By Tian-Quan Chen
This booklet provides the development of an asymptotic approach 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 degree, outlined as mathematics technique of the corresponding microscopic variables inside of a small local of the purpose, are random mostly. they're the easiest applicants for the macroscopic variables for turbulent flows. the result of the asymptotic approach for the Liouville equation unearths a few new phrases exhibiting the difficult interactions among the velocities and the interior energies of the turbulent fluid flows, which were misplaced within the classical conception of BBGKY hierarchy.
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Extra resources for A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos
E(ff-Y(xi s) ))-(/-l)(f/ s) -Y(x«))]^ y JF = 0. , 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.
We will not derive them until they are needed. The functional equation governing the evolution of the Hfunctional will be derived in the following section. 11). Thus we have dH = IdZ— dt 3 exp[A] - 27ri / dZFexp[A] ff ] T ^(y,u)p,-(y,u, Z)dydu. 7), is of the following form dH (dH\ ^ (dH\ dH _ (dl£\ (2E\ + (dH\ (dH> where ,„ „. 2), 33 CHAPTER 3. 2 (fX-Z-S-S-^-tCf),,. 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.
16)), which governs the evolution of the if-functional, is derived from the Liouville equation without any additional restrictions. 16) governs the evolutions of any i7-functionals corresponding to general solutions to the Liouville equation. It plays the role of the balance equations in the theory of Boltzmann equation, which are satisfied by the moments of the general solutions to the Boltzmann equation. The difference between them is that the iJ-functional equation is a closed equation, but the balance equations are not.
A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos by Tian-Quan Chen