A Non-Equilibrium Statistical Mechanics: Without the by Tian-Quan Chen

By Tian-Quan Chen

This booklet provides the development of an asymptotic process for fixing the Liouville equation, that's to a point an analogue of the Enskog–Chapman process for fixing the Boltzmann equation. as the assumption of molecular chaos has been given up on the outset, the macroscopic variables at some degree, outlined as mathematics technique of the corresponding microscopic variables within a small local of the purpose, are random normally. they're the simplest applicants for the macroscopic variables for turbulent flows. the result of the asymptotic strategy for the Liouville equation finds a few new phrases exhibiting the difficult interactions among the velocities and the inner energies of the turbulent fluid flows, that have been misplaced within the classical concept of BBGKY hierarchy.

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As a consequence of the assumption and the volume preservation of the flow on the phase space T, Liouville derived the following equation governing the evolution of the probability density F ( Z , t ) : dF ^ 8F J^f,- dF 0 n -at+E-r^ + Zi-wr - ,„„. 7) is called the Liouville equation. Being a probability density, F should be nonnegative and its integral over the phase space T should be unity. Furthermore we assume that F will decay sufficiently rapidly to zero as the total intermolecular potential tends to infinity.

G ipf. 4 is completed. Remark It is not difficult to show divergence of the integral 1 / d2H df^(0 47r2m BO2 d# 2 (9) exp(27riy • 0 a.. 6). The physical meaning of the inclusion of the last term 1 dH, 2-Tri de5 (©) dej , dy-d^{y>U\ . 40 CHAPTER 3. 6) is that the term corresponding to the self-interaction of one molecule must be excluded, because the intermolecular potential is equal to infinity at the origin: V>(o) = o. 6) denotes the "partie finie de Hadamard", of which the definition can be found in [69] and [38].

7) is called the Liouville equation. Being a probability density, F should be nonnegative and its integral over the phase space T should be unity. Furthermore we assume that F will decay sufficiently rapidly to zero as the total intermolecular potential tends to infinity. , whenever at least two hard cores of the molecules touch or overlap with each other). ) in the ball |x| < do- We can modify the value of the function i/>(x) in the ball |x| < d 0 arbitrarily without changing the value of the H functional and without violating the Liouville equation.

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