Scaling theory of nonlinear critical relaxation

Abstract

A scaling analysis of nonlinear critical slowing down on the basis of a Landau-type relaxation equation, shows that the critical exponents, ${\Delta }^{\mathrm{}\left(l\right)\mathrm{}}$ and ${\Delta }^{\mathrm{}\left(\mathrm{nl}\right)\mathrm{}}$ of the linear and nonlinear relaxation times of the order parameter are related by ${\Delta }^{\mathrm{}\left(\mathrm{nl}\right)\mathrm{}}={\Delta }^{\mathrm{}\left(l\right)\mathrm{}}-\beta$. Generally if $Q$ scales like $\Delta T^{{\beta }_Q}$ one has ${\Delta }_Q^{\mathrm{}\left(l\right)\mathrm{}}-{\Delta }_Q^{\mathrm{}\left(\mathrm{nl}\right)\mathrm{}}={\beta }_Q$ but a different relaxation may occur in systems with oscillatory modes.