Search results for "equation"

Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion

2016

In this paper the Turing pattern formation mechanism of a two components reaction-diffusion system modeling the Schnakenberg chemical reaction is considered. In Ref. (Madzavamuse et al., J Math Biol 70(4):709–743, 2015) it was shown how the presence of linear cross-diffusion terms favors the destabilization of the constant steady state. We perform the weakly nonlinear multiple scales analysis to derive the equations for the amplitude of the Turing patterns and to show how the cross-diffusion coefficients influence the occurrence of super-critical or sub-critical bifurcations. We present a numerical exploration of far from equilibrium regimes and prove the existence of multistable stationary…

A Langevin Approach to the Diffusion Equation

2002

We propose a generalized Langevin equation as a model for the diffusion equation of air pollution in the atmosphere. We write down a partial stochastic differential equation for the pollutant concentration, which we solve exactly obtaining the first and the second moment of the pollutant concentration. We obtain a linear multiplicative stochastic differential equation for the Fourier components of the concentration, which can be used to calculate higher moments of the concentration. We obtain the exact steady state solution in the case of neutral atmosphere and a general expression of the mean concentration as a function of the fluctuation intensity of the wind speed, the diffusion coeffici…

Acceleration of diffusion in randomly switching potential with supersymmetry

2004

We investigate the overdamped Brownian motion in a supersymmetric periodic potential switched by Markovian dichotomous noise between two configurations. The two configurations differ from each other by a shift of one-half period. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential. For the sawtooth potential, we obtain the exact expression for the effective diffusion coefficient, which is valid for the arbitrary mean rate of potential switchings and arbitrary intensity of white Gaussian noise. We…

The Fokker-Planck Equation

2009

Stochastic Kinetics with Wave Nature

2003

We consider stochastic second-order partial differential equations. We indroduce a noisy non-linear wave equation and discuss its connections, in particular via the Lorentz transformation, with known stochastic models.

Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

2018

This paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonallyinvariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $%L^{1}(\Omega \times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $%\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show that, under very natural assumptions, if an …

Studies of the hydrodynamic evolution of matter produced in fluctuations inp¯pcollisions and in ultrarelativistic nuclear collisions

1986

In this first paper of a series of two, we present a comprehensive study of the hydrodynamic evolution of matter produced in the central region of ultrarelativistic heavy-ion collisions and in high-multiplicity fluctuations of p-barp-italic collisions. We shall begin with a discussion of the limits of the applicability of a perfect-fluid hydrodynamic description of high-energy collisions. A simple bag-model equation of state is argued to have qualitative and semiquantitative features expected from lattice gauge theory and present theoretical understanding. We also discuss the boundary conditions for the perfect-fluid hydrodynamic equations, and what classes of simple events would correspond…

Dynamics of surface enrichment: A theory based on the Kawasaki spin-exchange model in the presence of a wall

1991

A mean-field theory is developed for the description of the dynamics of surface enrichment in binary mixtures, where one component is favored by an impenetrable wall. Assuming a direct exchange (Kawasaki-type) model of interdiffusion, a layerwise molecular-field approximation is formulated in the framework of a lattice model. Also the corresponding continuum theory is considered, paying particular attention to the proper derivation of boundary conditions for the differential equation at the hard wall. As an application, we consider the explicit solutions of the derived equations in the case where nonlinear effects can be neglected, studying the approach of an initially flat (homogeneous) co…

Self‐similar problems for modeling the surface chemical reactions with the gravitation

1998

The mathematical model of a chemical reaction which takes place on the surface of the uniformly moving vertically imbedded glass fibre material is considered. The effect of gravitation is taken into account. Boussinesq's and boundary layer fittings allow to derive boundary value problems for self‐similar systems of ordinary differential equations. First Published Online: 14 Oct 2010

Simulation of surface-controlled phase separation in slit pores: Diffusive Ginzburg-Landau kinetics versus Molecular Dynamics

2008

The phase separation kinetics of binary fluids in constrained geometry is a challenge for computer simulation, since nontrivial structure formation occurs extending from the atomic scale up to mesoscopic scales, and a very large range of time needs to be considered. One line of attack to this problem is to try nevertheless standard Molecular Dynamics (MD), another approach is to coarse-grain the model to apply a time-dependent nonlinear Ginzburg–Landau equation that is numerically integrated. For a symmetric binary mixture confined between two parallel walls that prefer one species, both approaches are applied and compared to each other. There occurs a nontrivial interplay between the forma…