By Tian-Quan Chen
This publication offers the development of an asymptotic strategy for fixing the Liouville equation, that's to some extent 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 often. they're the easiest applicants for the macroscopic variables for turbulent flows. the result of the asymptotic procedure for the Liouville equation finds a few new phrases displaying the complex interactions among the velocities and the inner energies of the turbulent fluid flows, that have been misplaced within the classical thought of BBGKY hierarchy.
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Extra info for A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos
The solutions of the Euler equations with random initial data. In other words, the Euler ^-functional equation governs the motion of turbulent inviscid flows. We have seen that the approach in the first seven chapters of this paper is acceptable: it almost coincides with the statistical theory of turbulence in the classical fluid dynamics. The only difference between them is as follows. , those with vanishing variances. In classical fluid dynamics, a general flow is laminar (or, deterministic) and turbulence phenomena is the consequence of the instability of a basic (laminar) flows.
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.
Hence Massignon's is far much better than those in the theory of BBGKY hierarchy in describing the reality. I think, ease in making calculations is the only cause why the local quantities were introduced in the theory of BBGKY hierarchy in a way inconvenient in describing the reality. Hence Massignon's correction should be considered reasonable. Of course, the complexity of calculations is the inevitable price for the reasonability. 2. 20). 20). 13) can be written as follows: N ' (s) N. w, Z / i r a .
A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos by Tian-Quan Chen