A Non-equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos

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World Scientific, 2003 - Science - 436 pages
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This 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."
 

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This book is awesome. In the 'terrified out of your skin' kind of way.
I'd like to start by saying I generally enjoy mathematically difficult books - Korner's 'Fourier Analysis', Arnold's 'Ordinary
Differential Equations' and Rudin's 'Principles of Mathematical Analysis' are some of my all-time favorites. And I do know some statistical mechanics - at least up to Chandler's 'Modern Statistical Mechanics' or Kardar's 'Statistical Physics of Particles.'
But this book is terrifying.
It's stated goal, if it actually achieves it, is quite significant: it hopes to derive the functional equations of fluid mechanics (for both laminar and turbulent regimes) from the first principles of non-equilibrium statistical mechanics.If the author has done this, it would be an incredibly significant achievement.
Unfortunately, I can't say if he has.
Because this book is terrifying.
Open to a random page and you'll probably see no text. Just an equation. More accurately, you'll probably see *half* of an equation - because the equations in this book regularly span multiple pages.
Even in the foreword it's acknowledged that in this 'commercial 21st century' this book is likely to only be read by a few 'connoisseurs'. The author thanks the publisher for likely taking a loss with this manuscript.
This books is truly one for the experts. If you, like me, were caught by the title and thought this might contain a rational introduction to non-equilibrium statistical mechanics, without all the fuss of ergodic hypotheses and the like, then please beware. This is effectively an extended paper on a very difficult topic. I salute the author for his work, trusting that he has indeed succeeded in his difficult task. But the only people he is writing to are experts with far more knowledge than I. Us mere humans must seek simpler texts.
 

Contents

Functional
27
AFunctional
69
Turbulent Gibbs Distributions
81
Euler AFunctional Equation
119
Functionals and Distributions
157
Local Stationary Liouville Equation
175
Second Order Approximate Solutions
227
A Finer Functional Equation
271
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
Bibliography
407
Index
415
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