The kinetic rate law for a Phi⁴ model in the order/disorder limit

Abstract

A kinetic rate law is established for a Phi ⁴ model on a lattice when the system is initially placed in a configurational state far away from thermal equilibrium. The discussion is based on a Fokker-Planck equation, which is used for describing the relaxation of the non-conserved order parameter. On applying a mean-field approximation, the N coupled integro-differential equations reduce to one self-consistent equation for the order parameter. The relaxation of the order parameter in a quenching procedure is next compared with a molecular dynamic simulation of the same approximate potential. The agreement is excellent. Finally, in the limit of a very deep on-site potential, the rate equation of the order parameter is shown to reduce to the well known Glauber equation for a two-state Ising system and by following the Kramers treatment, one also deduces the rate of jumps from one well to the other. The latter rate is found to be small, as expected. Also considered in brief is the conserved order parameter relaxation behaviour in the order/disorder limit, which is shown to yield the Kawasaki rate equation.