Many–Body Aspects of Approach to Equilibrium

In 1956 Mark Kac [5] invented a microscopic, linear model, from which the nonlinear Boltzmann equation describing the evolution of the velocity distribution for a system of colliding particles could be rigorously derived. Consider N particles in one dimension that interact through random collisions. We consider the spatially homogeneous case only, i.e., the case where the particles are uniformly distributed in configuration space, since our focus, as was Kac’s, is on the collision mechanism. The collisions in the model are binary. When the i-th and j-th particles collide, their pre–collisional velocities vi and vj are transformed into the post collisional velocities