Towards a full solution of the relativistic Boltzmann equation for quark-gluon matter on GPUs

Abstract

Many studies for nonequilibrium systems, e.g., the pre-equilibration puzzle in heavy-ion collisions, require solving the relativistic Boltzmann equation (BE) with the full collisional kernel to high precision. It is challenging to solve relativistic BE due to its high dimensional phase-space integrals and limited computing resources. We have developed a numerical framework for a full solution of the relativistic Boltzmann equations for the quark-gluon matter using the multiple graphics processing units (GPUs) on distributed clusters. Including all the 2 → 2 scattering processes of 3-flavor quarks and gluons, we compute the time evolution of distribution functions in both coordinate and momentum spaces for the cases of pure gluons, quarks, and the mixture of quarks and gluons. By introducing a symmetrical sampling method on GPUs which ensures the particle number conservation, our framework is able to perform the space-time evolution of quark-gluon system toward thermal equilibrium with high performance. We also observe that the gluons naturally accumulate in the soft region at the early time, which may indicate the gluon condensation.

JefiGPU: Jefimenko's Equations on GPU

“…In the current work, we have provided an implementation of the Jefimenko's equations on GPU based on our previous works [23,24,25]. Our code gives stable and accurate results compared with the theoretical calculations.…”

“…Except for the numerical difficulties, the source term ρ may not be easily available since it is experimentally hard to measure the charge density distributions [22]. To provide a distribution of the charge and current densities, one may need to solve the transport equations (e.g., the Boltzmann equation [23]) which is another challenge so far.…”

“…In this regard, we have applied the Jefimenko's equation on the state-ofart GPU based on our previous work ZMCintegral [24,25] and RBG [23]. Our present code mainly contributes to the field of Heavy-Ion collisions [26,27,28,29] where the electromagnetic fields in the early time significantly influence the evolution of the quark-gluon matter.…”

“…Our present code mainly contributes to the field of Heavy-Ion collisions [26,27,28,29] where the electromagnetic fields in the early time significantly influence the evolution of the quark-gluon matter. Our hope is to combine the Jefimenko's equations to the relativistic Boltzmann equation [23] and eventually answer the question of "early thermalization" [30,31,32]. Due to the high clock rate, high instruction per cycle [33], and multiple cores of the GPUs, more and more numerical algorithms that were hard in the past are now possible [24,34].…”

Two typical collective behaviors of the heavy ions expanding in cold plasma with ambient magnetic field

“…Since the contribution of charge separation may not be negligible, a full PIC (Particle-In-Cell) treatment [36] or the direct solving of the coupled relativistic Boltzmann-Maxwell's equations [37] are required. The hybrid simulation model used in this paper assumes that the electrons always appear in such a way that the ions are neutralized, and meanwhile, we have also assumed that displacement current can be neglected in the formulation.…”