{t}}_{1}$=0 or ${\mathit{t}}_{1}$\ensuremath{\gg}1. Our numerical simulations for the two-dimensional, conserved scalar order parameter show that \ensuremath{\lambda}\ensuremath{\approxeq}4 for ${\mathit{t}}_{1}$ in the scaling regime, consistent with our bound. The asymptotic decay when ${\mathit{t}}_{1}$=0, while exhibiting an unexpected sensitivity to the amplitude of the initial correlations, is slower than when ${\mathit{t}}_{1}$\ensuremath{\gg}1 and obeys the bound \ensuremath{\lambda}\ensuremath{\ge}d/2. \textcopyright{} 1996 The American Physical Society.