Velocity Distribution for Strings in Phase Ordering Kinetics

The continuity equations expressing conservation of string defect charge can be used to find an explicit expression for the string velocity field in terms of the order parameter in the case of an O(n) symmetric time-dependent Ginzburg-Landau model. This expression for the velocity is used to find the string velocity probability distribution in the case of phase-ordering kinetics for a nonconserved order parameter. For long times $t$ after the quench, velocities scale as $t^{-1/2}$. There is a large velocity tail in the distribution corresponding to annihilation of defects which goes as $V^{-(2d+2-n)}$ for both point and string defects in $d$ spatial dimensions.