Discrete Quantum Boltzmann Equation

In this work, we consider a Boltzmann equation for anyons. In particular, we study a general discrete velocity model of the equation, where the velocity variable is assumed to only take values from a given finite - such that the (finite) number of velocities is arbitrary - set of velocities. Included, as two limiting cases, is the discrete quantum Boltzmann equation (Nordheim-Boltzmann /Uehling-Uhlenbeck equation) for bosons and fermions. Mass, momentum, and energy are assumed to be conserved during collisions, and considering suitable discrete velocity models, they will also be the only collision invariants. The equilibrium distributions will be given by a transcendental equation, and only in some few cases - including the two limiting cases where they are Planckians - they will be explicitly expressed. However, there is an H-theorem, and therefore one can prove that for the spatially homogeneous equation, as time tends to infinity, as well as, for the steady equation in a half-space with slab-symmetry, as the space variable tends to infinity, the distribution function converges to an equilibrium distribution. Linearizing around an equilibrium distribution in a suitable way, we find that the obtained linearized operator has similar properties as the corresponding linearized operator for the discrete Boltzmann equation: e.g. it is symmetric and positive semi-definite. Hence, previously obtained results for the spatially homogeneous Cauchy problem and the steady half-space problem in a slab symmetry for the discrete Boltzmann equation, can be applied also in the considered quantum case.