Weiss mean-field approximation for multicomponent stochastic spatially extended systems

Svetlana E. Kurushina, Valerii V. Maximov, and Yurii M. Romanovskii
Phys. Rev. E 90, 022135 – Published 26 August 2014

Abstract

We develop a mean-field approach for multicomponent stochastic spatially extended systems and use it to obtain a multivariate nonlinear self-consistent Fokker-Planck equation defining the probability density of the state of the system, which describes a well-known model of autocatalytic chemical reaction (brusselator) with spatially correlated multiplicative noise, and to study the evolution of probability density and statistical characteristics of the system in the process of spatial pattern formation. We propose the finite-difference method for the numerical solving of a general class of multivariate nonlinear self-consistent time-dependent Fokker-Planck equations. We illustrate the accuracy and reliability of the method by applying it to an exactly solvable nonlinear Fokker-Planck equation (NFPE) for the Shimizu-Yamada model [Prog. Theor. Phys. 47, 350 (1972)] and nonlinear Fokker-Planck equation [Desai and Zwanzig, J. Stat. Phys. 19, 1 (1978)] obtained for a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1975)]. Taking the problems indicated above as an example, the accuracy of the method is compared with the accuracy of Hermite distributed approximating functional method [Zhang et al., Phys. Rev. E 56, 1197 (1997)]. Numerical study of the NFPE solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple “repumping” of probability density through bimodality. Additionally, we study the behavior of the order parameter of the system under consideration and show that the second type of solution arises in the supercritical region if noise intensity values are close to the values appropriate for the transition from bimodal stationary probability density for the order parameter to the unimodal one.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
20 More
  • Received 3 June 2014

DOI:https://doi.org/10.1103/PhysRevE.90.022135

©2014 American Physical Society

Authors & Affiliations

Svetlana E. Kurushina and Valerii V. Maximov

  • Physics Department, Samara State Aerospace University named after S.P. Korolyov, Moskovskoye Shosse 34, 443086 Samara, Russian Federation, and Mathematics Department, Samara State Transport University, First Bezimyannii Pereulok 18, 443066 Samara, Russian Federation

Yurii M. Romanovskii

  • Physics Department, Lomonosov Moscow State University, GSP-1, Leninskie Gory, 119991 Moscow, Russian Federation

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 90, Iss. 2 — August 2014

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×