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We propose a new spectral Lagrangian based deterministic solver for thenon-linear Boltzmann Transport Equation for Variable Hard Potential (VHP)collision kernels with conservative or non-conservative binary interactions.The method is based on symmetries of the Fourier transform of the collisionintegral, where the complexity in its computing is reduced to a separateintegral over the unit sphere $S^2$. In addition, the conservation of momentsis enforced by Lagrangian constraints. The resulting scheme, implemented infree space is very versatile and adjusts in a very simple manner, to severalcases that involve energy dissipation due to local micro-reversibility(inelastic interactions) or elastic model of slowing down process. Oursimulations are benchmarked with the available exact self-similar solutions,exact moment equations and analytical estimates for homogeneous Boltzmannequation for both elastic and inelastic VHP interactions. Benchmarking of thesimulations involves the selection of a time self-similar rescaling of thenumerical distribution function which is performed using the continuousspectrum of the equation for Maxwell molecules as studied first in\cite{asympGranular} and generalized to a wide range of related models in\cite{BCG}. The method also produces accurate results in the case of inelasticdiffusive Boltzmann equations for hard-spheres (inelastic collisions underthermal bath), where overpopulated non-Gaussian exponential tails have beenconjectured in computations by stochastic methods in \cite{vanNoi-Ernst,ernst-brito, MSS, GRW04} and rigourously proven in \cite{diffGran} and\cite{highEnergyTails}.

Spectral - Lagrangian methods for Collisional Models of Non - Equilibrium Statistical States