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Kramers' rate for systems with multiplicative noise.
Alexandre Rosas1, Italo'Ivo Lima Dias Pinto2, Katja Lindenberg2
1Departamento de Física, CCEN, Universidade Federal da Paraíba, Caixa Postal 5008, 58059-900, João Pessoa, Brazil.
Physical Review. E
|August 31, 2016
Summary
This study generalizes Kramers' rate for systems with multiplicative noise, finding the common expression incorrect. Numerical simulations of the Langevin equation confirm this new understanding of barrier crossing dynamics.
Area of Science:
- Chemical Kinetics
- Statistical Mechanics
- Nonlinear Dynamics
Background:
- Kramers' rate describes barrier crossing in systems with additive noise.
- Multiplicative noise effects on Kramers' rate are not well-established.
- Existing literature may misrepresent multiplicative noise dynamics.
Purpose of the Study:
- To generalize Kramers' rate for systems driven by multiplicative noise.
- To correct the commonly used, but inaccurate, expression for multiplicative noise in Kramers' rate calculations.
- To validate the proposed generalization through numerical simulations.
Main Methods:
- Derivation of a generalized Kramers' rate expression for multiplicative noise.
- Numerical integration of the Langevin equation for a quartic bistable potential.
- Comparison of simulation results with theoretical predictions.
Main Results:
- A novel, accurate expression for Kramers' rate in the presence of multiplicative noise is presented.
- The commonly used formula for multiplicative noise is demonstrated to be incorrect.
- Numerical results strongly corroborate the derived generalized rate.
Conclusions:
- The generalization of Kramers' rate for multiplicative noise is crucial for accurate modeling.
- The study corrects a significant misconception in the field of barrier crossing dynamics.
- This work provides a more reliable theoretical framework for systems with multiplicative noise.
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