  The Boltzmann equation is a powerful theoretical tool for modeling the
collective dynamics of quantum many-body systems subject to external
perturbations. Analysis of the equation gives access to linear response
properties including collective modes and transport coefficients, but often
proves intractable due to computational costs associated with multidimensional
integrals describing collision processes. Here, we present a method to resolve
this bottleneck, enabling the study of a broad class of many-body systems that
appear in fundamental science contexts and technological applications.
Specifically, we demonstrate that a Gaussian mixture model can accurately
represent equilibrium distribution functions, thereby allowing efficient
evaluation of collision integrals. Inspired by cold atom experiments, we apply
this method to investigate the collective behavior of a quantum Bose-Fermi
mixture of cold atoms in a cigar-shaped trap, a system that is particularly
challenging to analyze. We focus on monopole and quadrupole collective modes
above the Bose-Einstein transition temperature, and find a rich phenomenology
that spans interference effects between bosonic and fermionic collective modes,
dampening of these modes, and the emergence of hydrodynamics in various
parameter regimes. These effects are readily verifiable experimentally.
