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

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 stochastic 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.

[1]  J. C. Lewis.  On the Einstein-Stokes diffusion coefficient for Brownian motion in two dimensions , 1973 .

[2]  Gammon,et al.  Light-scattering measurements of nonequilibrium fluctuations in a liquid mixture. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Stefano Olla,et al.  Hydrodynamics and large deviation for simple exclusion processes , 1989 .

[4]  M. Rex,et al.  Dynamical density functional theory for colloidal dispersions including hydrodynamic interactions , 2009, The European physical journal. E, Soft matter.

[5]  R. Kapral,et al.  Molecular theory of translational diffusion: Microscopic generalization of the normal velocity boundary condition , 1979 .

[6]  T. R. Kirkpatrick,et al.  Generic Long-Range Correlations in Molecular Fluids , 1994 .

[7]  D. Cannell,et al.  Fractal fronts of diffusion in microgravity , 2011, Nature communications.

[8]  D. Brogioli,et al.  Diffusive mass transfer by nonequilibrium fluctuations: Fick's law revisited. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Ann S. Almgren,et al.  An unsplit, higher-order Godunov method using quadratic reconstruction for advection in two dimensions , 2011 .

[10]  Alejandro L. Garcia,et al.  Enhancement of diffusive transport by non-equilibrium thermal fluctuations , 2011, 1103.5244.

[11]  Eric Vanden-Eijnden,et al.  NUMERICAL TECHNIQUES FOR MULTI-SCALE DYNAMICAL SYSTEMS WITH STOCHASTIC EFFECTS ⁄ , 2003 .

[12]  Leslie Greengard,et al.  Accelerating the Nonuniform Fast Fourier Transform , 2004, SIAM Rev..

[13]  S. Adzhiev,et al.  Entropy in the sense of Boltzmann and Poincaré , 2014, Contemporary Mathematics. Fundamental Directions.

[14]  Aleksandar Donev,et al.  On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics , 2009 .

[15]  Grigorios A. Pavliotis,et al.  Multiscale Methods: Averaging and Homogenization , 2008 .

[16]  Andrew J. Majda,et al.  Stochastic models for selected slow variables in large deterministic systems , 2006 .

[17]  A. Donev,et al.  Reversible Diffusion by Thermal Fluctuations , 2013, 1306.3158.

[18]  Thomas G. Kurtz,et al.  A limit theorem for perturbed operator semigroups with applications to random evolutions , 1973 .

[19]  E Weinan,et al.  A general strategy for designing seamless multiscale methods , 2009, J. Comput. Phys..

[20]  R. Khas'minskii,et al.  Principle of Averaging for Parabolic and Elliptic Differential Equations and for Markov Processes with Small Diffusion , 1963 .

[21]  Eric Vanden Eijnden.  Numerical techniques for multi-scale dynamical systems with stochastic effects , 2003 .

[22]  U. Frisch,et al.  Lagrangian method for multiple correlations in passive scalar advection , 1998 .

[23]  H. Jonker,et al.  Incompressibility of the Leray-α model for wall-bounded flows , 2006 .

[24]  M. Giglio,et al.  Nonequilibrium fluctuations in time-dependent diffusion processes , 1998 .

[25]  David I. Ketcheson,et al.  Communications in Applied Mathematics and Computational Science , 2012 .

[26]  Boyce E. Griffith,et al.  Efficient Variable-Coefficient Finite-Volume Stokes Solvers , 2013, 1308.4605.

[27]  D. Dean.  LETTER TO THE EDITOR: Langevin equation for the density of a system of interacting Langevin processes , 1996, cond-mat/9611104.

[28]  P. Español,et al.  Derivation of dynamical density functional theory using the projection operator technique. , 2009, The Journal of chemical physics.

[29]  Andrew J. Archer,et al.  Dynamical density functional theory for interacting Brownian particles: stochastic or deterministic? , 2004, cond-mat/0405603.

[30]  Aleksandar Donev,et al.  The Stokes-Einstein relation at moderate Schmidt number. , 2013, The Journal of chemical physics.

[31]  Juan J de Pablo,et al.  Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry. , 2007, Physical review letters.

[32]  J. Bechhoefer,et al.  EXPERIMENTS ON TRACER DIFFUSION IN THIN FREE-STANDING LIQUID-CRYSTAL FILMS , 1997 .

[33]  John B. Bell,et al.  Staggered Schemes for Fluctuating Hydrodynamics , 2012, Multiscale Model. Simul..

[34]  LETTER TO THE EDITOR: On the controversy over the stochastic density functional equations , 2000, cond-mat/0003339.

[35]  T. Schaffter.  Numerical Integration of SDEs: A Short Tutorial , 2010 .

[36]  Boyce E. Griffith,et al.  An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner , 2009, J. Comput. Phys..

[37]  I. Oppenheim.  Beyond Equilibrium Thermodynamics , 2006 .

