Maximum Entropy Principle for Lattice Kinetic Equations

where ci is the D-dimensional vector of the ith link, and i ­ 1, . . . , b. The right-hand side of Eq. (1) is the LBGK collision integral Di , the function N eq i is the local equilibrium, and v $ 0 is a dimensionless parameter. As the state of the lattice is updated long enough, the dynamics of Ni becomes governed by macroscopic equations for a finite system of local averages. Depending on the geometry of the lattice, the averages of interest are rsr, td ­ Pb i Nisr, td, rusr, td ­ Pb i ciNisr, td, and 2rEsr, td ­ Pb i c 2 i Nisr, td. Functions r, u, and E are lattice analogs of the hydrodynamic quantities (local density, average velocity, and energy). If it is possible to cast the lattice macroscopic equations into the form of NavierStokes equations, then hydrodynamics is implemented in a fairly simple fully discrete kinetic picture (1). The central issue of the LBGK method is the local equilibrium. Variational approach to the construction of N eq i amounts to a maximization of a strictly concave function SfNg ­ Pb i­1 FsNid, subject to certain constraints. The minimal set of constraints consists of the hydrodynamic constraints which fix r and u. Taking Fsxd ­ 2x ln x, the formal solution is N eq i ­ expsa 1 b ? cid. However, for lattices of interest, Lagrange multipliers a and b cannot be explicitly expressed in terms of r and u. This applies to all functions S, closely related to the Boltzmann entropy, and it has led to a perturbation technique through a low Mach number expansion (LMN) around the zero-flow equilibrium N eq i su ­ 0d ­ b21r. At present, most of the definitions of N eq i originate from the LMN expansions, and are motivated by a matching to the Navier-