### Kinetic Theory beyond the Stosszahlansatz

Gregor Chliamovitch, Orestis Malaspinas, Bastien Chopard
2017 Entropy
In a recent paper (Chliamovitch, et al., 2015) , we suggested using the principle of maximum entropy to generalize Boltzmann's Stosszahlansatz to higher-order distribution functions. This conceptual shift of focus allowed us to derive an analog of the Boltzmann equation for the two-particle distribution function. While we only briefly mentioned there the possibility of a hydrodynamical treatment, we complete here a crucial step towards this program. We discuss bilocal collisional invariants,
more » ... onal invariants, from which we deduce the two-particle stationary distribution. This allows for the existence of equilibrium states in which the momenta of particles are correlated, as well as for the existence of a fourth conserved quantity besides mass, momentum and kinetic energy. Entropy 2017, 19, 381 2 of 12 entropy factorization lends itself nicely to generalization. This maximum entropy ansatz on the three-particle distribution then allows closing the BBGKY hierarchy at the second order and deriving a kinetic equation describing the evolution of the two-particle distribution. Once the kinetic equation is set up, it becomes possible to follow the usual steps leading to the equilibrium distribution and macroscopic balance equations. There is however on the road a subtlety related to the definition of collisional invariants appropriate to the bilocal events under consideration here, and it happens that, besides conservation of (bilocal) mass, momentum and kinetic energy, it is necessary to consider a fourth invariant that eventually accounts for the momentum correlation of particles. The aim of this paper is to discuss these conceptual points in detail. In Sections 2 and 3, we derive the BBGKY equation at the second order. In Section 4, we introduce the maximum entropy ansatz for the three-particle distribution and the resulting closure of the hierarchy. Bilocal collisional invariants are discussed in Section 5, equilibrium distributions in Section 6, and balance equations in Section 7. We conclude with some remarks of a more philosophical flavour, emphasizing among others that what appears to be intuitive when working in the one-particle description does not necessarily hold any longer in the two-particle description. Liouville Equation and BBGKY Hierarchy Let us consider N particles of mass m, whose coordinates in phase space are their positions x i and momenta p i . It will be convenient to define a condensed notation ξ i = (x i , p i ). We let f N (ξ 1 , ..., ξ N , t) denote the joint distribution function characterizing the system; f N obeys Liouville's equation