We present a numerical scheme to solve the Wigner equation, based on a lattice discretization of momentum space. The moments of the Wigner function are recovered exactly, up to the desired order given by the number of discrete momenta retained in the discretisation, which also determines the accuracy of the method. The Wigner equation is equipped with an additional collision operator, designed in such a way as to ensure numerical stability without affecting the evolution of the relevant moments of the Wigner function. The lattice Wigner scheme is validated for the case of quantum harmonic and anharmonic potentials, showing good agreement with theoretical results. It is further applied to the study of the transport properties of one and two dimensional open quantum systems with potential barriers. Finally, the computational viability of the scheme for the case of threedimensional open systems is also illustrated.
In this work we introduce a new semi-implicit second order correction scheme to the kinetic Kohn-Sham lattice model. The new approach is validated by performing realistic exchange-correlation energy calculations of atoms and dimers of the first two rows of the periodic table finding good agreement with the expected values. Additionally we simulate the ethane molecule where we recover the bond lengths and compare the results with standard methods. Finally, we discuss the current applicability of pseudopotentials within the lattice kinetic Kohn-Sham approach.PACS numbers: 47.11.Qr, 31.15.X-Central to many theoretical and practical problems in molecular and condensed matter physics, quantum chemistry, and material science is the solution of the Schrödinger equation for systems of electrons in molecules and crystals. It is well known that this problem scales exponentially [1] in the number of electrons and is in general a non trivial task. This has led to the development of a number of approximate solution methods with different degrees of success in different scenarios: density matrix renormalisation group (DMRG) methods[2] for 1D model systems, Hartree Fock [3,4], quantum Monte Carlo[5], exact diagonalization, and Kohn-Sham density functional theory (DFT) [6,7].The DFT formalism has proven to be one of the most versatile methods for ground state electronic calculations in spite of its known shortcomings [8]. This is due to the fact that it is formally exact, and that it allows to calculate several physical quantities such as bond lengths, bonding energies, ionization energies, etc. More specifically, within the Kohn-Sham DFT theory the manyelectron Schrödinger equation is mapped to an auxiliary problem of non-interacting electrons subject to an external potential V xc [ρ] that depends on the electron density ρ. V xc [ρ] can be obtained from a universal, albeit, unknown energy functional of the electronic density, and in practice several highly accurate approximations have been proposed over the years.Recently [9] it was shown that the Kohn-Sham equations that describe a system of electrons, can be recovered from an underlaying kinetic model described by the Boltzmann equation. This connection opens the door to the possibility of including exchange and correlation corrections into electronic calculations based on a kinetic perspective. Furthermore, it also allows the use of efficient Lattice Boltzmann (LB) methods[10] to solve the Kohn-Sham equations. LB methods, are well known numerical tools in the area of computational fluid dynamics. In recent years, however, their use has * sosergio@ethz.ch † mmendoza@ethz.ch ‡ hjherrmann@ethz.ch
We study the rectification of a two-dimensional thermal gas in a channel of asymmetric dissipative walls. For an ensemble of smooth Lennard-Jones particles, our numerical simulations reveal a nonmonotonic dependence of the flux on the thermostat temperature, channel asymmetry, and particle density, with three distinct regimes. Theoretical arguments are developed to shed light on the functional dependence of the flux on the model parameters.
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