[38]  M. Giglio,et al.  Universal behavior of nonequilibrium fluctuations in free diffusion processes , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[39]  Paul J. Atzberger,et al.  A note on the correspondence of an immersed boundary method incorporating thermal fluctuations with Stokesian-Brownian dynamics , 2007 .

[40]  M. Giglio,et al.  Giant fluctuations in a free diffusion process , 1997, Nature.

[41]  Transport of a Passive Tracer by an Irregular Velocity Field , 2004 .

[42]  C. Gardiner.  Adiabatic elimination in stochastic systems. I: Formulation of methods and application to few-variable systems , 1984 .

[43]  Jan V. Sengers,et al.  Hydrodynamic Fluctuations in Fluids and Fluid Mixtures , 2006 .

[44]  I. Oppenheim,et al.  Bilinear Hydrodynamics and the Stokes-Einstein Law , 1973 .

[45]  Paul J. Atzberger,et al.  Stochastic Eulerian Lagrangian methods for fluid-structure interactions with thermal fluctuations , 2009, J. Comput. Phys..

[46]  D. J. O H.  Accelerated Stokesian Dynamics simulations , 2022 .

[47]  Burkhard Dünweg,et al.  Implicit and explicit solvent models for the simulation of a single polymer chain in solution: Lattice Boltzmann versus Brownian dynamics. , 2009, The Journal of chemical physics.

[48]  F. Detcheverry,et al.  Thermal fluctuations of hydrodynamic flows in nanochannels. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  Boyce E. Griffith,et al.  Temporal integrators for fluctuating hydrodynamics , 2012, 1212.1033.

[50]  Self-Similar Decay in the Kraichnan Model of a Passive Scalar , 1999, chao-dyn/9910022.

[51]  E Weinan,et al.  Heterogeneous multiscale methods: A review , 2007 .

[52]  Thomas G. Fai,et al.  Low Mach Number Fluctuating Hydrodynamics of Diffusively Mixing Fluids , 2012, 1212.2644.

[53]  A. Majda,et al.  SIMPLIFIED MODELS FOR TURBULENT DIFFUSION : THEORY, NUMERICAL MODELLING, AND PHYSICAL PHENOMENA , 1999 .

[54]  Pedro Tarazona,et al.  Dynamic density functional theory of fluids , 1999 .

[55]  F. Scheffold,et al.  A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient. , 2012, The Journal of chemical physics.

[56]  I. Ial,et al.  Nature Communications , 2010, Nature Cell Biology.

[57]  P. Español,et al.  On the definition of discrete hydrodynamic variables. , 2009, The Journal of chemical physics.

[58]  P. Español,et al.  Microscopic derivation of discrete hydrodynamics. , 2009, The Journal of chemical physics.

[59]  Darryl D. Holm,et al.  The Navier–Stokes-alpha model of fluid turbulence , 2001, nlin/0103037.

[60]  G. Pavliotis,et al.  General dynamical density functional theory for classical fluids. , 2012, Physical review letters.

[61]  F. Detcheverry,et al.  Thermal fluctuations in nanofluidic transport. , 2012, Physical review letters.

[62]  E. Vanden-Eijnden,et al.  Dynamic density functional theory with hydrodynamic interactions and fluctuations. , 2014, The Journal of chemical physics.

[63]  Aleksandar Donev,et al.  Diffusive transport by thermal velocity fluctuations. , 2011, Physical review letters.

[64]  Boyce E. Griffith,et al.  Inertial coupling method for particles in an incompressible fluctuating fluid , 2012, 1212.6427.

[65]  M. Rex,et al.  Dynamical density functional theory with hydrodynamic interactions and colloids in unstable traps. , 2008, Physical review letters.

[66]  Boyce E. Griffith,et al.  Brownian dynamics without Green's functions. , 2013, The Journal of chemical physics.

[67]  John B. Bell,et al.  A Three-Dimensional, Unsplit Godunov Method for Scalar Conservation Laws , 2011, SIAM J. Sci. Comput..

[68]  G. Papanicolaou.  Some probabilistic problems and methods in singular perturbations , 1976 .

[69]  C. W. Gardiner,et al.  Adiabatic elimination in stochastic systems. III. Application to renormalization-group transformations of the time-dependent Ginsburg-Landau model , 1984 .

[70]  Aleksandar Donev,et al.  On the Accuracy of Finite-Volume Schemes for Fluctuating Hydrodynamics , 2009, 0906.2425.

[71]  D. Cannell,et al.  Nondiffusive decay of gradient-driven fluctuations in a free-diffusion process. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[72]  Large deviations from the hydrodynamical limit for a system of independent brownian particles , 1990 .

[73]  J. Schuster,et al.  Optical detection of heterogeneous single molecule diffusion in thin liquid crystal films. , 2010, Physical chemistry chemical physics : PCCP.