Tian-Quan Chen's A Non-Equilibrium Statistical Mechanics: Without the PDF

By Tian-Quan Chen

This ebook offers the development of an asymptotic strategy for fixing the Liouville equation, that is to a point an analogue of the Enskog–Chapman process 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 quite often. they're the easiest applicants for the macroscopic variables for turbulent flows. the result of the asymptotic procedure for the Liouville equation unearths a few new phrases displaying the problematic interactions among the velocities and the inner energies of the turbulent fluid flows, which were misplaced within the classical conception of BBGKY hierarchy.

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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) . 1. 5 is completed. ^J^dej. 6 is completed. 9) 9u Proof 3ff\ N Y2 / dZFexp[A](-2mm)vr 94(y,u)6(xl-y)exp(-2mu-v,) — (xi)dydu CHAPTER 3. , . , „ . \&U, . 7 is completed. 8 is completed.

Therefore the assumption previously made about the subdivision of the space into a large number of cubes is consistent with the classical thermodynamics and fluid dynamics. In the sequel, we assume that the space occupied by the fluid has been subdivided into a large number of cubes in the manner stated above. 12) 16 CHAPTER 1. INTRODUCTION K denoting the side length of the small cube and 0 < k < K, i = 1,2,3. 11) is negligibly small. 11) and rewrite the equation thus obtained in the following way: d dx ( s ) N.

A Jdw|s), F = 0. 29) Neglecting the higher order infinitesimal W. £1=2VU^Wnt^dx] W, a-i) a ax{'> AT. 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. , [28]), 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.

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