Parallelized Transport and Corrections to Equilibrium Phase Space Distributions
Abstract: When modeling a heavy ion collision, hydrodynamics is not applicable in the regime where deviations from local thermal equilibrium are not guaranteed to be small. Further, the fact that experiments detect particles warrants a switch from a ﬂuid dynamical picture to a particle picture. One approach to modeling particles with non-equilibrium dynamics utilizes the relativistic Boltzmann Transport Equation (BTE). The BTE describes the evolution of particle phase space distributions via collision terms for various scattering processes. While the elastic 2 → 2 collision terms are useful to study the approach to thermal equilibrium, the radiative 2 ↔ 3 collision terms are necessary to study the approach to chemical equilibrium in systems with changing particle number.
The existing MPC/Grid code solves the BTE by discretizing space into cells on a grid, and simulating propagation and interactions of point particles within each cell. Within a single time step in the evolution of the system, every particle in a given cell is equally likely to interact with every other particle in the cell. Within each cell in the grid, the code loops through all possible pairs and triplets and generates a random number for each. The random number is compared to a collision probability in order for a decision to be made as to whether or not a collision occurs. The 2 → 3 collision probability is determined via a 2D integral for each pair. This integral takes up a signiﬁcant amount of calculation time, because it needs to be carried out serially for every single pair in each cell in the system.
The grid nature of the algorithm requires high enough particles per cell and small enough time steps in order to obtain realistic results. Increasing the number of test particles is computationally expensive, especially with the current algorithm for 2 → 3 collision checks. To improve the feasibility of simulations with large numbers of test particles, we created a parallelized version of this code using Message Passing Interface (MPI). This version divides the spatial grid into “subsystems” that are allocated to diﬀerent cores.
While the MPI parallelization did result in a speed up, the integrals for collision checks are still carried out in serial loops within each subsystem. The 2D integral that is carried out for every single pair presents an opportunity for parallelization using GPUs. We want to investigate the computational eﬃciency of such a parallelization. A speed up via GPUs would allow for complex systems to be studied within reasonable amounts of time.
The ﬁrst application of such a parallelized code would be to expand an existing study. We have already studied the evolution of deviations from equilibrium in phase space distributions in a 1D expanding system. We compared this evolution to that predicted by various models based purely on hydrodynamic ﬁelds. This study was limited to 2 → 2 interactions only. Around 300 million particles were required in order for the results to be trusted. Including 2 ↔ 3 interactions and transverse expansion would describe the behavior of a more realistic system without particle number conservation. However, these eﬀects are expected to require even higher particle statistics, thus calling for further speed ups in MPC/Grid and motivating us to attempt a speed up via GPUs.