Hamilton-Jacobi method for molecular distribution function in a chemical oscillator

TL;DR

This paper applies the Hamilton-Jacobi method to solve chemical Fokker-Planck equations, deriving formulas for probability distributions in transient, steady, and oscillatory states, with practical numerical evaluation and validation against simulations.

Contribution

It introduces a Hamilton-Jacobi-based approach to analyze chemical oscillators, providing explicit formulas for distributions and phase diffusion, applicable to transient and oscillatory regimes.

Findings

Derived a compact formula for conditional probability distribution.

Obtained phase diffusion constant along periodic orbits.

Validated formulas with Monte Carlo simulations for Brusselator.

Abstract

Using the Hamilton-Jacobi method, we solve chemical Fokker-Planck equations within the Gaussian approximation and obtain a simple and compact formula for a conditional probability distribution. The formula holds in general transient situations, and can be applied not only for a steady state but also for a oscillatory state. By analyzing the long time behavior of the solution in the oscillatory case, we obtain the phase diffusion constant along the periodic orbit and the steady distribution perpendicular to it. A simple method for numerical evaluation of these formulas are devised, and they are compared with Monte Carlo simulations in the case of Brusselator as an example. Some results are shown to be identical to previously obtained expressions.