A Lattice-Boltzmann Method for electrons in metals

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In the 80's a numerical method was developed to solve the Boltzmann equation with the BGK (Bhatnagar, Gross
and Krook) collision term. This method, based on the discretization of the phase space, was very successful in
solving various problems of fuid mechanics, including problems with complex geometry, interfacial phenomena
and multicomponent fluids. Known as the LBM - Lattice Boltzmann Method - it describes the evolution of
a set of statistical distributions of particles defined on a regular space lattice in which each site has a finite
number of velocities directed to neighbouring sites. The advantage over other methods lies in the simplicity of
its dynamics and especially the flexibility for implementation in parallel computing. In recent years, there has
been a great interest in the construction of an LBM able to describe fluids that are not described by Maxwell-
Boltzmann distribution, like semi-classical fluids (described by Fermi-Dirac and Bose-Einstein distribution) and
relativistic fluids (described by Maxwell-Juttner distribution).

In this presentation we derive a general mathematical framework that leads to new LBM models associated
to generic equilibrium distribution functions. This framework is based on our discovery of a new polynomial
basis in Euclidean space which yields the Hermite polynomial basis in the special limit that the weight function
becomes the Gaussian function. The equilibrium function is expanded in this new basis and we discuss the
order that must be considered to obtain the correct conservation laws. We also obtain the discrete lattices
associated to the new polynomial basis. As an application, we construct a LBM capable of describing electrons
in the Fermi surface and show some numerical simulations. This particular LBM is a very promising one since it
could be used to describe the conduction of electrons in arbitrary geometries, something of interest in condensed
matter and also in industrial applications.