Solving the Fokker-Planck kinetic equation on a lattice

Weiss mean-field approximation for multicomponent stochastic spatially extended systems

We develop a mean-field approach for multicomponent stochastic spatially extended systems and use it to obtain a multivariate nonlinear self-consistent Fokker-Planck equation defining the probability density of the state of the system, which describes a well-known model of autocatalytic chemical reaction (brusselator) with spatially correlated multiplicative noise, and to study the evolution of probability density and statistical characteristics of the system in the process of spatial pattern formation. We propose the finite-difference method for the numerical solving of a general class of multivariate nonlinear self-consistent time-dependent Fokker-Planck equations. We illustrate the accuracy and reliability of the method by applying it to an exactly solvable nonlinear Fokker-Planck equation (NFPE) for the Shimizu-Yamada model [Prog. Theor. Phys. 47, 350 (1972)] and nonlinear Fokker-Planck equation [Desai and Zwanzig, J. Stat. Phys. 19, 1 (1978)] obtained for a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1975)]. Taking the problems indicated above as an example, the accuracy of the method is compared with the accuracy of Hermite distributed approximating functional method [Zhang et al., Phys. Rev. E 56, 1197 (1997)]. Numerical study of the NFPE solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple "repumping" of probability density through bimodality. Additionally, we study the behavior of the order parameter of the system under consideration and show that the second type of solution arises in the supercritical region if noise intensity values are close to the values appropriate for the transition from bimodal stationary probability density for the order parameter to the unimodal one.

Lattice Fokker Planck for dilute polymer dynamics

We show that the actual diffusive dynamics, governing the momentum relaxation of a polymer molecule, and described by a Fokker-Planck equation, may be replaced by a BGK-type relaxation dynamics without affecting the slow (Smoluchowski) dynamics in configuration space. Based on the BGK-type description, we present a lattice-Boltzmann (LB) based direct discretization approach for the phase-space description of inertial polymer dynamics. We benchmark this formulation by determining the bulk rheological properties for both steady and time-dependent shear and extensional flows at moderate to large Weissenberg numbers. Finally, we compare the usefulness of the different discrete velocity models, typically used in the LB framework, for solving diffusive dynamics based on the Fokker-Planck equation.

Stabilized lattice Boltzmann-Enskog method for compressible flows and its application to one- and two-component fluids in nanochannels

A numerically stable method to solve the discretized Boltzmann-Enskog equation describing the behavior of nonideal fluids under inhomogeneous conditions is presented. The algorithm employed uses a Lagrangian finite-difference scheme for the treatment of the convective term and a forcing term to account for the molecular repulsion together with a Bhatnagar-Gross-Krook relaxation term. In order to eliminate the spurious currents induced by the numerical discretization procedure, we use a trapezoidal rule for the time integration together with a version of the two-distribution method of He et al. [J. Comput. Phys. 152, 642 (1999)]. Numerical tests show that, in the case of a one-component fluid in the presence of a spherical potential well, the proposed method reduces the numerical error by several orders of magnitude. We conduct another test by considering the flow of a two-component fluid in a channel with a bottleneck and provide information about the density and velocity field in this structured geometry.

A lattice Boltzmann method for dilute polymer solutions

We present a lattice Boltzmann approach for the simulation of non-Newtonian fluids. The method is illustrated for the specific case of dilute polymer solutions. With the appropriate local equilibrium distribution, phase-space dynamics on a lattice, driven by a Bhatnagar-Gross-Krook (BGK) relaxation term, leads to a solution of the Fokker-Planck equation governing the probability density of polymer configurations. Results for the bulk rheological characteristics for steady and start-up shear flow are presented, and compare favourably with those obtained using Brownian dynamics simulations. The new method is less expensive than stochastic simulation techniques, particularly in the range of small to moderate Weissenberg numbers (Wi).

Bridging molecular and continuous descriptions: the case of dynamics in clays

The theory of transport in porous media such as clays depends on the level of description. On the macroscopic scale,hydrodynamics equations are used. These continuous descriptions are convenient to model the fluid motion in a confined system. Nevertheless, they are valid only if the pores of the material are much larger than the molecular size of the components of the system. Another approach consists in using molecular descriptions. These two methods which correspond to different levels of description are complementary. The link between them can be clarified by using a coarse-graining procedure where the microscopic laws are averaged over fast variables to get the long time macroscopic laws. We present such an approach in the case of clays. Firstly, we detail the various levels of description and the relations among them, by emphasizing the validity domain of the hydrodynamic equations. Secondly, we focus on the case of dehydrated clays where hydrodynamics is not relevant. We show that it is possible to derive a simple model for the motion of the cesium ion based on the difference on time scale between the solvent and the solute particles.

Kinetic theory of correlated fluids: from dynamic density functional to Lattice Boltzmann methods

Using methods of kinetic theory and liquid state theory we propose a description of the nonequilibrium behavior of molecular fluids, which takes into account their microscopic structure and thermodynamic properties. The present work represents an alternative to the recent dynamic density functional theory, which can only deal with colloidal fluids and is not apt to describe the hydrodynamic behavior of a molecular fluid. The method is based on a suitable modification of the Boltzmann transport equation for the phase space distribution and provides a detailed description of the local structure of the fluid and its transport coefficients. Finally, we propose a practical scheme to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the Lattice Boltzmann method.

Phase-space approach to dynamical density functional theory

The authors consider a system of interacting particles subjected to Langevin inertial dynamics and derive the governing time-dependent equation for the one-body density. They show that, after suitable truncations of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and a multiple time scale analysis, they obtain a self-consistent equation involving only the one-body density. This study extends to arbitrary dimensions previous work on a one-dimensional fluid and highlights the subtleties of kinetic theory in the derivation of dynamical density functional theory.

A multiscale approach to ion diffusion in clays: Building a two-state diffusion–reaction scheme from microscopic dynamics

The mobility of particles is generally lowered by the presence of a confining medium, both because of geometrical effects, and because of the interactions with the confining surfaces, especially when the latter are charged. The water/mineral interface plays a central role in the dynamics of ions. The ionic mobility in clays is often understood as an interplay between the diffusion of mobile ions and their possible trapping at the mineral surfaces. We describe how to build a two-state diffusion-reaction scheme from the microscopic dynamics of ions, controlled by their interaction with a mineral surface. The starting point is an atomic description of the clay interlayer using molecular simulations. These provide a complete description of the ionic dynamics on short time and length scales. Using the results of these simulations, we then build a robust mesoscopic (Fokker-Planck) description. In turn, this mesoscopic description is used to determine the mobility of the ions in the interlayer. These results can then be cast into a diffusion-reaction scheme, introducing in particular the fraction of mobile ions, or equivalently the distribution coefficient Kd. This coefficient is of great importance in characterizing electrokinetic phenomena in porous materials.

Second-order lattice Fokker-Planck algorithm from the trapezoidal rule

The well-established procedure of deriving lattice Boltzmann schemes using the trapezoidal rule for time integration is generalized to the recently introduced lattice Fokker-Planck method.