Origin and growth of the disorder within an ordered state of the spatially extended chemical reaction model

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We now review the main points of mean-field approximation (MFA) in its application to multicomponent stochastic reaction-diffusion systems.

We present the chemical reaction model under study — brusselator. We write the kinetic equations of reaction supplementing them with terms that describe the diffusion of the intermediate components and the fluctuations of the concentrations of the initial products. We simulate the fluctuations as random Gaussian homogeneous and spatially isotropic fields with zero means and spatial correlation functions with a non-trivial structure. The model parameter values correspond to a spatially-inhomogeneous ordered state in the deterministic case.

In the MFA we derive single-site two-dimensional nonlinear self-consistent Fokker–Planck equation in the Stratonovich's interpretation for spatially extended stochastic brusselator, which describes the dynamics of probability distribution density of component concentration values of the system under consideration. We find the noise intensity values appropriate to two types of Fokker–Planck equation solutions: solution with transient bimodality and solution with the multiple alternation of unimodal and bimodal types of probability density. We study numerically the probability density dynamics and time behavior of variances, expectations, and most probable values of component concentrations at various noise intensity values and the bifurcation parameter in the specified region of the problem parameters.

Beginning from some value of external noise intensity inside the ordered phase disorder originates existing for a finite time, and the higher the noise level, the longer this disorder “embryo” lives. The farther away from the bifurcation point, the lower the noise that generates it and the narrower the range of noise intensity values at which the system evolves to the ordered, but already a new statistically steady state. At some second noise intensity value the intermittency of the ordered and disordered phases occurs. The increasing noise intensity leads to the fact that the order and disorder alternate increasingly.

Thus, the scenario of the noise induced order–disorder transition in the system under study consists in the intermittency of the ordered and disordered phases.

Keywords: mean field approximation, reaction-diffusion systems, nonlinear self-consistent Fokker–Planck equation, dynamical phase transitions, disorder
Citation in English: Kurushina S.E., Shapovalova E.A. Origin and growth of the disorder within an ordered state of the spatially extended chemical reaction model // Computer Research and Modeling, 2017, vol. 9, no. 4, pp. 595-607
Citation in English: Kurushina S.E., Shapovalova E.A. Origin and growth of the disorder within an ordered state of the spatially extended chemical reaction model // Computer Research and Modeling, 2017, vol. 9, no. 4, pp. 595-607
DOI: 10.20537/2076-7633-2017-9-4-595-607
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