Kinetic equations for site populations from the Fokker-Planck equation

A new method is proposed for the calculation of kinetic coefficients from Fokker-Planck (FP) equations. Starting from the time scale separation between the kinetic modes of the FP operator and the other faster eigenmodes associated to the local equilibration of the coordinates, a set of site-localizing functions is introduced for the ensemble of stable states of the system by means of linear combinations of the kinetic eigenfunctions. They allow the mapping of a nonequilibrium distribution onto a set of site populations which obey to rate equations of kinetic type. Such a procedure is easily implemented by using numerical eigenfunctions of the time evolution operator, so providing a set of transition rates which leads to the same decay rates of the FP kinetic modes. Several examples are considered in order to illustrate the typical results of the method. Particularly interesting is a two-dimensional model for the chain of two bistable oscillators bound to a wall. A new kind of kinetic processes is recovered, besides ordinary transitions associated to saddle point crossings. They represent displacements localized within the chain without saddle point crossing, with analogy to crank-shaft transitions of polymers.