Boltzmann Equation Solver Adapted to Emergent Chemical Non-equilibrium

We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow for emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature $T(t)$ and phase space occupation factor $\Upsilon(t)$. In this first paper we address (effectively) massless fermions and derive dynamical equations for $T(t)$ and $\Upsilon(t)$ such that the zeroth order term of the basis alone captures the particle number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to easily represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component ($e^\pm$-annihilation).