A reversible mesoscopic model of diffusion in liquids: from giant fluctuations to Fick's law

Abstract

We study diffusive mixing in the presence of thermal fluctuations under the assumption of large Schmidt number. In this regime we obtain a limiting equation that contains a diffusive thermal drift term with diffusion coefficient obeying a Stokes-Einstein relation, in addition to the expected advection by a random velocity. The overdamped limit correctly reproduces both the enhanced diffusion in the ensemble-averaged mean and the long-range correlated giant fluctuations in individual realizations of the mixing process, and is amenable to efficient numerical solution. Through a combination of Eulerian and Lagrangian numerical methods we demonstrate that diffusion in liquids is not most fundamentally described by Fick's irreversible law; rather, diffusion is better modeled as reversible random advection by thermal velocity fluctuations. We find that the diffusion coefficient is effectively renormalized to a value that depends on the scale of observation. Our work reveals somewhat unexpected connections between flows at small scales, dominated by thermal fluctuations, and flows at large scales, dominated by turbulent fluctuations.

To appear in JSTAT. Shorter version available as ArXiv:1306.3158

References in corpus (4)

• Dynamical density functional theory for interacting Brownian particles: stochastic or deterministic?
• Dynamical density functional theory with hydrodynamic interactions and colloids in unstable traps
• Brownian Dynamics without Green's Functions
• Simple Systems with Anomalous Dissipation and Energy Cascade

Cited by in corpus (30)

• Perspective: Dissipative Particle Dynamics
• Classical dynamical density functional theory: from fundamentals to applications
• Brownian Dynamics without Green's Functions
• Dynamic Density Functional Theory with hydrodynamic interactions and fluctuations
• Stochastic Simulation of Reaction-Diffusion Systems: A Fluctuating-Hydrodynamics Approach
• Non-equilibrium Casimir-like Forces in Liquid Mixtures
• Low Mach Number Fluctuating Hydrodynamics of Multispecies Liquid Mixtures
• Multiscale temporal integrators for fluctuating hydrodynamics
• On Dean-Kawasaki Dynamics with Smooth Drift Potential
• Low Mach Number Fluctuating Hydrodynamics for Electrolytes
• Finite element discretization of non-linear diffusion equations with thermal fluctuations
• Efficient Reactive Brownian Dynamics
• Coupling a nano-particle with isothermal fluctuating hydrodynamics: Coarse-graining from microscopic to mesoscopic dynamics
• Low Mach Number Fluctuating Hydrodynamics of Binary Liquid Mixtures
• The Five Problems of Irreversibility
• A Discrete Ion Stochastic Continuum Overdamped Solvent Algorithm for Modeling Electrolytes
• Hydrodynamic fluctuations in quasi-two dimensional diffusion
• Dynamic density functional theory for drying colloidal suspensions: Comparison of hard-sphere free-energy functionals
• Stochastic dynamics and thermodynamics around a metastable state based on the linear Dean-Kawasaki equation
• Weak error analysis for a nonlinear SPDE approximation of the Dean-Kawasaki equation
• Turbulent diffusion of chemically reacting gaseous admixtures
• Fluctuating hydrodynamics of dilute electrolyte solutions: systematic perturbation calculation of effective transport coefficients governing large-scale dynamics
• Lattice Boltzmann Simulations of Non-Equilibrium Fluctuations in a Non-Ideal Binary Mixture
• Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise
• Emergence of long-range non-equilibrium correlations in free liquid diffusion
• Non-Gaussian statistics of concentration fluctuations in free liquid diffusion
• Fluctuation-enhanced electric conductivity in electrolyte solutions
• Numerical approximation of singular-degenerate parabolic stochastic PDEs
• A Hybrid Algorithm for Systems of Non-interacting Particles with an External Potential
• Hydrodynamic interactions in a binary-mixture colloidal monolayer