A Lattice Boltzmann Method for Electrons in Metals
R.C.V. Coelho1; M.M. Doria1; A.S. Ilha2
1 UFRJ; 2 Inmetro
Resumo
In the 80s 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 fluid 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 neighboring 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 LBM's 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, for instance, by Maxwell-Juttner distribution). In this work we derive a general mathematical framework that leads to new LBM's associated to generic equilibrium distribution functions and we apply this model to electrons in metals. 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, as the shock tube test and the Poiseuille flow. 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 ind
Palavras-chave: Fluidodynamics, Plasma and Turbulence Modeling, Numerical Simulation and Optimization ; Nonlinear Dynamics in Thermal and Fluid Sciences