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

By Tian-Quan Chen

This booklet provides the development of an asymptotic approach for fixing the Liouville equation, that's to some extent an analogue of the Enskog–Chapman approach 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 within a small local of the purpose, are random in most cases. they're the simplest applicants for the macroscopic variables for turbulent flows. the result of the asymptotic approach for the Liouville equation unearths a few new phrases displaying the elaborate interactions among the velocities and the inner energies of the turbulent fluid flows, which were misplaced within the classical concept of BBGKY hierarchy.

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0-S(©> del uexp(27riy-0 $ ^ *i(y> u> ' ) y j ( y ) . 1. i(y)ii-Y(y) 1 /" d H T ^Pf. / ^ f ( O - ^ r ( e ) u e x p ( 2 7 r i y . O ] [ > ( y . M ) ^ ( y ) . 10 is completed. ri£ • y)

3 is completed. 4). 1. 5) U MW1RL<*>"\ where V>(£) denotes the Fourier transform of the function ip(x) — ip{\x\): VK£) = / dx^(x)exp(-27rix-^) and 2 pf. riy • £)<5(u) t^i d2H exp(27riy • 0 » 47r2m d0 2 (©) . g u ( y > u ) - exp(-27riy • £)<5(u) 1 dff a^ 27ri 96»5 (©) 0y • dxi(y ,u)]|. 6) Proof TV £ y dZ Fexp[^](-27rmi)v i • J J 0 4 (y, u ) ^ ' 1 J2 ~ y ) exp(-27riu • v,) W dydu I dZ Fexp[A] 2TT miv, • / / " 0 4 (y, u) 1 x ( *^ y) exp(-27riu • v,) * -X>(|x,-x f c |) + tf(x,) dydu k±l -27rmi /" dZ Fexp[i4] ff | ^ ( y , u) • j ^ U,«(xj - y) exp(-27riu • v,) 38 CHAPTER 3.

Actually any convex linear combinations of turbulent Gibbs distributions themselves are turbulent Gibbs distributions too. 33) is a subset of the set of the turbulent Gibbs distributions and the latter is far much wider than the former. 2. OUTLINE OF THE BOOK 23 linear combinations of local Gibbs distributions should be included in the class of the distributions describing turbulence phenomena (of inviscid flows). Hence, among the molecular distributions, the turbulent Gibbs distributions might be the best candidate for describing turbulence phenomena (of inviscid flows).

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