Relativistic Maxwellian mixture model

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 estimation of the relativistic Maxwellian distribution. We then introduce a relativistic Maxwellian mixture model (R-MMM), which is a weighted sum of relativistic Maxwellian distributions. We develop an expectation-maximization algorithm for estimating the parameters of R-MMM, namely, the mixing proportion, the bulk velocity, and the temperature of each component. 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 with the initial temperature of PIC simulation.