Relativistic Lattice Boltzmann Methods: Theory and Applications

TL;DR

This paper develops a comprehensive relativistic Lattice Boltzmann method (RLBM) for dissipative hydrodynamics, bridging ultra-relativistic and non-relativistic regimes, with detailed derivations, calibration, and validation against known results.

Contribution

It introduces a unified, dimension-independent RLBM scheme with systematic derivation of transport coefficients and validation in relativistic regimes, advancing simulation accuracy and theoretical understanding.

Findings

Transport coefficients as a function of relaxation time derived for multiple dimensions

Validation of RLBM against analytic solutions for shock propagation in relativistic fluids

Calibration of simulation parameters with quantitative agreement to theoretical models

Abstract

We present a systematic account of recent developments of the relativistic Lattice Boltzmann method (RLBM) for dissipative hydrodynamics. We describe in full detail a unified, compact and dimension-independent procedure to design relativistic LB schemes capable of bridging the gap between the ultra-relativistic regime, $k_{\rm B} T \gg mc^2$, and the non-relativistic one, $k_{\rm B} T \ll mc^2$. We further develop a systematic derivation of the transport coefficients as a function of the kinetic relaxation time in $d=1,2,3$ spatial dimensions. The latter step allows to establish a quantitative bridge between the parameters of the kinetic model and the macroscopic transport coefficients. This leads to accurate calibrations of simulation parameters and is also relevant at the theoretical level, as it provides neat numerical evidence of the correctness of the Chapman-Enskog procedure. We present an extended set of validation tests, in which simulation results based on the RLBMs are compared with existing analytic or semi-analytic results in the mildly-relativistic ($k_{\rm B} T \sim mc^2$) regime for the case of shock propagations in quark-gluon plasmas and laminar electronic flows in ultra-clean graphene samples. It is hoped and expected that the material collected in this paper may allow the interested readers to reproduce the present results and generate new applications of the RLBM scheme.