Exact solution for the spatially inhomogeneous nonlinear Kac model of the Boltzmann equation

Abstract

We study the spatially inhomogeneous Kac model of the nonlinear Boltzmann equation in 1+1+1 dimensions (velocity v, time t, position x). We obtain an exact solution which is the product of a Maxwellian with a time-dependent width by a second-order polynomial in the velocity variable. The solution satisfies a specular reflection condition at the boundary x=x0. The position x0−x appears linearly and only in the odd part of the velocity distribution, the range of x0−x being arbitrarily large but finite in order to maintain the positivity of the distribution. The local density is spatially homogeneous. Further, a particular linear relation between the moments of the cross section must be satisfied. The most general Maxwellian width has two relaxation times and their ratio is a function of the moments of the cross sections. Depending on whether this ratio is larger than or smaller than 1 we find contraction or expansion. The solution relaxes towards a Maxwellian equilibrium solution. Studying the Tjon overpopulation effect of high velocity particles, we find that it depends weakly on the initial condition, and strongly on both the microscopic model of cross section and on the ratio of the two relaxations times. We give a simple criterion (linked to the distinction between contraction and expansion) for the existence of the effect. Theoretically and numerically we test its validity and its failure.