Exact soliton-like probability measures for interacting jump processes

TL;DR

This paper analytically solves a nonlinear master equation for interacting Markov jump processes, revealing a soliton-like stationary distribution whose existence depends on the interaction strength, and identifies a critical bifurcation point.

Contribution

It provides an exact analytical solution for soliton-like probability measures in interacting jump processes, highlighting the phase transition driven by interaction strength.

Findings

Existence of soliton-like stationary distributions depends on interaction strength.

A critical threshold for the destruction of the soliton-like solution is explicitly calculated.

The nonlinear master equation is solved analytically in the time-asymptotic regime.

Abstract

The cooperative dynamics of a 1-D collection of Markov jump, interacting stochastic processes is studied via a mean-field approach. In the time-asymptotic regime, the resulting nonlinear master equation is analytically solved. The nonlinearity compensates jumps induced diffusive behavior giving rise to a soliton-like stationary probability density. The soliton velocity and its sharpness both intimately depend on the interaction strength. Below a critical threshold of the strength of interactions, the cooperative behavior cannot be sustained leading to the destruction of the soliton-like solution. The bifurcation point for this behavioral phase transition is explicitly calculated.