Paper detail

Application of Bogolyubov's approach to the derivation of kinetic equations for dissipative systems

The main goal of the present article is to extend the Bogolyubov method for deriving kinetic equations to dissipative many-body systems. The basic conjecture underlying the Bogolyubov approach is the functional hypothesis, according to which, the many-particle distribution functions are assumed to be functionals of the one-particle distribution function on kinetic time scales. Another ingredient in the Bogolyubov approach is the principle of the spatial weakening of correlations, which reflects statistical independence of physical values at distant spatial points. One can consider it as a reasonable mixing property of many-particle distribution functions. The motivation behind the generalization of Bogolyubov's approach to (classical) many-body dissipative systems is the wish to describe the dynamics of granular systems, in particular granular fluids. To this end we first define a general dissipative fluid through a dissipation function, thereby generalizing the commonly employed models for granular fluids. Using the Bogolyubov functional hypothesis we show how a reduction of the pertinent BBGKY hierarchy can be achieved. The method is then employed to cases which can be treated perturbatively, such as those in which the interactions are weak or the dissipation is small or the particle density is small. Kinetic descriptions are obtained in all of these limiting cases. As a test case, we show that the Bogolyubov method begets the now standard inelastic Boltzmann equation for dilute monodisperse collections of spheres whose collisions are characterized by a fixed coefficient of normal restitution. Possible further applications and implications are discussed.