Interrelations between Stochastic Equations for Systems with Pair Interactions

TL;DR

This paper explores various stochastic equations derived from master equations for systems with pair interactions, clarifying their interrelations, validity conditions, and proposing a self-consistent solution method, with extensions to multi-system interactions.

Contribution

It systematically derives and interrelates multiple stochastic equations for pair-interacting systems from a master equation framework, including new conditions and solution procedures.

Findings

Derived four types of stochastic equations from master equations.

Clarified conditions under which these equations are valid.

Proposed a self-consistent solution procedure.

Abstract

Several types of stochastic equations are important in thermodynamics, chemistry, evolutionary biology, population dynamics and quantitative social science. For systems with pair interactions four different types of equations are derived, starting from a master equation for the state space: First, general mean value and (co)variance equations. Second, Boltzmann-like equations. Third, a master equation for the configuration space allowing transition rates which depend on the occupation numbers of the states. Fourth, a Fokker-Planck equation and a ``Boltzmann-Fokker-Planck equation''. The interrelations of these equations and the conditions for their validity are worked out clearly. A procedure for a selfconsistent solution of the nonlinear equations is proposed. Generalizations to interactions between an arbitrary number of systems are discussed.