Influence of collisions on the fluctuations of a distribution function in a rarefied gas

Collision-induced (nonlinear) corrections to a correlator of a fluctuating component of a distribution function in an equilibrium state of a rarefied gas are studied in the framework of an extended Boltzmann equation, taking into account the nonlocality of a two-particle collision. The calculation of a nonlinear correction to the Kadomtsev's correlator (i.e. the one obtained in the linear approximation) encounters a power divergence when performing integration over small spatial scales. The divergence stems from the fact that the thermodynamic (equal-time) correlator, which is a boundary value for the kinetic (different-time) one, is delta-correlated with respect to the particles' velocities. Cutting off the divergent integrals on a distance scale of the order of a particle's size, we reveal that the nonlinear corrections are of the same order as the Kadomtsev's term. With the use of a graph technique, it proves to be possible to single out the most divergent terms in all orders of the perturbation theory, and to obtain, in a comformable approximation, a closed system of nonlinear integral equations for the full correlator (i.e. the one renormalized on account of the nonlinear corrections) and an auxiliary Green's function. Qualitative analysis of that system enables us to obtain some inferences concerning the behaviour of the full correlator.