Stochastic fluctuations and stability in birth-death population dynamics: two-component Langevin equation in path-integral formalism

TL;DR

This paper investigates how stochastic fluctuations influence the stability of a birth-death process involving two particle types, using a path-integral Langevin approach to reveal fluctuation-induced stable states.

Contribution

It introduces a path-integral formalism for a two-component Langevin equation in birth-death dynamics, showing how stochastic fluctuations generate a new driving force leading to stable equilibrium states.

Findings

Stochastic fluctuations can induce nontrivial stable equilibrium states.

A minimum of two stochastic variables is required for such equilibrium formation.

The Langevin equation analysis reveals fluctuation-driven order in population dynamics.

Abstract

We discuss the stochastic process of creation and annihilation of particles, i.e., the $A^{n} \rightleftarrows B$ process in which $n$ particles $A$s and one particle $B$ are transformed to each other. Considering the case that the stochastic fluctuations are dependent on the numbers of $A$ and $B$, we apply the Langevin equation for the stochastic time-evolution of the numbers of $A$ and $B$. We analyze the Langevin equation in the path-integral formalism, and show that the new driving force is generated dynamically by the stochastic fluctuations. We present that the generated driving force leads to the nontrivial stable equilibrium state. This equilibrium state is regarded as the new state of order which is induced effectively by stochastic fluctuations. We also discuss that the formation of such equilibrium state requires at least two stochastic variables in the stochastic processes.