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We studied bistable systems with multiplicative noise, finding corrections to the Kramers escape rate. Our path integral method, using instantons and local time reparametrization, accurately models this complex stochastic dynamics.

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Area of Science:

  • Statistical Physics
  • Nonlinear Dynamics
  • Stochastic Processes

Background:

  • Bistable systems are fundamental in various scientific fields.
  • Understanding the influence of multiplicative noise is crucial for accurate modeling.
  • The Kramers escape rate is a key parameter for system stability and transition dynamics.

Purpose of the Study:

  • To develop a theoretical framework for bistable systems driven by multiplicative noise.
  • To compute conditional probabilities and escape rates within the weak noise approximation.
  • To investigate corrections to the Kramers escape rate due to state-dependent diffusion.

Main Methods:

  • Path integral representation of overdamped Langevin dynamics.
  • Weak noise approximation and saddle-point analysis of functional integrals.
  • Introduction of local time reparametrization for fluctuation integration.
  • Numerical simulations to validate theoretical predictions.

Main Results:

  • Identified a saddle-point solution involving instantons and anti-instantons.
  • Developed a method to compute fluctuations using Gaussian integrals.
  • Derived corrections to the Kramers escape rate influenced by the diffusion function.
  • Theoretical results were consistent with numerical simulation outcomes.

Conclusions:

  • The study provides a robust method for analyzing bistable systems with multiplicative noise.
  • State-dependent diffusion significantly impacts escape rates, offering insights into system dynamics.
  • The findings contribute to a deeper understanding of stochastic processes in complex systems.