Probabilistic ballistic annihilation with continuous velocity distributions

We investigate the problem of ballistically controlled reactions where particles either annihilate upon collision with probability $p$, or undergo an elastic shock with probability $1-p$. Restricting to homogeneous systems, we provide in the scaling regime that emerges in the long time limit, analytical expressions for the exponents describing the time decay of the density and the root-mean-square velocity, as continuous functions of the probability $p$ and of a parameter related to the dissipation of energy. We work at the level of molecular chaos (non-linear Boltzmann equation), and using a systematic Sonine polynomials expansion of the velocity distribution, we obtain in arbitrary dimension the first non-Gaussian correction and the corresponding expressions for the decay exponents. We implement Monte-Carlo simulations in two dimensions, that are in excellent agreement with our analytical predictions. For $p<1$, numerical simulations lead to conjecture that unlike for pure annihilation ($p=1$), the velocity distribution becomes universal, i.e. does not depend on the initial conditions.