Technical Report Arap-7-p-1 the Foundations of Nonequilibrium Statistical Mechanics

In this paper a novel philosophy of irreversible dynamics is formulated, This philosophy stems from the fact that the kinetic equations available (Boltzmann, Landau,, Bogolubov-Balescu-Lenard) are essentially exact and cannot be improved. That isv for kinetic gases (those whose behavior is characterized by that of one typical particle) these equations constitute closed, statistically complete knowledge, This thesis is demonstrated by using atstechnique that separates completely the different time components exhibited by the evolution of a gas when an appropriate parameter (characteristic of the regime in which the gas is found) is small. The expansion in this parameter is pushed up to its breaking point marked by the presence of an intrinsic divergence. With our technique we can pinch off the series at this point and remain with a closed, finite system of equations. The argument is made more compelling by the fact that the same divergence occurs for all gaseous regimes (shortrange« weak-couplings dilute weak-coupling and Debye). We furthermore prove that not all isolated gases will eventually become kinetic In fact,, the necessary and sufficient conditions that the initial departure from equilibrium must satisfy for kineticity to eventually set in ("absence of parallel motion") are deduced directly from the Liouvllle equation. When statistical information about a g*s is needed beyond that afforded by the knowledge of the motion of the average particle a new asymptotic expansion of the Liouville equation must be devised. Thus,, for example, the preponderance of three-body collisions demands knowledge of the evolution of the average pair of particles. For this situation a pair-kinetic expansion is Introduced. A hierarchy of increasingly more informative descriptions of a gas is thus envisaged. As a by-product of our analysis,, (i) we have found the limits a *n(*?r**-i-4"*''*C^ "~**"i **'« '"""i T* *\ 1 « * H'« +\ "•'» ''•''■'^ y 'H'.'»iH-y^-r ^"v«'y ^y n.r^ti^T» i "X" y—y—TTi."»"^"jITIT "V^*" ^"i" .V" ■" ■ * ■"■* n^'t^«-»""« *>•* tfww ■ t<m y^ g ^ r" ■ ■ ^ ■ ■*' <r"nr— V— g « ir«'nm n .BHIIU.-IT.MI i^fwwiff» w»iiiiiyi inywiLi^ of validity of Bogolubov's assumption of synchronization and of his boundary condition and (ii) we have proved the equivalence of Kirkwood's time averaging procedure (made systematic) with Bogolubov's technique.