Luo, "Theory of the lattice Boltzmann method: From the

Boltzmann equation to the lattice

Boltzmann equation," Physical Review, vol.

One of the advantages of such an approach to the derivation of the

Boltzmann equation is an opportunity to construct the kinetic equation, involving correlations at initial time, in particular, that can characterize the condensed states.

Mauri, "Phonon-limited resistivity of graphene by first-principles calculations: Electron-phonon interactions, strain-induced gauge field, and

Boltzmann equation," Physical Review B--Condensed Matter and Materials Physics, vol.

The Lattice

Boltzmann equation (LBE) can be determined as

Research on the Gevrey regularity of the

Boltzmann equation can be traced back to the work of Ukai [6], who constructed a unique local solution in Gevrey space for both spatially homogeneous and inhomogeneous noncutoff

Boltzmann equations.

Romano, "Maximum entropy moment system of the semiconductor

Boltzmann equation using Kane's dispersion relation," Continuum Mechanics and Thermodynamics, vol.

(1981; 1989) introduced a "manifestly covariant" relativistic

Boltzmann equation that shows a different mean energy-temperature relation in the ultra-relativistic regime of infinite temperature.

In addition, we also utilize MI schemes to handle nonlinear problems on analysis of numerical results for semiconductor equations [21-23] and the Poisson

Boltzmann equation [24].

of Paris-Sud, France) and Querlioz (Stanford U., US) suggest replacing the

Boltzmann equation with the Wigner transport equation, which is similar, except in the influence of the potential whose rapid space variations generate quantum effects.

Unlike traditional CFD methods, LBM solves the discrete

Boltzmann equation to simulate the flow of a fluid by using collision models, such as Bhatnagar-Gross-Krook (BGK).

usually characteristic of the

Boltzmann equation (where it describes collisions between particles), integrates the contribution from nonlinear interactions into changes in the wave fields, [T.sub.1234] is the interaction coefficient, [DELTA]k = 0, [DELTA][omega] = 0 are the resonance conditions for the wave vectors and angular frequencies, respectively, [S.sub.diss] = - [[gamma].sub.D] (k) N (k) reflects dissipation of wave energy due to different reasons and [S.sub.In] = [beta](k)N(k) expresses the wind input to the wave systems.

The solutions of the

Boltzmann equation problem, beyond current computational capabilities, describe the location of gas molecules probabilistically and predict the likelihood that a molecule will reside at any particular location and have a particular momentum at any given time in the future.