M. Malek Mansour

Asymptotic properties of markovian master equations

Abstract An asymptotic method of analysis of fluctuations in systems far from equilibrium is developed. A systematic singular perturbative expansion of the equation for the generating function is set up, using as smallness parameter the inverse of the size of the system. Static and time-dependent properties are analysed before, near and at a bifurcation point, both for homogeneous and inhomogeneous fluctuations. The connection with critical phenomena near equilibrium is also discussed.

On the validity of hydrodynamics in plane Poiseuille flows

Microscopic simulations of plane Poiseuille flow for a dilute gas are presented. Although the flow is laminar (Reynolds number ≈10) and sub-sonic, the temperature and pressure profiles measured in the simulations differ qualitatively from the hydrodynamic predictions. The results are in agreement with a recent theoretical analysis based on the asymptotic solution of the BGK model of the Boltzmann equation.

Stochastic theory of adiabatic explosion

A stochastic description of an exothermic reaction leading to adiabatic explosion is set up. The numerical solution of the master equation reveals the appearance of a long tail and of multiple humps of the probability distribution, which subsist for a certain period of time. During this interval the system displays a markedly chaotic behavior, reflecting the random character of the ignition process. An analytical description of this transient evolution is developed, using a piecewise linear approximation of the transition rates. A comparison with other transient phenomena observed in stochastic theory is carried out.

Microscopic simulation of chemical oscillations in homogeneous systems

A microscopic computer experiment is set up to investigate the statistical properties of far from equilibrium homogeneous chemical systems undergoing instabilities. Sustained periodic behavior of limit cycle type is observed. Both the frequency and the amplitude of the oscillations are found to be in good agreement with the macroscopic description. A comparison with the stochastic theory of chemical systems based on master equation formalism is also carried out. The dynamical and static correlation functions obtained by these two procedures are in very good agreement.

Microscopic simulation of chemical bistability in homogeneous systems

Microscopic simulation is used to clarify the status of stochastic theories of homogeneous chemical systems operating in the multiple steady state region. The results demonstrate the failure of the Langevin approach, but show excellent agreement with the master equation formulation.

Asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise. II: Transition to a limit cycle

We apply the singular perturbation technique, developed in the companion paper, to the study of the fluctuations at the onset of a limit cycle, both for the cases of a soft and a hard transition. The technique and results are illustrated on the Poincaré model (soft transition) and on the Van der Pol oscillator (hard transition).

Microscopic Simulation of Chemical Systems

Abstract Microscopic simulations of chemical systems under nonequilibrium constraints are reported. The main differences with molecular dynamics simulations of simple non-reactive fluids are underlined. The relevance of the new information available from the simulation in testing theoretical ideas is assessed.

Temperature profile of a dilute gas undergoing a plane Poiseuille flow

The temperature and the pressure profile of a dilute or moderately rarefied gas undergoing a plane Poiseuille flow are calculated and compared with simulation data for a gas of hard spheres as first presented by Malek Mansour, Baras and Garcia (Physica A 240 (1997) 225). The Boltzmann equation is solved by the moment method. The desired solution including terms up to second order in the applied force driving the flow, requires the use of a 19 moments approximation. It comprises results based on somewhat simpler 13 moments approximation which are qualitatively reasonable for the present problem.

Asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise. I: Cusp bifurcation

We show how a singular perturbation technique based on the introduction of properly scaled variables enables us to derive the asymptotic properties of coupled Langevin equations in the limit of weak noise. This technique can be applied when the macroscopic steady state is asymptotically or marginally stable. In the close vicinity of a cusp bifurcation point, a simple prescription for the adiabatic elimination of the fast variable is established. The critical variable exhibits amplified non-Gaussian fluctuations on a slow time scale. The properties of the fast variable depend on the nonlinearity of the system under consideration. Because of its coupling to the critical variable, it may exhibit amplified fluctuations of non-Gaussian nature.

Nonlinear analysis of the shearing instability in granular gases

It is known that a finite-size homogeneous granular fluid develops a hydrodynamiclike instability when dissipation crosses a threshold value. This instability is analyzed in terms of modified hydrodynamic equations: first, a source term is added to the energy equation which accounts for the energy dissipation at collisions and the phenomenological Fourier law is generalized according to previous results. Second, a rescaled time formalism is introduced that maps the homogeneous cooling state into a nonequilibrium steady state. A nonlinear stability analysis of the resulting equations is done which predicts the appearance of flow patterns. A stable modulation of density and temperature is produced that does not lead to clustering. Also a global decrease of the temperature is obtained, giving rise to a decrease of the collision frequency and dissipation rate. Good agreement with molecular dynamics simulations of inelastic hard disks is found for low dissipation.