Pair annihilation of pointlike topological defects in the ordering process of quenched systems

Abstract

The annihilation process of pointlike topological defects moving irreversibly is studied as a model for the growth of order in quenched systems with O(N) symmetry in D=N dimensions. On the basis of the time-dependent Ginzburg-Landau model (TDGL), the defect picture is shown to be valid in any dimensions. We consider defects with pairwise power-type interaction. By a mean-field assumption we find that the total number N(t) of defects decreases as N(t)\ensuremath{\propto}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathit{D}/(\mathrm{\ensuremath{\beta}}+1)}$, where \ensuremath{\beta} is the exponent indicating the separation dependence of the force. We show analytically that this power law is valid for \ensuremath{\beta}D\ensuremath{\le}2\ensuremath{\beta} and confirm the analysis by the simulation of molecular dynamics of defects for various \ensuremath{\beta}'s in two dimensions. The distribution of defect density is well represented by the binomial distribution function in this parameter region. To keep the distribution uniform is a key of exhibiting mean-field behavior. The TDGL model realizes the mean-field power law only in two dimensions.