[P-EM16] Space Plasma Physics: Theory and Simulation

[PEM16-P08] Relativistic Maxwellian mixture model

*Genta Ueno¹, Seiji Zenitani² (1.The Institute of Statistical Mathematics, 2.Kobe University)

Keywords:Particle-in-cell method, Special relativity, Particle distribution functions, Statistical models, EM algorithm, Maxwell-Boltzmann distribution

For analyzing distribution functions of relativistic plasma, we propose a mixture model composed of relativistic Maxwellian distributions. We first summarize the basic properties of the relativistic Maxwellian distribution, including the derivation of the normalization constant when there is a bulk velocity. We also examine the maximum-likelihood (ML) estimation of the relativistic Maxwellian distribution, and derive simple equations that are satisfied by the maximum likelihood estimators (MLEs) of bulk velocity and temperature. We then introduce a relativistic Maxwellian mixture model (R-MMM), which is a weighted sum of relativistic Maxwellian distributions. We develop an EM (expectation--maximization) algorithm for estimating the parameters of R-MMM, namely the mixing proportion, the bulk velocity, and the temperature of each component. In particular, we derive M-step equations whose solution is guaranteed to maximize the conditional expectation of the complete-data log-likelihood. To initialize the parameters for the EM algorithm, we divide data into groups whose number corresponds to that of the components, and in each group we use ML estimates or estimates by the method of moments. We apply a two-component R-MMM to a distribution function by a particle-in-cell (PIC) simulation of relativistic pair plasma, and separate the simulated distribution function into two components. We find that one component has a large bulk velocity while the other is almost stagnant, and that the two components have almost the same temperatures, which is also consistent to the initial temperature of the PIC simulation. Based on the parameters, we can infer large-scale plasma environments such shocks and discontinuities.

Reference

Genta Ueno and Seiji Zenitani, "Relativistic Maxwellian mixture model", Physics of Plasmas 28, 122106 (2021) https://doi.org/10.1063/5.0059126