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Related Experiment Videos

Rate processes in a delayed, stochastically driven, and overdamped system

Guillouzic1, L'Heureux, Longtin

  • 1Ottawa-Carleton Institute for Physics, University of Ottawa, Ontario, Canada.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|October 14, 2000
PubMed
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This study introduces a time scales approximation for stochastic delay differential equations, simplifying complex systems. The findings reveal how noise and delay influence transition rates and system stability.

Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Stochastic Processes

Background:

  • Stochastic delay differential equations (SDDEs) present challenges in analysis.
  • A Fokker-Planck formulation for SDDEs has been recently proposed.
  • Simplifying approximations are needed for complex multistable systems.

Purpose of the Study:

  • Introduce and validate a separation of time scales approximation for Fokker-Planck equations derived from SDDEs.
  • Analyze the dynamics of a particle in a delayed quartic potential.
  • Investigate the influence of noise and delay on transition rates and system stability.

Main Methods:

  • Developed a separation of time scales approximation for Fokker-Planck equations.
  • Applied the approximation to a particle in a delayed quartic potential.

Related Experiment Videos

  • Determined steady-state probability density via numerical simulations and small delay expansion.
  • Calculated transition rates and mean first passage times.

    • Main Results:

    • The approximation simplifies multistable systems, yielding phenomenological rate laws.
    • Transition rates and mean first passage times depend on noise variance and steady-state probability density.
    • Numerical determination of steady-state probability yields more accurate results than small delay expansion.
    • Transition rates and mean first passage times follow Arrhenius' law for small noise variance, even with large delays.
    • Deterministic unbounded solutions coexist with bounded solutions; noise increases the transition rate to unbounded solutions with delay.

    Conclusions:

    • The separation of time scales approximation is valid for SDDEs.
    • Accurate steady-state probability density is crucial for precise rate calculations.
    • System dynamics exhibit complex behavior including coexistence of bounded and unbounded solutions.
    • Delay significantly impacts the rate of transition to unbounded solutions in the presence of noise.