Exact Exponent $\lambda$ of the Autocorrelation Function for a Soluble Model of Coarsening

Abstract

The exponent $\lambda$ that describes the decay of the autocorrelation function $A(t)$ in a phase ordering system, $A(t)\sim L^{-(d-\lambda )}$, where $d$ is the dimension and $L$ the characteristic length scale at time $t$, is calculated exactly for the time-dependent Ginzburg-Landau equation in $d=1$. We find $\lambda =0.3993835\dots$. We also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as $M(t)\sim L^{\mu }$ and prove that for the $1d$ Ginzburg-Landau equation, $\mu =\lambda$, exemplifying a general result.

8 pages, latex, no figures