By Tian-Quan Chen
This booklet provides the development of an asymptotic process 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 within a small local of the purpose, are random commonly. 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 elaborate interactions among the velocities and the interior energies of the turbulent fluid flows, that have been misplaced within the classical concept of BBGKY hierarchy.
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Additional info for A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos
1 Hydrodynamic Random Fields The 6TV-dimensional phase space for a system of N particles (molecules) is denoted by T = V w x R 3iV , where V denotes the space occupied by the N particle system. The representative point of T is Z = (xi, x 2 , • • -, x # ; v 1 ; v 2 , • • •, v ^ ) , where Xj = (XJI,XJ2,XJ3) and Vj = (VJI,VJ2,VJ3) denote the position and velocity of the j " 1 particle respectively. 3) where m denotes the mass of the particle ( molecular mass ), f (x 7 — Xfc)= the intermolecular force of the fcth molecule exerted on the j t h molecule, tp the intermolecular potential, Y the external force field.
Y(x) = 0 and the space V occupied by the N particles is of finite volume. 2. OUTLINE OF THE BOOK 21 particles is negligible. , ), in order to avoid the boundary effect it is frequently to treat a system of infinitely many particles in the whole space R 3 with periodic structure in the space R 3 instead of a finite particle system, but we just treat finite particle systems with vanishing boundary effects. , a first order asymptotic solution to the Liouville equation, where T/v(- • •; • • •, • • •, • • •; • • •) denotes a function of hv + 1 arguments, v — |V|/K3 being the number of cubes into which the space occupied by the fluid is divided.
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= !
A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos by Tian-Quan Chen