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

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.

Stochastic limit-cycle oscillations of a nonlinear system under random perturbations

“…Instead, the deterministic behavior and the local fluctuation of a jump Markov process can be obtained by the Ω expansion with respect to two different scales. In addition to the Ω expansion, another approach by the WKB approximation has been applied to give a full analysis of master equations [22,8,28,42]. In this method, the WKB ansatz is assumed for the solution of the Kramers-Moyal expansion of a master equation without truncating higher-order terms, so this method has no problem unlike the Kramers-Moyal Fokker-Planck equation.…”

Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system

“…For an orbit that converges to a linearly stable, attractive equilibrium, this neighborhood was computed in 1810 by Laplace [1,2] and is today known as a solution to the Lyapunov equation [3], or the Ornstein-Uhlenbeck process [4]: for a 1dimensional flow, the deterministic equilibrium point is smeared into a Gaussian probability density centered on it, whose covariance Q = −D/λ is a balance of the expansion rate D (diffusion constant) against the contraction rate λ < 0. Fokker-Planck equation [5] generalizations to higher-dimensional stable equilibria and limit cycles (stable periodic orbits) are immediate, provided proper care is taken of the diffusion along the periodic orbit [6,7].…”

“…I C, our goal here is to determine the stationary distribution ρ(x). For that purpose, we use as basis Gaussian ellipsoids that satisfy the local stationarity condition (7) in each neighborhood of the optimal partition. A set of Gaussians centered at every point in the state space M forms an overcomplete, non-orthogonal basis for functions in L 2 (M), as is well known from the study of coherent states of quantum harmonic oscillators [28].…”

Intrinsic fluctuations of chemical reactions with different approaches

“…Alternatively, equivalent to Fokker-Planck equation one can also use stochastic differential equation, called Langevin equation, to study the stochastic effect 13,14 ; moreover, the equation is often linearized about the equilibrium state of meanfield equation for analytic study. Along with this main stream, novel methods have been developed for specific studies [15][16][17][18][19] . For example, based on the chemical master equation Gaspard used the Hamilton-Jacobi method to give a formalism for the study of oscillating reactions 15 , and Nakanishi and et al employed the formalism to analyze the molecular density distribution in a chemical oscillator 16 .…”

“…Along with this main stream, novel methods have been developed for specific studies [15][16][17][18][19] . For example, based on the chemical master equation Gaspard used the Hamilton-Jacobi method to give a formalism for the study of oscillating reactions 15 , and Nakanishi and et al employed the formalism to analyze the molecular density distribution in a chemical oscillator 16 . Among different approaches, it is essential to understand the adequacy and the limitation of a method.…”