Multiplicative stochastic processes, Fokker-Planck equations, and a possible dynamical mechanism for critical behavior

Abstract

A derivation of the Fokker‐Planck equations for additive stochastic processes is given which involves treating the continuity equation in the configuration space representation of the additive stochastic process as a multiplicative stochastic process. The average of the continuity equation becomes the Fokker‐Planck equation. A presentation of the ``multiplicative stochastic, Markov approximation'' follows. This approximation is applied to the analysis of the dynamics of a heavy particle in a molecular fluid as described by Hamilton's equations. The nonperturbative approximation technique leads to the Fokker‐Planck equation for simple Brownian motion. As part of the analysis, ``intrinsic diffusion'' is discovered and used to show ergodicity for the autocorrelation formula which appears during the Brownian motion calculation. An account of how these methods might be used to study the dynamical origins of critical behavior is given.