A Non-equilibrium Statistical Mechanics: Without the Assumption of Molecular ChaosThis book presents the construction of an asymptotic technique for solving the Liouville equation, which is to some degree an analogue of the EnskogOCoChapman technique for solving the Boltzmann equation. Because the assumption of molecular chaos has been given up at the outset, the macroscopic variables at a point, defined as arithmetic means of the corresponding microscopic variables inside a small neighborhood of the point, are random in general. They are the best candidates for the macroscopic variables for turbulent flows. The outcome of the asymptotic technique for the Liouville equation reveals some new terms showing the intricate interactions between the velocities and the internal energies of the turbulent fluid flows, which have been lost in the classical theory of BBGKY hierarchy. Contents: H -Functional; H -Functional Equation; K -Functional; Some Useful Formulas; Turbulent Gibbs Distributions; Euler K -Functional Equation; Functionals and Distributions; Local Stationary Liouville Equation; Second Order Approximate Solutions; A Finer K -Functional Equation. Readership: Researchers in mathematical and statistical physics." |
Contents
Foreword | 1 |
HFunctional | 13 |
HFunctional Equation | 47 |
KFunctional | 69 |
Some Useful Formulas | 77 |
Turbulent Gibbs Distributions | 81 |
Euler KFunctional Equation | 119 |
Functionals and Distributions | 157 |
A Finer KFunctional Equation | 271 |
75 | 314 |
Conclusions | 339 |
A Some Facts About Spherical Harmonics | 347 |
A List of Spherical Harmonics | 349 |
Products of Some Spherical Harmonics | 369 |
Derivatives of Some Spherical Harmonics | 402 |
407 | |
Common terms and phrases
approximate solution arcsin assumption asymptotic BBGKY hierarchy Boltzmann equations classical fluid dynamics cube Ct defined denotes derivatives dw₂ dx(t dz(t dZTN EJFGH Et,int Et,kin Euler equations Euler K-functional equation exp[A expression Əti Ətj Ətk finer K-functional equation fluid dynamics fluid flow fluid particle following form Gibbs canonical Gibbs distribution Ty Gibbs mean governing the evolution Gross Determinism heat energy Hence inside the cube intermolecular potential energy JFGH Lemma Liouville equation macroscopic molecular chaos molecules inside momentum density N₁ N₂ Navier-Stokes equations orthonormal system perturbed phase space Precisely speaking present book Proof Proposal of Gross Proposition quantities right hand side spherical harmonics Theorem total intermolecular potential tuple turbulent Gibbs distribution variables vector velocities zỗ(y Σ Σ ΣΣ ΣΣΣ ΣΣΣΣ ак ән дн дс ду მხ