Convergent Generalization of the Boltzmann Equation for a Hard-Sphere Lorentz Gas

A generalization of the Boltzmann equation for a classical Lorentz gas with hard-core interaction is presented. The $N$-body streaming operator is evaluated directly from the dynamics, thereby avoiding the binary collision expansion. A cluster expansion is developed in a form that results in exponential decay of the dynamical correlations and regularizes all divergent diagrams. Virtual collisions, represented by virtual binary kernels, are related to configuration-space restrictions, which in turn are responsible for the collisional damping. A prescription is given for the convergent $l$-body collision integral.