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Self-similar renormalization approach to barrier crossing processes
1Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu, Tokyo 182-8585.
Summary
A new renormalization method accurately calculates Brownian particle escape rates over barriers. This approach provides a unified formula valid across weak and strong friction regimes, improving upon existing theories for activated rate processes.
Area of Science:
- Statistical mechanics
- Nonlinear dynamics
- Theoretical physics
Background:
- Thermally activated escape is crucial in various physical and chemical processes.
- Existing theories for calculating escape rates often rely on approximations valid only in specific friction regimes (weak or strong).
- Divergent series in field-theoretical models pose challenges for accurate analytical solutions.
Purpose of the Study:
- To apply a novel algebraic self-similar renormalization method to determine the escape rate of a Brownian particle over complex potential barriers.
- To develop a unified rate formula applicable to both underdamped and overdamped Brownian motion.
- To validate the new formula by comparing its predictions with existing theories and numerical results for various barrier shapes and heights.
Main Methods:
- Application of an algebraic self-similar renormalization technique to summation of divergent series.
- Utilizing the Mel'nikov-Meshkov result for underdamped Brownian motion.
- Employing an inverse friction expansion of the Fokker-Planck equation for strong friction.
- Testing the derived overall rate formula against Brownian motion in bistable potentials with parabolic, cusped, and quartic barriers.
Main Results:
- A unified rate formula for Brownian escape over arbitrary barriers was constructed.
- The formula demonstrates agreement with established results in both weak friction (diffusion equation in energy variables) and strong friction (Smoluchowski equation) limits.
- The proposed formula provides a reasonable description of activated rate processes, even for low barriers, and shows superior agreement with numerical rates compared to other interpolation formulas.
Conclusions:
- The algebraic self-similar renormalization method offers a powerful tool for analyzing complex dynamical systems like Brownian motion.
- The developed unified rate formula represents a significant improvement for predicting activated escape rates across different friction regimes.
- This approach enhances the understanding and modeling of barrier crossing phenomena in diverse scientific disciplines.