# Deterministic Computation of the Low Probability Tail of the Velocity Distribution Due to Particle Collisions in Spatially Homogeneous Plasmas

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## Abstract

We accurately and efficiently compute the low-probability and high-energy tail of the velocity probability density function (velocity pdf) of the homogeneous Boltzmann equation for applications to non-equilibrium plasmas. Gas-phase chemistry and surface kinetics are largely driven by collision processes between high energy electrons in the plasma and molecules in the gas phase. Such reactions in the gas phase are determined by the overlap between the electron velocity pdf and the electron-impact cross sections of the various species. The problem of accurately calculating the electron velocity pdf, especially in the low probability and high energy tail of the velocity pdf, is a particularly challenging one. We focus on computing the collision operator in the Boltzmann equation by combining a cylindrically symmetric assumption for the velocity pdf with a Fourier transform method. Under these assumptions, the computational cost of the method for computing the collision operator is reduced to O(N5) where N is the number of the discrete grid points in each velocity dimension. Furthermore, we prove that if the initial velocity pdf is cylindrically symmetric, then the solution to the homogeneous Boltzmann equation is also cylindrically symmetric.To further compute the low probability and high energy tail of the velocity pdf accurately, we introduce an analytical approximation to the collision operator. We derive an upper bound for the relative error of this analytical approximation under the assumption that the velocity pdf is bounded above by a Maxwell-Boltzmann distribution. We also establish a version of this estimate in the case that the input velocity pdf does not decay exponentially over the entire velocity space, but rather has a floor due to the numerical error in the computation of the Fourier transform. Finally, we compute the solutions of the homogeneous Boltzmann equation in the case of three different cylindrically symmetric initial velocity pdfs. For each of these initial conditions, we perform a simulation study with different sets of discretization parameters to investigate the trade off between the accuracy and efficiency of the computation of the collision operator. Then we compute the solution of the homogeneous Boltzmann equation and study the convergence of the solution to a Maxwellian velocity pdf over time.