Integer lattice gas with Monte Carlo collision operator recovers the lattice Boltzmann method with Poisson-distributed fluctuations

Asymptotic freedom in the lattice Boltzmann theory

Asymptotic freedom is a feature of quantum chromodynamics that guarantees its well posedness. We derive an analog of asymptotic freedom enabling unconditional linear stability of lattice Boltzmann simulation of hydrodynamics. We further demonstrate the validity of the derived conditions via the special case of the equilibrium based on entropy maximization, which is shown to be uniquely renormalizable.

Overrelaxation in a diffusive integer lattice gas

One of the most striking drawbacks of standard lattice gas methods over lattice Boltzmann methods is a much more limited range of transport parameters that can be achieved. It is common for lattice Boltzmann methods to use over-relaxation to achieve arbitrarily small transport parameters in the hydrodynamic equations. Here, we show that it is possible to implement over-relaxation for integer lattice gases. For simplicity, we focus here on lattice gases for the diffusion equation. We demonstrate that adding a flipping operation to lattice gases results in a multirelaxation time lattice Boltzmann scheme with over-relaxation in the Boltzmann limit.

Integer lattice gas with a sampling collision operator for the fluctuating diffusion equation

We developed an integer lattice gas method for the fluctuating diffusion equation. Such a method is unconditionally stable and able to recover the Poisson distribution for the microscopic densities. A key advance for integer lattice gases introduced in this paper is a sampling collision operator that replaces particle collisions with sampling from an equilibrium distribution. This can increase the efficiency of our integer lattice gas by several orders of magnitude.

Molecular dynamics lattice gas equilibrium distribution function for Lennard-Jones particles

The molecular dynamics lattice gas (MDLG) method maps a molecular dynamics (MD) simulation onto a lattice gas using a coarse-graining procedure. This is a novel fundamental approach to derive the lattice Boltzmann method (LBM) by taking a Boltzmann average over the MDLG. A key property of the LBM is the equilibrium distribution function, which was originally derived by assuming that the particle displacements in the MD simulation are Boltzmann distributed. However, we recently discovered that a single Gaussian distribution function is not sufficient to describe the particle displacements in a broad transition regime between free particles and particles undergoing many collisions in one time step. In a recent publication, we proposed a Poisson weighted sum of Gaussians which shows better agreement with the MD data. We derive a lattice Boltzmann equilibrium distribution function from the Poisson weighted sum of Gaussians model and compare it to a measured equilibrium distribution function from MD data and to an analytical approximation of the equilibrium distribution function from a single Gaussian probability distribution function. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Nonuniqueness of fluctuating momentum in coarse-grained systems

Coarse-grained descriptions of microscopic systems often require a mesoscopic definition of momentum. The question arises as to the uniqueness of such a momentum definition at a particular coarse-graining scale. We show here that particularly the fluctuating properties of common definitions of momentum in coarse-grained methods like lattice gas and lattice Boltzmann do not agree with a fundamental definition of momentum. In the case of lattice gases, the definition of momentum will even disagree in the limit of large wavelength. For short times we derive analytical representations for the distribution of different momentum measures and thereby give a full account of these differences.

Large Fluctuations in Nonideal Coarse-Grained Systems

Using the recently introduced molecular dynamics lattice gas approach, we test fluctuations of coarse-grained quantities. We show that as soon as the system can no longer be considered an ideal gas fluctuations fail to diminish upon coarse graining as is usually expected. These results suggest that current approaches to simulating fluctuating hydrodynamics may have to be augmented to achieve quantitative results for systems with a nonideal equation of state. The molecular dynamics lattice gas method gives a guidance to the exact nature of the fluctuation in such systems.