Formal and substantial analogies between the Boltzmann equation of classical statistical physics and the Dirac equation for relativistic quantum mechanics are discussed, with special emphasis on the Majorana representation of the Dirac equation, in which all matrices are real. It is shown that this property is instrumental in turning the Boltzmann-Dirac analogy into an efficient computational m...

A modified lattice Boltzmann model based on the two-dimensional, nine-velocity lattice-Bhatnagar-Gross-Krook fluid is presented for axisymmetric flows. A spatially and temporally varying source term is incorporated into the evolution equation for the momentum distribution function on a two-dimensional Cartesian lattice. The precise form of the source term is derived through a Chapman-Enskog ana...

We consider the simultaneous diffusion and homogenization limit of the linear Boltzmann equation in a periodic medium in the regime where the mean free path is much smaller than the lattice constant. The resulting equation is a diffusion equation, with an averaged diffusion matrix that is formally obtained by first performing the diffusion limit and then the homogenization one. The rigorous pro...

A lattice Boltzmann model for binary mixtures is formally derived from a two-fluid kinetic model for Maxwell molecules by discretizing the Boltzmann equation. In this model, collisions among the same and different species are treated separately, as non-linear BGK terms, enabling the independent management of the species viscosities and thermal conductivities from the binary diffusion coefficien...

The lattice Boltzmann equation ~LBE! is directly derived from the Boltzmann equation by discretization in both time and phase space. A procedure to systematically derive discrete velocity models is presented. A LBE algorithm with arbitrary mesh grids is proposed and a numerical simulation of the backward-facing step is conducted. The numerical result agrees well with experimental and previous n...

A method for the discretization of the Boltzmann equation in velocity space via a Galerkin procedure with Hermite polynomials as trial and test functions is proposed. This procedure results in a set of partial differential equations, which is an alternative to the lattice-Boltzmann equations. These PDEs are discretized using an explicit finite difference scheme and a numerical example shows the...

We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to...