Fluctuating Lattice Boltzmann

– The lattice Boltzmann algorithm efficiently simulates the Navier Stokes equation of isothermal fluid flow, but ignores thermal fluctuations of the fluid, important in mesoscopic flows. We show how to adapt the algorithm to include noise, satisfying a fluctuation-dissipation theorem (FDT) directly at lattice level: this gives correct fluctuations for mass and momentum densities, and for stresses, at all wavevectors k. Unlike previous work, which recovers FDT only as k → 0, our algorithm offers full statistical mechanical consistency in mesoscale simulations of, e.g., fluctuating colloidal hydrodynamics. The lattice Boltzmann equation (LBE) is a widely used lattice formulation of fluid mechanics [1]. It offers a faithful discretization of the Navier Stokes equation of isothermal, incompressible fluid flow, and is very well adapted to parallel computation [2]. While used for large-scale fluid dynamics simulations such as flows around aircraft [3], the LBE approach is particularly adapted to simulating mesoscopic problems [4]. These include, e.g., porous medium flows and flows of complex and multicomponent fluids with microstructure [5–8]. The latter can be modelled using various extensions of the basic algorithm for a single component fluid as considered here [8–10]. However, the Navier Stokes equation, and with it the LBE, ignores thermal fluctuations. While these may safely be ignored in macroscopic fluid-dynamical flows, at mesoscopic length scales they form an essential part of the physics [11]. This applies even in linear problems such as the Brownian motion of a colloidal particle suspended in a simple fluid: if that fluid is simulated using the LBE, no Brownian motion occurs [8]. Fluctuations are also central to nonlinear phenomena such as mode-coupling effects and long-time tails [12]. By the same token, extensions of the LBE to fluid mixtures [9] and amphiphilic solutions [10] cannot address critical phenomena, where fluctuations dominate. In this letter we present a fluctuating LBE (FLBE). This offers a fully consistent dis-cretization of the equations of fluctuating nonlinear hydrodynamics for an isothermal fluid, opening the way to more accurate and efficient simulation of many of the mesoscale physics problems mentioned previously, such as colloid hydrodynamics. Its generalization to multi-component fluids is conceptually straightforward; we pursue this elsewhere [13]. Our work also raises broader issues for numerical statistical mechanics: how best to implement fluctuation-dissipation theorems (FDTs), derived in the continuum with respect for appropriate conservation laws, in a system discretized in space and time [14]. We contend that accuracy and efficiency are best combined …