Dissipativity and the spectral property of the collision operator

It has been shown by the Brussels School that there is a physical representation in which the evolution of the density matrix reveals the dissipative structure. In this article, we present a two-dimensional model of the collision operator with a singularity at the origin which characterizes a long-range force. The dissipativity and the approach to equilibrium in the physical representation are studied and it is shown that the second law may be established in spite of the failure of Boltzmann's kinetic equation.