The Master kinetic equation for the statistical treatment of the Boltzmann-Sinai classical dynamical system

Abstract

In this investigation, exact particular realizations are sought for the microscopic statistical description which is associated with the classical dynamical system (CDS) formed by N identical smooth hard spheres subject to elastic collisions (S N -CDS). The problem is posed in the framework of the ab initio statistical description of S N -CDS recently developed. It is shown that the Liouville equation associated with SN-CDS admits an exact particular solution for the N-body probability density function (PDF). This is factorized in terms of the i-th particle 1-body PDF (for all i = 1, N) via suitable weighting factors, which are denoted here as particle occupation coefficients. The latter are found to depend functionally only on the 1-body PDFs which are associated with each of the remaining particles belonging to S N -CDS. Furthermore, the 1-body PDF is proved to obey a well-defined statistical equation, referred to here as Master kinetic equation. This is an exact kinetic equation which takes into account the occurrence of configuration-space correlations due to the finite size of the extended particles, while depending functionally on the same 1-body PDF only. The asymptotic approximation of the Master equation, which holds in validity of the Boltzmann-Grad limit, is shown to recover in a suitable asymptotic sense the customary Boltzmann equation. Finally, a critical analysis is presented of the original and modified versions of the Enskog kinetic equation, as well as of some of the non-linear kinetic approaches formulated in the past for dense granular gases. Their conditions of validity and main differences with respect to the present theory are pointed out.