By Tian-Quan Chen
This ebook provides the development of an asymptotic strategy for fixing the Liouville equation, that is to a point an analogue of the Enskog–Chapman method 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 as a rule. they're the easiest applicants for the macroscopic variables for turbulent flows. the result of the asymptotic strategy for the Liouville equation finds 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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Additional resources for A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos
32) will be modified as follows: C e -/3(H-(iV m |u| 2 )) (132) / where N denotes the number of molecules in the system. 32)' the Gibbs (canonical) distribution too. Now we consider a fluid continuum as a system of subsystems. Each subsystem corresponds to the aggregate of molecules in a small cube of the fluid continuum. 2), and that the subsystems are statistically independent. m|u„P)] s = (*i,s 2 ,*3). 33) and the convex linear combinations of several local Gibbs distributions with different parameters are special cases of the turbulent Gibbs distributions.
W12 ,(»> - =A 12 = Ax< s ». , the size of the fluid particle in the classical fluid dynamics) is an infinitesimal (in comparison with the macroscopic length scale) and the intermolecular potential ip is assumed to depend on the parameter K and the molecular mass m in the following way: y,(|x|)=m~f |Xi (1-18)! The external force Y is assumed to be of the form: Y(x) = mT(x), (1-18)2 18 CHAPTER!. INTRODUCTION where 5 and T are functions independent of n and m. Moreover we assume that the distribution function F depends on K in such a way that dF(Z-t) dy\a) = 1 « f0r/>2.
At the end of Chapter IV, the connections between the derivatives of Hfunctional and those of AT-functional, which is a restriction of the iJ-functional, are shown. These connections will be used in the derivation of a functional equation, which the /^-functional for inviscid flows should satisfy. In Chapter V, a class of distributions, called the turbulent Gibbs distributions, is introduced as a class of asymptotic solutions to the Liouville equation. The turbulent Gibbs distributions are generalizations of the convex linear combinations of local Gibbs canonical ensembles.