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Kramers' law for a bistable system with time-delayed noise
D Goulding1, S Melnik, D Curtin
1Tyndall National Institute, Lee Maltings, Cork, Ireland.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 13, 2007
Summary
This study extends the Kramers
Area of Science:
- Statistical physics
- Nonlinear dynamics
- Stochastic processes
Background:
- The Kramers' escape problem describes particle escape from a potential well.
- Classical models often assume uncorrelated noise.
- Bistable systems are fundamental in many physical and chemical processes.
Purpose of the Study:
- To extend the Kramers' escape problem to systems with delayed noise.
- To analyze the switching dynamics in a bistable system subjected to correlated noise.
- To develop a theoretical framework for understanding noise effects beyond the white noise approximation.
Main Methods:
- Utilized the Langevin equation to model the bistable system dynamics.
- Analytically calculated switching rates for different time intervals.
- Introduced effective potential and diffusion coefficient to account for delayed noise effects.
Main Results:
- Demonstrated that delayed noise significantly alters switching dynamics.
- The distribution of switching times exhibits piecewise exponential decay.
- Derived analytical expressions for switching rates considering noise correlation.
Conclusions:
- The classical Kramers' problem can be extended to include delayed noise.
- Delayed noise introduces non-Markovian effects, necessitating modified theoretical approaches.
- The findings provide insights into stochastic processes in systems with memory.
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