Cercignani, "On the

Boltzmann equation for rigid spheres," Transport Theory and Statistical Physics, vol.

The

Boltzmann equations for transport are obtained in the thermodynamic limit and under some conditions such that the interactions involving more than two particles can be neglected, the collisions are elastic and involve only uncorrelated particles.

The combination of streaming and collision steps in a 9-velocity square lattice modified the lattice

Boltzmann equation as

Under the hypotheses of 0 < s < 1/2, [gamma] [greater than or equal to] 0, [gamma] + 2s < 1, and the modified kinetic factor [PHI]([absolute value of v]) = [(1 + [[absolute value of v].sup.2]).sup.[gamma]/2], they showed the Gevrey smooth property for this type of solutions to the Cauchy problem of the nonlinear homogeneous

Boltzmann equation. By using the original definition of kinetic factor, Zhang and Yin [10] extended the above result in a general framework: 0 < s < 1/2 and -1 < [gamma] + 2s < 1.

A manifestly covariant relativistic

boltzmann equation for the evolution of a system of events.

A stable discretization of the lattice

Boltzmann equation for simulation of incompressible two-phase flows at high density ratio.

Using modern mathematical techniques from the fields of partial differential equations and harmonic analysis - many of which were developed during the last five to 50 years, and thus relatively new to mathematics - the mathematicians proved the global existence of classical solutions and rapid time decay to equilibrium for the

Boltzmann equation with long-range interactions.

The time evolution of the PVDF obeys a differential form of the lattice

Boltzmann equation and its local collidion operator is simplified using the Bhatnagar Gross Krook (BGK) model (7) with the single relaxation time approximation as

The new method relies on a different equation, called the

Boltzmann equation, which is typically employed to predict the behaviors of molecules in gases and liquids.

This paper describes a multilevel algorithm for solving the multi-group, anisotropic scattering

Boltzmann equation formulated with a first-order system least-squares methodology.

[V.sub.j] relationship to the

Boltzmann equation yielded [V.sub.o] values [greater than] 80 mV (see figure legend) confirming the relatively weak [V.sub.j] gating d escribed earlier.