Variational theory of activated rate processes for an arbitrary barrier

Area of Science:

• Statistical physics
• Chemical kinetics
• Theoretical chemistry

Background:

• Brownian motion describes random particle movement.
• Thermally activated escape is crucial in chemical reactions and molecular processes.
• The Fokker-Planck equation models these systems.

Purpose of the Study:

• To develop a novel method for calculating the escape rate of Brownian particles over smooth potential barriers.
• To accurately determine the least nonzero eigenvalue of the Fokker-Planck equation for systems with moderate to large friction.
• To validate the new method against existing approaches and numerical simulations.

Main Methods:

• The problem is framed as an eigenproblem of the Fokker-Planck equation.
• A Rayleigh-quotient-based perturbation method is employed for moderate and large friction regimes.
• The method is tested on bistable potentials with parabolic and quartic barriers.

Main Results:

• An accurate expression for the least nonzero eigenvalue was derived.
• The proposed method yields unprecedented accuracy across all barrier heights.
• Performance is validated against variational methods and numerical simulations.

Conclusions:

• The developed perturbation method provides a highly accurate way to calculate Brownian particle escape rates.
• This approach is effective for diverse barrier landscapes, including low-barrier limits.
• The findings offer a significant advancement in understanding thermally activated processes.