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

By Tian-Quan Chen

This publication provides the development of an asymptotic method for fixing the Liouville equation, that's to a point an analogue of the Enskog-Chapman method for fixing the Boltzmann equation. as the assumption of molecular chaos has been given up on the outset, the macroscopic variables at some extent, outlined as mathematics technique of the corresponding microscopic variables within a small local of the purpose, are random in most cases. they're the simplest applicants for the macroscopic variables for turbulent flows. the end result of the asymptotic procedure for the Liouville equation unearths a few new phrases exhibiting the tricky interactions among the velocities and the inner energies of the turbulent fluid flows, which were misplaced within the classical idea of BBGKY hierarchy.

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Grad [33] said that statistical mechanics is a sort of asymptotic mechanics. Hence what we are interested in is not the behavior of a single turbulent Gibbs distribution, but the asymptotic behavior of a sequence (precisely speaking, a net or a filter) of turbulent Gibbs distributions. The mathematical subtleties of the theory will be touched upon in Chapters VI and VIII. 19)), is derived from the //-functional equation under the assumption that the probability distribution on the phase space takes the form of a turbulent Gibbs distribution just as the Euler equations are derived from the balance equations under the condition that the probability distribution on one-molecule phase space takes the form of a local Maxwellian distribution in the classical theory of Boltzmann equations.

2), and that the subsystems are statistically independent. m|u„P)] s = (*i,s 2 ,*3). 33) and the convex linear combinations of several local Gibbs distributions with different parameters are special cases of the turbulent Gibbs distributions. Actually any convex linear combinations of turbulent Gibbs distributions themselves are turbulent Gibbs distributions too. 33) is a subset of the set of the turbulent Gibbs distributions and the latter is far much wider than the former. 2. OUTLINE OF THE BOOK 23 linear combinations of local Gibbs distributions should be included in the class of the distributions describing turbulence phenomena (of inviscid flows).

L)N. -l)N. l x W12 ,(»> - =A 12 = Ax< s ». , the size of the fluid particle in the classical fluid dynamics) is an infinitesimal (in comparison with the macroscopic length scale) and the intermolecular potential ip is assumed to depend on the parameter K and the molecular mass m in the following way: y,(|x|)=m~f |Xi (1-18)! The external force Y is assumed to be of the form: Y(x) = mT(x), (1-18)2 18 CHAPTER!. INTRODUCTION where 5 and T are functions independent of n and m. Moreover we assume that the distribution function F depends on K in such a way that dF(Z-t) dy\a) = 1 « f0r/>2.

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