Lattice Boltzmann model for resistive relativistic magnetohydrodynamics

In this paper, we develop a lattice Boltzmann model for relativistic magnetohydrodynamics (MHD). Even though the model is derived for resistive MHD, it is shown that it is numerically robust even in the high conductivity (ideal MHD) limit. In order to validate the numerical method, test simulations are carried out for both ideal and resistive limits, namely the propagation of Alfvén waves in the ideal MHD and the evolution of current sheets in the resistive regime, where very good agreement is observed comparing to the analytical results. Additionally, two-dimensional magnetic reconnection driven by Kelvin-Helmholtz instability is studied and the effects of different parameters on the reconnection rate are investigated. It is shown that the density ratio has a negligible effect on the magnetic reconnection rate, while an increase in shear velocity decreases the reconnection rate. Additionally, it is found that the reconnection rate is proportional to σ-1/2, σ being the conductivity, which is in agreement with the scaling law of the Sweet-Parker model. Finally, the numerical model is used to study the magnetic reconnection in a stellar flare. Three-dimensional simulation suggests that the reconnection between the background and flux rope magnetic lines in a stellar flare can take place as a result of a shear velocity in the photosphere.

Gaussian quadrature and lattice discretization of the Fermi-Dirac distribution for graphene

We construct a lattice kinetic scheme to study electronic flow in graphene. For this purpose, we first derive a basis of orthogonal polynomials, using as the weight function the ultrarelativistic Fermi-Dirac distribution at rest. Later, we use these polynomials to expand the respective distribution in a moving frame, for both cases, undoped and doped graphene. In order to discretize the Boltzmann equation and make feasible the numerical implementation, we reduce the number of discrete points in momentum space to 18 by applying a Gaussian quadrature, finding that the family of representative wave (2+1)-vectors, which satisfies the quadrature, reconstructs a honeycomb lattice. The procedure and discrete model are validated by solving the Riemann problem, finding excellent agreement with other numerical models. In addition, we have extended the Riemann problem to the case of different dopings, finding that by increasing the chemical potential the electronic fluid behaves as if it increases its effective viscosity.

Hydrodynamic model for conductivity in graphene

Based on the recently developed picture of an electronic ideal relativistic fluid at the Dirac point, we present an analytical model for the conductivity in graphene that is able to describe the linear dependence on the carrier density and the existence of a minimum conductivity. The model treats impurities as submerged rigid obstacles, forming a disordered medium through which graphene electrons flow, in close analogy with classical fluid dynamics. To describe the minimum conductivity, we take into account the additional carrier density induced by the impurities in the sample. The model, which predicts the conductivity as a function of the impurity fraction of the sample, is supported by extensive simulations for different values of ε, the dimensionless strength of the electric field, and provides excellent agreement with experimental data.