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

作者：Tessarotto Massimo^*; Cremaschini Claudio

来源：European Physical Journal Plus, 2014, 129(7): 157.

DOI：10.1140/epjp/i2014-14157-4

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 S-N-CDS admits an exact particular solution for the N-body probability density 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.

• 出版日期2014-7-28