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Liouville Equation for Turbulent Gases
Author(s) -
Wilhelm H. E.
Publication year - 1980
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/andp.19804920402
Subject(s) - physics , turbulence , phase space , mathematical physics , ideal gas , distribution function , compressibility , amplitude , scalar (mathematics) , statistical mechanics , dissipation , hamiltonian (control theory) , entropy (arrow of time) , fourier transform , gaussian , classical mechanics , quantum mechanics , thermodynamics , mathematics , geometry , mathematical optimization
Abstract The Hamiltonian H and Hamilton equations of motion are derived for the Fourier amplitudes pk and qk of the canonical conjugate fields p and q defined by p = mν and ∂ q /∂ t = n ν where ν( r , t ) and n ( r, t ) are the velocity and density fields of ideal, compressible gases in the state of fully developed turbulence. A Liouville equation is presented for the distribution function f ( pk , qk ; { k }) in the multidimensional phase space formed by the scalar components of the set { k } of wave mode vectors pk and qk . Though the theory is highly idealized since viscous and thermal dissipation are not taken into account, it may be considered as a first attempt at extending statistical mechanics to random continuous media. As an application, that stationary solution f = f ( H ) of the Liouville equation is calculated, which maximizes the turbulence entropy. It is shown that the distribution of the velocity and density fluctuations of compressible gases is Gaussian in fully developed turbulence, in agreement with the experiments.