Stochastic dynamics of N bistable elements with global time-delayed interactions: towards an exact solution of the master equations for finite N

TL;DR

This paper analyzes the stochastic behavior of finite networks of bistable elements with delayed interactions, providing a partial analytical solution for the stationary state in the case of two elements under certain symmetry conditions.

Contribution

It offers a novel partial analytical approach to the master equations for finite N bistable elements with time delay, focusing on the two-element case.

Findings

Partial analytical solution for two-element case

Time-symmetry condition for transition rates

Stationary state characterization under specific conditions

Abstract

We consider a network of N noisy bistable elements with global time-delayed couplings. In a two-state description, where elements are represented by Ising spins, the collective dynamics is described by an infinite hierarchy of coupled master equations which was solved at the mean-field level in the thermodynamic limit. For a finite number of elements, an analytical description was deemed so far intractable and numerical studies seemed to be necessary. In this paper we consider the case of two interacting elements and show that a partial analytical description of the stationary state is possible if the stochastic process is time-symmetric. This requires some relationship between the transition rates to be satisfied.