Fluctuating hydrodynamics of multi-species reactive mixtures

Physical Review E, 2014

In this paper we discuss the formulation of the fluctuating Navier-Stokes (FNS) equations for multi-species, non-reactive fluids. In particular, we establish a form suitable for numerical solution of the resulting stochastic partial differential equations. An accurate and efficient numerical scheme, based on our previous methods for single species and binary mixtures, is presented and tested at equilibrium as well as for a variety of non-equilibrium problems. These include the study of giant nonequilibrium concentration fluctuations in a ternary mixture in the presence of a diffusion barrier, the triggering of a Rayleigh-Taylor instability by diffusion in a four-species mixture, as well as reverse diffusion in a ternary mixture. Good agreement with theory and experiment demonstrates that the formulation is robust and can serve as a useful tool in the study of thermal fluctuations for multispecies fluids. The extension to include chemical reactions will be treated in a sequel paper.

Physica A: Statistical Mechanics and its Applications, 1997

We have used the thermodynamical description of a chemical reaction as a diffusion process along an internal coordinate to analyze fluctuations in the density of the constituents, which are treated under the framework of fluctuating hydrodynamics. We then obtain a Langevin equation for the density, as a function of the internal coordinate, whose stochastic source statisfies a fluctuation-dissipation theorem. After contraction of the description, by means of integration in the internal coordinate, we derive the Langevin equation for the concentration of reactants and products as well as the statistical properties of the random source which agree with the corresponding results obtained by means of Keizer's theory. Application of the formalism is illustrated by considering particular cases. An extension to coupled chemical reactions is also discussed.

Journal of Statistical Physics, 1991

The one-dimensional single-species diffusion-limited-coagulation process with irreversible random particle input (A~A + A reversibly and B~A irreversibly), under the influence of external fluctuations in the system parameters, is formulated in terms of a closed and linear partial-differential equation. Our theoretical treatment includes both internal fluctuations and external noise simultaneously and without approximation, allowing investigation of the interplay of their effects on the macroscopic behavior of this diffusion-reaction system. For the reversible model with the rate of the A~A+ A reaction fluctuating between two values as a Markov stochastic process, we solve the system exactly. We observe that spatially homogeneous macroscopic fluctuations in the system parameters can induce microscopic spatial correlations in the nonequilibrium steady state. Direct Monte Carlo simulations of the microscopic dynamics are presented, confirming the theoretical analysis and directly illustrating the externalnoise-induced spatial correlations.

Communications in Applied Mathematics and Computational Science, 2014

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations represent a coarse-graining of the microscopic dynamics of the fluid molecules in both space and time and eliminate the fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatiotemporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions and construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation-of-state constraint. The resulting spatiotemporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients and investigate the validity of common simplifications such as neglecting the spatial nonhomogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular-dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing.

2013

The dynamics of a binary system with non conserved order parameter under a plain shear flow with rate γ is solved analytically in the large-N limit. A phase transition is observed at a critical temperature Tc(γ). After a quench from a high temperature equilibrium state to a lower temperature T a nonequilibrium stationary state is entered when T> Tc(γ), while aging dynamics characterizes the phases with T ≤ Tc(γ). Two-time quantities are computed and the off-equilibrium generalization of the fluctuation-dissipation theorem is provided.

arXiv (Cornell University), 2022

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory.

Physics Letters A, 1979

A modification is introduced in a previously derived master equation describing a fluctuating Boltzmann gas in a lumped phase space in order to obtain from it conservation equations in a correct local form.

2002

The dynamics of a binary system with non conserved order parameter under a plain shear flow with rate γ is solved analytically in the large-N limit. A phase transition is observed at a critical temperature T c (γ). After a quench from a high temperature equilibrium state to a lower temperature T a nonequilibrium stationary state is entered when T > T c (γ), while aging dynamics characterizes the phases with T ≤ T c (γ). Two-time quantities are computed and the off-equilibrium generalization of the fluctuation-dissipation theorem is provided. 05.70.Ln; 64.75.+g; 83.50.Ax Typeset using REVT E X

Physical Review A, 1992

The one-dimensional single-species diffusion-limited-coagulation process with irreversible random particle input (A~A + A reversibly and B~A irreversibly), under the influence of external fluctuations in the system parameters, is formulated in terms of a closed and linear partial-differential equation. Our theoretical treatment includes both internal fluctuations and external noise simultaneously and without approximation, allowing investigation of the interplay of their effects on the macroscopic behavior of this diffusion-reaction system. For the reversible model with the rate of the A~A+ A reaction fluctuating between two values as a Markov stochastic process, we solve the system exactly. We observe that spatially homogeneous macroscopic fluctuations in the system parameters can induce microscopic spatial correlations in the nonequilibrium steady state. Direct Monte Carlo simulations of the microscopic dynamics are presented, confirming the theoretical analysis and directly illustrating the externalnoise-induced spatial correlations.

The Journal of Chemical Physics, 2007

In this paper a simple reaction-diffusion system, namely a binary fluid mixture with an association-dissociation reaction between the two components, is considered. Fluctuations at hydrodynamic spatiotemporal scales when a temperature gradient is present in this chemically reacting system are studied. First, fluctuating hydrodynamics when the system is in global equilibrium ͑isothermal͒ is reviewed. Comparing the two cases, an enhancement of the intensity of concentration fluctuations in the presence of a temperature gradient is predicted. The nonequilibrium concentration fluctuations are spatially long ranged, with an intensity depending on the wave number q. The intensity exhibits a crossover from a ϰq −4 to a ϰq −2 behavior depending on whether the corresponding wavelength is smaller or larger than the penetration depth of the reacting mixture. This opens a possibility to distinguish between diffusion-or activation-controlled regimes of the reaction by measuring these fluctuations. In addition, the possible observation of these fluctuations in nonequilibrium molecular dynamics simulations is considered.