Statistical Mechanics of the Fluctuating Lattice Boltzmann Equation

TL;DR

This paper introduces a new fluctuating lattice Boltzmann equation formulation that aligns with statistical mechanics and hydrodynamics, enabling thermodynamically consistent simulations of thermal and multiphase flows.

Contribution

It presents a generalized lattice-gas model with stochastic collision rules satisfying detailed balance, leading to a consistent fluctuating lattice Boltzmann framework.

Findings

Thermal fluctuations are controlled by particle number at lattice sites.

All non-conserved modes must be thermalized to satisfy detailed balance.

The derived fluctuating hydrodynamics are thermodynamically consistent.

Abstract

We propose a new formulation of the fluctuating lattice Boltzmann equation that is consistent with both equilibrium statististical mechanics and fluctuating hydrodynamics. The formalism is based on a generalized lattice-gas model, with each velocity direction occupied by many particles. We show that the most probable state of this model corresponds to the usual equilibrium distribution of the lattice Boltzmann equation. Thermal fluctuations about this equilibrium are controlled by the mean number of particles at a lattice site. Stochastic collision rules are described by a Monte Carlo process satisfying detailed balance. This allows for a straightforward derivation of discrete Langevin equations for the fluctuating modes. It is shown that all non-conserved modes should be thermalized, as first pointed out by Adhikari et al.; any other choice violates the condition of detailed balance. A Chapman-Enskog analysis is used to derive the equations of fluctuating hydrodynamics on large length and time scales; the level of fluctuations is shown to be thermodynamically consistent with the equation of state of an isothermal, ideal gas. We believe this formalism will be useful in developing new algorithms for thermal and multiphase flows.