The Classical Limit for the Uehling–uhlenbeck Operator by D. Benedetto and M. Pulvirenti

Consider a classical system of N identical particles. We are interested in a situation where the number of particles N is very large and the interaction strength quite moderate. In addition we look for a reduced or macroscopic description of the system. According to the general prescription of the kinetic theory, we introduce r > 0, a small parameter expressing the ratio between the macro and the micro scales. The weakness of the interaction is expressed by assuming that the potential is O( √ r). Since many of the physical quantities of interest are varying on a macroscopic scale and are almost constant on the microscopic scale, we rescale the equation of motion. Then the behavior of the one-particle distribution function f(x, v) (being x the position and v the velocity of a test particle) in the limit r → 0, N = O(r−3), is expected to solve the Fokker–Plank–Landau nonlinear diffusion equation: