Title
"Exact solutions of the Boltzmann-Enskog kinetic equation for elastic and inelastic hard spheres"
Anton Trushechkin (Steklov Mathematical Institute, Russian Academy of Sciences)
Abstract
We find exact solutions for the nonlinear integro-differential Boltzmann-Enskog kinetic equation (for both elastic and inelastic hard spheres). These solutions are particle-like (analogous to multi-soliton solutions of the Korteweg-de Vries equation). To our awareness, these are the first known smooth exact solutions of the Boltzmann-Enskog equation.
The constructed solutions are regularized versions of the so called microscopic solutions of the Boltzmann-Enskog equation discovered by N.N. Bogolyubov [1] (see also [2]). The microscopic solutions have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. However, Bogolyubov defined these solutions at the "physical level" of rigour. In particular, he did not discuss the products of delta-functions in the collision integral. Here we give a rigorous sense to microscopic solutions as a limiting case of the constructed smooth delta-like solutions.
Finally, we give some comments on the reversibility paradox and entropy production.
The presentation is based on paper [3], some preliminary results are given in [4].
[1] N.N. Bogolyubov, "Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls", Theor. Math. Phys. 24(2) 804-807 (1975).
[2] N.N. Bogolubov, N.N. Bogolubov Jr., Introduction to Quantum Statistical Mechanics (World Scientific, Singapore, 2010).
[3] A.S. Trushechkin, "Microscopic solutions of kinetic equations and the irreversibility problem", Proc. Steklov Inst. Math. (2013), to be published
[4] A.S. Trushechkin, "Derivation of the particle dynamics from kinetic equations", p-Adic Numbers, Ultrametric Analysis and Applications 4(2) 130-142 (2012); arXiv.org:1201.3607 [math-ph].
