Equilibrium Stochastic Delay Processes

TL;DR

This paper introduces a new class of nonlinear stochastic delay processes that obey fluctuation theorems, converge to equilibrium, and are useful for analyzing complex delay systems in physics and engineering.

Contribution

It presents a novel class of equilibrium stochastic delay processes with nonlinear forces, extending analysis beyond linear models and providing exact constraints for nonlinear delay problems.

Findings

Processes obey fluctuation theorems

Converge to Boltzmann equilibrium at long times

Serve as a basis for perturbative analysis of nonlinear delays

Abstract

Stochastic processes with temporal delay play an important role in science and engineering whenever finite speeds of signal transmission and processing occur. However, an exact mathematical analysis of their dynamics and thermodynamics is available for linear models only. We introduce a class of stochastic delay processes with nonlinear time-local forces and linear time-delayed forces that obey fluctuation theorems and converge to a Boltzmann equilibrium at long times. From the point of view of control theory, such ``equilibrium stochastic delay processes'' are stable and energetically passive, by construction. Computationally, they provide diverse exact constraints on general nonlinear stochastic delay problems and can, in various situations, serve as a starting point for their perturbative analysis. Physically, they admit an interpretation in terms of an underdamped Brownian particle that is either subjected to a time-local force in a non-Markovian thermal bath or to a delayed feedback force in a Markovian thermal bath. We illustrate these properties numerically for a setup familiar from feedback cooling and point out experimental implications.