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Physically consistent numerical solver for time-dependent Fokker-Planck equations.

Viktor Holubec1,2, Klaus Kroy1, Stefano Steffenoni1,3

  • 1Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany.

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|April 20, 2019
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Summary
This summary is machine-generated.

We developed a thermodynamically consistent method to solve Fokker-Planck equations (FPE) for stochastic processes. This approach accurately calculates system dynamics and thermodynamic properties, validated by heat engine simulations.

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Area of Science:

  • Statistical Physics
  • Computational Physics
  • Chemical Physics

Background:

  • Fokker-Planck equations (FPE) describe stochastic processes but are challenging to solve, especially for time-dependent systems.
  • Existing methods may struggle with thermodynamic consistency and calculating key observables like fluctuating currents and work.

Purpose of the Study:

  • To present a novel, thermodynamically consistent numerical method for solving time-dependent Fokker-Planck equations (FPE) for overdamped stochastic processes.
  • To enable computation of transition/steady-state behavior and large-deviation functions for observables like particle current, heat, and work.

Main Methods:

  • Approximating the FPE with a master equation using transition rates that satisfy local detailed balance.
  • Directly computing the time-ordered exponential of the discretized configuration space propagator by summing over all paths.
  • Preserving solution positivity, normalization, and ensuring physically reasonable entropy production.

Main Results:

  • The method provides accurate time-dependent solutions for FPE, capturing both transient and steady-state behaviors.
  • It allows for the calculation of moment-generating and large-deviation functions for various observables.
  • The approach was validated against Brownian dynamics simulations of a model heat engine.

Conclusions:

  • The proposed method offers a robust and thermodynamically consistent way to solve time-dependent FPE.
  • It is versatile, applicable to diverse stochastic processes and capable of computing complex observables.
  • This technique holds significant potential for applications in statistical physics and beyond.