By Tian-Quan Chen

ISBN-10: 9812383786

ISBN-13: 9789812383785

ISBN-10: 9812795197

ISBN-13: 9789812795199

This e-book offers the development of an asymptotic process for fixing the Liouville equation, that is to some extent an analogue of the Enskog–Chapman strategy 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 commonly. they're the simplest applicants for the macroscopic variables for turbulent flows. the end result of the asymptotic strategy for the Liouville equation finds a few new phrases displaying the complicated interactions among the velocities and the interior energies of the turbulent fluid flows, which were misplaced within the classical idea of BBGKY hierarchy.

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**Extra resources for A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos**

**Sample text**

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]. 6). 3) consecutively. 7) 5 Proof \dtj13 = J2 fdZFexp{A}(-2irmi)vi = J2 [dZFexp[A](2irmi)vt • ff 65(y, u ) ^ ^ — ^ exp(-27riu • v,)dydu • ff fl6(y, u ) d S ^ ~ Y^ exp(-27riu • vf)dydu l=i N = Y^ fdZFexp[A}(-2irmi)xi • ff " ^ ^ V^y^Ei r*i /7a^(y,u) = 22 J dZFexp[A]mJJ —^T^( x S(xt - y) exp(-27riu • vt)dydu , ' ~ v) .

25) As was pointed out by Grad ([33]), statistical mechanics concerns itself with the asymptotic behavior of the iV-particle system as N —> oo under certain specific conditions. 26) s and Ki < NK6 < K 2 , where K, Ki and K2 denote three positive constants. 27) 20 CHAPTER 1. 13) can be rewritten as follows: (S) N. r/-l w / 7 +£ ti^dx? £ + ^V £ F ^ dt JV. i-1 £f< s >-(Z-l)f< (s) +E femyFi fc=l d dx fc-1ax! a Jdw|s), F = 0. 29) Neglecting the higher order infinitesimal W. £1=2VU^Wnt^dx] W, a-i) a ax{'> AT.

On K and U, i = 1,2,3. 12). 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.

### A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos by Tian-Quan Chen

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