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

Sarah A M Loos1, Sabine H L Klapp1

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Summary

This study introduces a method to accurately describe classical stochastic systems with time delays using Fokker-Planck equations. The approach provides a closed-form solution for probability density, improving upon existing approximations for nonlinear systems.

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Area of Science:

  • Statistical Physics
  • Nonlinear Dynamics
  • Stochastic Processes

Background:

  • Classical stochastic systems with time delays exhibit non-Markovian dynamics.
  • This leads to infinite hierarchies of Fokker-Planck equations, posing analytical challenges.

Purpose of the Study:

  • To develop a method for closing the Fokker-Planck hierarchy at the one-time level for systems with discrete time delay.
  • To accurately determine the steady-state probability density of nonlinear stochastic systems with time delays.

Main Methods:

  • Linearization of deterministic forces in the Fokker-Planck hierarchy.
  • Derivation of a closed equation for the one-time probability density.
  • Comparison with quasiexact numerical solutions of Langevin equations.

Main Results:

  • The proposed method accurately predicts the probability density for nonlinear systems with delay.
  • The approach significantly outperforms small-delay and perturbation-theoretical approximations.
  • The method allows for the calculation of transport quantities like escape times via Kramers approximation.

Conclusions:

  • The developed approach provides an accurate and efficient method for analyzing stochastic systems with time delays.
  • This technique is applicable to a broad range of nonlinear deterministic force models.
  • It offers improved predictions compared to existing approximation methods.