ShARK: A Stochastic Transport Framework for the Relativistic Relaxation Time Approximation Boltzmann Equation
Abstract
We show that the Anderson-Witting relaxation-time approximation (relativistic Bhatnagar-Gross-Krook (BGK) equation) emerges as the N→ ∞ limit of a stochastic N-body particle gas, in which collisions perform a full microcanonical momentum redraw that enforces local energy-momentum conservation at every event. Using the RAMBO algorithm to sample the Lorentz-invariant phase space, we derive the finite-N single-particle momentum spectrum and show that it converges to the Jüttner-Boltzmann equilibrium distribution as the local particle number grows. We couple this relaxation kernel to an advection-relaxation splitting scheme to construct ShARK (Stochastic Advection Relaxation Kinetics), a 3D Monte Carlo relativistic solver for the relaxation time approximation Boltzmann equation. The framework is conceptually related to lattice Boltzmann methods, but formulated in continuous momentum space to avoid the accuracy loss caused by finite-order momentum discretizations far from local equilibrium. We validate the numerical results against analytical solutions for two highly symmetric conformal expansions, the Bjorken and Gubser flows. The framework provides a first-principles route to far-from-equilibrium relativistic transport, with applications ranging from heavy-ion collisions to expanding astrophysical plasmas.
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BibTeX
@misc{silva2026shark,
title = {{ShARK: A Stochastic Transport Framework for the Relativistic Relaxation Time Approximation Boltzmann Equation}},
author = {Tiago Nunes da Silva and Jadna L. Barauna and Giorgio Torrieri},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35148},
eprint = {2609.35148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35148},
}
RIS
TY - UNPB
TI - ShARK: A Stochastic Transport Framework for the Relativistic Relaxation Time Approximation Boltzmann Equation
AU - Tiago Nunes da Silva
AU - Jadna L. Barauna
AU - Giorgio Torrieri
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35148
UR - https://arxiv.org/abs/2609.35148
AB - We show that the Anderson-Witting relaxation-time approximation (relativistic Bhatnagar-Gross-Krook (BGK) equation) emerges as the N→ ∞ limit of a stochastic N-body particle gas, in which collisions perform a full microcanonical momentum redraw that enforces local energy-momentum conservation at every event. Using the RAMBO algorithm to sample the Lorentz-invariant phase space, we derive the finite-N single-particle momentum spectrum and show that it converges to the Jüttner-Boltzmann equilibrium distribution as the local particle number grows. We couple this relaxation kernel to an advection-relaxation splitting scheme to construct ShARK (Stochastic Advection Relaxation Kinetics), a 3D Monte Carlo relativistic solver for the relaxation time approximation Boltzmann equation. The framework is conceptually related to lattice Boltzmann methods, but formulated in continuous momentum space to avoid the accuracy loss caused by finite-order momentum discretizations far from local equilibrium. We validate the numerical results against analytical solutions for two highly symmetric conformal expansions, the Bjorken and Gubser flows. The framework provides a first-principles route to far-from-equilibrium relativistic transport, with applications ranging from heavy-ion collisions to expanding astrophysical plasmas.
ER -