# Force-linearization closure for non-Markovian Langevin systems with time delay

## TL;DR

This paper introduces a linearization-based closure method for non-Markovian Langevin systems with delay, providing accurate steady-state probability densities and improving upon existing approximations.

## Contribution

A novel linearization closure approach for the Fokker-Planck hierarchy in delayed Langevin systems, enabling accurate steady-state density calculations.

## Key findings

- Accurately predicts steady-state densities for delayed nonlinear systems.
- Outperforms small-delay and perturbation approximations.
- Applicable to a wide class of nonlinear force models.

## Abstract

This paper is concerned with the Fokker-Planck (FP) description of classical stochastic systems with discrete time delay. The non-Markovian character of the corresponding Langevin dynamics naturally leads to a coupled infinite hierarchy of FP equations for the various $n$-time joint distribution functions. Here we present a novel approach to close the hierarchy at the one-time level based on a linearization of the deterministic forces in all members of the hierarchy starting from the second one. This leads to a closed equation for the one-time probability density in the steady state. Considering two generic nonlinear systems, a colloidal particle in a sinusoidal or bistable potential supplemented by a linear delay force, we demonstrate that our approach yields a very accurate representation of the density as compared to quasi-exact numerical results from direct solution of the Langevin equation. Moreover, the results are significantly improved against those from a small-delay approximation and a perturbation-theoretical approach. We also discuss the possibility of accessing transport-related quantities, such as escape times, based on an additional Kramers approximation. Our approach applies to a wide class of models with nonlinear deterministic forces.