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

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)∝${\mathit{t}}^{\mathrm{-}\mathit{D}/(\mathrm{\beta }+1)}$, where β is the exponent indicating the separation dependence of the force. We show analytically that this power law is valid for βD≤2β and confirm the analysis by the simulation of molecular dynamics of defects for various β’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.