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The Boltzmann equation is a powerful theoretical tool for modeling thecollective dynamics of quantum many-body systems subject to externalperturbations. Analysis of the equation gives access to linear responseproperties including collective modes and transport coefficients, but oftenproves intractable due to computational costs associated with multidimensionalintegrals describing collision processes. Here, we present a method to resolvethis bottleneck, enabling the study of a broad class of many-body systems thatappear in fundamental science contexts and technological applications.Specifically, we demonstrate that a Gaussian mixture model can accuratelyrepresent equilibrium distribution functions, thereby allowing efficientevaluation of collision integrals. Inspired by cold atom experiments, we applythis method to investigate the collective behavior of a quantum Bose-Fermimixture of cold atoms in a cigar-shaped trap, a system that is particularlychallenging to analyze. We focus on monopole and quadrupole collective modesabove the Bose-Einstein transition temperature, and find a rich phenomenologythat spans interference effects between bosonic and fermionic collective modes,dampening of these modes, and the emergence of hydrodynamics in variousparameter regimes. These effects are readily verifiable experimentally.

Accelerating analysis of Boltzmann equations using Gaussian mixture models: Application to quantum Bose-Fermi mixtures