as D3Q19) are provided.
1.3 Basics of Semiclassical LB method
Although Boltzmann Equation has been successfully applied to many eld like dilute gas dynamics, multi-scale simulations, however, in recent years, micro and nano technology has been emerged quickly, and the transport phenomena in semiconductors at low temperatures is very important. There have been a successful theory in statistical mechanics which can predict the transport coe cients like shear viscosity and thermal conductivity of ideal quantum
uid such like electrons in the metal. The questions arise of whether quantum systems like that can be described similar to the one developed for the clas-sical counterpart. When solving these kinds of problems, clasclas-sical Boltzmann Equation is not enough and it require quantum mechanical treatments. In quantum mechanics identical particles are absolutely indistinguishable from one another and N-particle system can be described by a wave function with permutation symmetry. In nature, it is found that particles with antisymmet-ric wave functions are called fermions which obey FD statistics and particles with symmetric wave functions are called bosons which obey BE statistics.
The statistical properties of fermion and boson systems are profoundly dier-ent at low temperature. However, in the classical limit, both quantum dis-tributions reduce to the Maxwell-Boltzmann(MB) distribution. Boltzmann equation describes the dynamic behavior of ideal gas by a single-particle dis-tribution function.
In general, there are three strategies to take for statistically treating a quantum system [34]. One is to use a kinetic equation governing the density matrix, another one is to use a kinetic equation with the Wigner distribution
function, the last one, which is also the method used in this thesis, is to assume a semiclassical kinetic equation such as the UUB [21][22] kinetic equation as a generalization of the classical one. In UUB equation, the collision term of the Boltzmann equation is rewritten with the quantum particle scattering form.
The equilibrium distribution to semiclassical Boltzmann equation is the BE or FD distributions. The particles obey BE distribution are called bosons and FD distribution for fermions, no third category has yet been found.
Considering of the recently successful developing on LB method and well deriving semiclassical Boltzmann equation, it is natural to extend the conven-tional LB method to the semiclassical LB method for dealing with dierent quantum statistics. Although UUB equation introduced above has considered the quantum statistics and would reveal quantum eects in speci c problems.
However, dealing with UUB equation is still a challenge. In this dissertation, we proposed a new semiclassical LB method which extends well known LB method to solve semiclassical Boltzmann equation with BGK approximation.
In [28], a new semiclassical LB-BGK method had been developed, and this method would describe quantum systems in dierent approaches. The follow-ing works about rare ed channel ow [35] and axisymmetric ow of quantum statistics [36] present dierent applications of this new method. The idea of extending the conventional LB method to semiclassical LB method is to adopt the Grad's moment method to nd solutions to semiclassical Boltz-mann equation by expanding f(x, ζ, t) in terms of Hermite polynomials. And in simulations, the N-th nite order truncated distribution function fN was considered. the details will be shown in chapter 3.
It is worth mentioning here the dierences between the present method and quantum LB method which is proposed by Succi and Benzi [37][38]. In the quantum LB method, Dirac equation which is the most general equation
de-1.3. Basics of Semiclassical LB method 7
scribing single particle motion in compliance with quantum theory and special relativity is solved by LB method. In [37], the procedure builds on a formal analogy between the Dirac equation and a special discrete kinetic equation known as LB method, it was then shown that the non-relativistic Schr¨odinger equation ensues from the Dirac equation under an adiabatic assumption that is formally similar to the one which takes the Boltzmann equation to the Navier-Stokes equations in kinetic theory. In [38], it was further shown that by a proper resort to operator splitting methods, the Dirac equation can be in-tegrated as a sequence of three one dimensional LB equation evolving complex valued distribution function. In these works, LB method with complex distri-bution function is treated as a numerical tool for solving a complex equation.
By using multiscale technique and the Chapman-Enskog expansion on com-plex variables, the comcom-plex partial dierential equations could be recovered.
Recent years, this procedure is applied for solving many dierent equations, for examples, the one-dimensional nonlinear Dirac equation[39], the nonlin-ear convection-diusion equations in [40] and the complex Ginzburg-Landau equation in [41]. It should be also noticed that several quantum lattice gas cellular automata methods [37][42][43][44] have been recently presented which applying and extending the concept of classical lattice gas cellular automata models to treat the time evolution of wave functions for spinning particles and the Schrödiger equation or the Dirac equation directly. For a more detailed review, see [45]. However, present semiclassical LB method associated with the works presented in [25][26][27] are based on the semiclassical kinetic de-scription. i.e., the particle motion (velocity or momentum) and position are treated in classical mechanics manner while the particles can be of quantum statistics. The procedure and physical meanings are much dierent.