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
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  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.