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Numerical path integral calculation of the probability function and exit time: an application to non-gradient drift

Fernando Mora1,2, Pierre Coullet2, Sergio Rica1

  • 1Facultad de Ingeniería y Ciencias and UAI Physics Center, Universidad Adolfo Ibáñez, Santiago, Chile.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|November 14, 2018
PubMed
Summary

This study presents numerical solutions for stochastic processes with non-gradient drift Langevin forces and noise. The method accurately tracks probability density functions and calculates exit times, showing excellent agreement with theory in the weak noise limit.

Keywords:
mean first passage timenonlinear physicspath integral methodstochastic processtransitions induced by noise

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Area of Science:

  • Physics
  • Chemistry
  • Biology
  • Non-equilibrium thermodynamics

Background:

  • Stochastic processes are fundamental to understanding systems with inherent randomness.
  • Non-gradient drift Langevin forces and noise are common in physical and chemical systems.
  • Accurate computation of probability density functions and exit times is crucial for system analysis.

Purpose of the Study:

  • To develop numerical solutions for stochastic processes involving non-gradient drift Langevin forces and noise.
  • To accurately follow the temporal evolution of probability density functions.
  • To compute exit times for systems with arbitrary noise characteristics.

Main Methods:

  • Utilizing the path integral representation of stochastic processes.
  • Implementing numerical solutions for non-gradient drift Langevin dynamics.
  • Comparing numerical results with theoretical calculations.

Main Results:

  • The developed numerical method successfully tracks the temporal evolution of probability density functions.
  • Exit times were computed accurately for systems with arbitrary noise.
  • Excellent agreement was observed between numerical solutions and theoretical calculations in the weak noise limit.

Conclusions:

  • The path integral approach provides a robust framework for solving complex stochastic processes.
  • The numerical solutions are reliable for analyzing systems with non-gradient drift and noise.
  • This method offers a valuable tool for studying dissipative structures in non-equilibrium systems.