By Tian-Quan Chen

ISBN-10: 9812383786

ISBN-13: 9789812383785

ISBN-10: 9812795197

ISBN-13: 9789812795199

This ebook provides the development of an asymptotic method for fixing the Liouville equation, that's 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 inside of a small local of the purpose, are random typically. they're the simplest applicants for the macroscopic variables for turbulent flows. the result of the asymptotic procedure for the Liouville equation unearths a few new phrases exhibiting the difficult 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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**Extra resources for A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos**

**Example text**

3)! 1. i N N GEE *(i*i-*i))=£|£(i*« \ j=i Xfcl). 4) 2it\debK ' Proof dH at ) -±J dZ F exp[A](-27rmi)v i /ll l=1J 3 / / E Wy>u'^)^(gx~y) exp(-27riu • vfiYjMdydu 36 CHAPTER 3. H-FUNCTIONAL EQUATION N = ^2 dZFexp[ J 4](27rmi)vj //E0i(y>u>*)a*(yy) N r -27rmi V J=I 86, •lit ex / ^ 3 N = m J2 f dZ F exp[A] ff <9u = - m ^ y dZFexp[A]vi (y, u, t)6(xi - y) exp(-27riu • vj)Y^(y)dydu T[dy N P(-27riu • v,)Yj(x,)4ydu r rr dZFexp[A}jJ j i ( y , u , t)5(x, - y) exp(-27riu • vj) )5^(y)dydu 3 ^ i=i £ g2^ ^—|^(y,u,i)J(xi-y)exp(-27riu-v,)yj(y)dydu i=i 2m 86$ ' ^ <-]3= !

H-FUNCTIONAL EQUATION 1 N "l\ - ^ V ( | x j - x f c | ) + £/(xO jdydu k*l N m JdZFexplA] Jj ^(y,u) • £ / (<5(x, - y ) A (exp(-27riu • v<)) JV 5>(|x,-x fe |) + t/(xO = —m j dZ F exp[A] J J ^ - ^ (y, u) £ dydu ( 5(xj - y) exp(-27riu • v. 1. riu • vfidydu N f x / exp(-27ri£ • y)S(u) ^ J fc S(xk - y) exp(-27riu • vk)dydu —1 + --(e) 27Ti9fl6 e l/«*0{*=w< > '"<*>«$=<*"> 9# 2 exp(27riy • 0 ^ — | ^ ( y > u ) > exp(-27riy • £)<5(u) 1 dH del (y,u) 27ri d0 5 (©) dy • du 1 •]} + 8fi2 ^ / ^ 2 ^ ( G ) 2 50 exp(27riy • g) g .

2. OUTLINE OF THE BOOK 25 ary Liouville equation obtained in the Chapter IX and, therefore, a form of the second order approximate solution of the Liouville equation (with undetermined coefficients) can be explicitly written down. In Chapter XI a functional equation governing the evolution of the AT-functional and finer than the Euler JiT-functional equation is derived with aid of the second order approximate solutions to the Liouville equation obtained in Chapter X. A lot of new transport phenomena emerge in the calculations in Massignon's theoretical framework.

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

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