Mohammad-Ali Malek-Mansour

A master equation description of local fluctuations

A theory of fluctuations of macrovariables in nonequilibrium systems based on a nonlinear master equation is outlined. This equation takes into account, via a “mean field” type of approximation, the effect of the spatial extension of fluctuations. A comparison with the birth and death formalism reveals several unsatisfactory features of the latter.

Systematic analysis of the multivariate master equation for a reaction-diffusion system

The multivariate master equation for a reaction-diffusion system is analyzed using a singular perturbation approach. It is shown that in the vicinity of a bifurcation leading to two simultaneously stable steady states, the steady-state probability distribution reduces asymptotically to the exponential of the Landau-Ginzburg functional. On the other hand, for a system displaying quadratic nonlinearities and an absorbing state, critical behavior is ruled out.

The influence of external noise on non-equilibrium phase transitions

The method of Ito stochastic differential equations is used to analyze the influence of external noise in non-equilibrium phase transitions. It is found that external noise deeply affects the behaviour of the system and gives rise to new phenomena not predicted by the deterministic analysis.

A new approximation scheme for the study of fluctuations in nonuniform nonequilibrium systems

Abstract A new approximation scheme for solving the multivariate master equation is presented. The results are compared with those of computer experiments. A very good agreement is obtained, even in the neighbourhood of the critical point where a nonequilibrium phase transition occurs.

The Onset of Instabilities in Nonequilibrium Systems

The nonlinear master equation previously proposed by Malek-Mansour and Nicolis is applied to the analysis of unstable transitions leading to temporally or spatially organized patterns. Thecorrelation length of the destabilizing fluctuations is determined, and a number of striking analogies with equilibrium phase transitions are pointed out.

Mean-field theory of critical behaviour and first order transition: A comparison between the Master Equation and the nonlinear Fokker-Planck equation

The results of the nonlinear Fokker-Planck equation formalism, which was recently derived as an asymptotic representation of the Master Equation for large system size, are compared with the exact solution of the Master Equation for the Schlögl model. As far as critical behaviour is concerned complete agreement is found. Furthermore at the first order transition points the nonlinear Fokker-Planck equation is shown to constitute an excellent approximation.

Fluctuations and the onset of instabilities in nonequilibrium systems

Abstract A nonlinear master equation describing the nucleation of critical fluctuations leading to an instability and subsequently to a dissipative structure is derived. It is suggested that the formation of these structures bears strong analogies with first order phase transitions.

On the diffusion operator of the multivariate master equation

The eigenfunctions of the diffusion operator of the multivariate master equation, describing reaction diffusion systems, are calculated for various boundary conditions. This serves as a starting point for a systematic study of the general solution of the master equation. As a first application a perturbation expansion in the inverse of the diffusion constant is carried out.

On the absorbing zero boundary problem in birth and death processes

The probability of absorption as well as the mean absorption time and its variance, have been calculated for a model which does not satisfy detailed balance. The explicit calculation of the variance is helpful in understanding the different predictions between the deterministic and the stochastic descriptions. These results can be obtained directly from the master equation by Laplace transform techniques.

An asymptotic expansion of the nonlinear master equation

Recently, a nonlinear master equation has been suggested to account for the effect of diffusion in the fluctuations of nonlinear systems away from equilibrium. An asymptotic expansion of the solutions of this master equation in the inverse of the diffusion constant is presented. The applicability of the method is illustrated with several examples of model chemical reactions.