A Lattice Gas Cellular Automaton with Tomographic Dynamics

Abstract

We study the tomographic effect in lattice gas cellular automata, which are simple discrete models for hydrodynamics. The tomographic effect is predicted in two-dimensional Fermi liquids whenever head-on scattering dominates, leading to inefficient angular relaxation due to Pauli blocking. As a result, odd-parity deformations of the Fermi surface are anomalously long-lived compared to even-parity ones. We find an analogous effect in two-dimensional triangular lattice gas automata, which exhibit a single discrete odd-parity mode that we call the hexapolar mode. We analyze the relaxation times of the linearized lattice gas dynamics, and find that at low densities, head-on collisions dominate and the odd-parity mode is long-lived, resulting in a two-step relaxation of the system. At higher densities, three-body interactions set the dominant decay channel and the odd-parity mode becomes short-lived. We then analyze a reduced model of the lattice gas with a long-lived hexapolar mode, and derive the equilibrium distribution via entropy maximization. Using a small perturbation approximation and a multi-scale expansion, we derive a set of macroscopic equations that incorporate the hexapolar mode. We show that the odd-parity mode couples anisotropically to the momentum, leading to direction-dependent dynamics. We analyze the linear hydrodynamic modes and solve for the velocity profile in a channel (Poiseuille flow). We then compare our findings with the expected behavior for a two-dimensional Fermi liquid that exhibits the tomographic effect. Using the linearized kinetic equation for a two-dimensional homogeneous electron gas, we derive a set of linearized macroscopic equations. We conclude by discussing the limitations of the model due to its discrete nature and lack of full rotational symmetry.