Stochastic theory of the kinetics of phase transitions

Abstract

We present a stochastic theory of the kinetics of phase transitions in univariant, nonuniform systems. We assume a master equation and a relation of the transition probability to the free energy [J. S. Langer, Ann. Phys. (N.Y) 65, 53 (1971)]. The free energy is taken to be of the Cahn–Hilliard form. By means of path integral methods we obtain a formal solution from which we derive a deterministic differential equation for the most probable variation of the density distribution in time. This equation is of the Landau–Ginzburg type. We show the existence of a Liapunoff function for this kinetic equation, which is then used to derive simply the classical result of nucleation theory for the critical radius of a droplet. Next we show from the kinetic equation that if the structure of the free energy is such that a phase transition occurs and metastable states are possible, then the kinetics of the decay to the stable and metastable states is nonlinear, of the cubic type. Finally we derive from the kinetic equation solutions for solitary waves which describe the motion of the interface in evaporation and condensation.