Spectral formulation and WKB approximation for rare-event statistics in reaction systems

Area of Science:

• Statistical physics
• Chemical kinetics
• Quantum mechanics

Background:

• Calculating probabilities of rare events (large deviations from the mean) in systems of reacting particles with infinite-range interaction is crucial.
• Master equations are often used to describe such systems.
• Existing semiclassical approximations have limitations.

Purpose of the Study:

• To develop a spectral formulation and stationary WKB approximation for calculating rare event probabilities.
• To compare the stationary WKB approximation with a time-dependent semiclassical approximation.
• To benchmark the approximations using an exactly solvable binary annihilation reaction.

Main Methods:

• Spectral formulation
• Stationary WKB approximation
• Comparison with time-dependent semiclassical approximation
• Exact solution for 2A-->0 reaction

Main Results:

• The stationary WKB approximation provides accurate probabilities for rare events.
• The developed spectral formulation offers a robust framework for analyzing large deviations.
• The binary annihilation reaction benchmark confirms the validity of the stationary WKB approximation.

Conclusions:

• The stationary WKB approximation is a powerful tool for studying rare events in reacting particle systems.
• The spectral formulation provides new insights into the dynamics of large deviations.
• This work offers a valuable alternative to existing semiclassical methods.