Statistical Fluid Mechanics: Mechanics of Turbulence

A. S. Monin and A. M. Yaglom, The MIT Press, Cambridge, Massachusetts, Volume 2, 1975

It is really a shame that I had been holding a wrong comprehension about locally homogeneous turbulence that means turbulence is homogeneous locally. I had never thought under this condition a turbulence flow will be overall homogeneous, since there is no change from different loci. In fact, the locally homogeneous random field is a counterpoint of the random field with stationary increments. A better and unambiguous term should be random field with homogeneous increments (page 93). Then the underlying philosophy for Kolmogorov to consider locally homogeneous turbulence is easily appreciated, after all, it is a kind of linear approximation in space! This is natural idea, however, with amazing success.

Addendum

Soon after putting down the above words, I realized that I was wrong again! In fact, Kolmogorov’s locally isotropic homogeneous turbulence theory has nothing to do with the locally homogeneous random field, i.e., random field with homogeneous increments. Why had I changed my idea so easily and even found some “sound reasons” to support the new idea? Ridiculous!

Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach

Roger M. Samelson and Stephen Wiggins, Springer, New York, 2006

This is a short book. I have just skipped through half of the book, very briefly. Long time ago, I once picked up Wiggins’ dynamics book, but had not finished it. I think his book is one of the classics in dynamics. However, the present book about Lagrangian transport seems just a short collection of simple dynamical models with analysis based on classic dynamics theory. I knew a little about dynamics, now I know a little more, except for much more intuition on the KAM theorem.

Accidentally, I have a new notion that perhaps only seldom scholars can really produce abundant academic fruits, most will resell the best of their achievements or knowledge in various disguised forms. If it is the truth, then I should be a bit more optimistic that I am not the lonely one who has limited ability to survey the extensive and profound nature.

Lagrangian Fluid Dynamics

Andrew Bennett, Cambridge University Press, Cambridge, 2006

In the preface, the author described the motivation and ulterior motive to write the first book in fluid dynamics mainly from a Lagrangian perspective. I agree with many of the author’s believes that Lagrangian method can take advantage of many analytical results that are not available for Eulerian method, despite generally the numerical disadvantage of Lagrangian method. The author also commented that stochastical differential equation (SDE) is only a beautiful mathematical theory for diffusion problem. I have not realized that, since I am not so confident about the whole content of  SDE. But having retrospected all my knowledge of SDE, I must admit that the author’s comment has a good possibility to be true. There is another point of view that I share with the author that it seems today’s students are good at computer skills, while an earlier generation has excellent ability on mathematical analysis. The author claimed that he falls between the two generations, which is my aim also.

In chapter 2, when proving the equivalence of single-point single-time statistics between Lagrangian and Eulerian perspectives, the author referred the outline to Lumley (1962, An approach to the Eulerian-Lagrangian problem. Journal of Mathematical Physics, 3, 309-312). But Lumley’s paper does not imply the proof at all. In fact, I found quite similar proof in Monin and Yaglom’s book (1971, Statistical Fluid Mechanics: Mechanics of Turbulence, MIT Press), where the reference is another paper of Lumley (1962, The mathematical nature of the problem of relating Lagrangian  and Eulerian statistical functions in turbulence, Mecanique de la turbulence (Coll. Intern. du CNRS a Marseille), Paris, Ed. CNRS). I have not read the last conference paper, but infer from the title that it is quite plausible that Lumley had presented his proof in the conference paper.

Catastrophe Theory

V. I. Arnol’d, Springer-Verlag, third, Berlin, 1992

The booklet is really interesting, although I just read several pages. I had thought Arnol’d book with such a startling title must be very hard to read. But the fact is quite contrary (perhaps reading more will change my altitude).

The Whitney’s theory demonstrates that for a smooth mapping from a surface to a plane there are generally only two kinds of singularities, fold and cusp. The idea is easily understood after grasping the definitions and meditating on a transparent torus. Admittedly, a torus or a inflated tire is not so interesting at all. However, the application of the Whitney’s theory to describe the relation between technical proficiency, enthusiasm and achievement of a scientist is amazing. Simple and convincing! In fact, I even got some philosophical include from this simple example: hold on, and one day all things will change abruptly. Believing in this philosophy is very encouraging. So what I am pursuing in academia probably is not climbing a mounting that I cannot see the peak, but digging in a mine, the gold is likely buried only one inch down. So persisting in promoting the technical proficiency is not futile. I am just waiting for the catastrophe.

Making Globalization Work

Joseph E. Stiglitz, W. W. Norton & Company, New York, 2006

After encountered too many typos I realized the e-book perhaps is not the original version but a scan-OCR one. The author, a Nobel laureate, showed much concern about global poverty and inequality, and criticized many policies of the United States and EU that had deepened the level of poverty and inequality other than ameliorate it. I have somewhat changed my prejudice that the think tanks in western are just the spokesmen of capitalists.

A Modern Course in Statistical Physics

L. E. Reichl, John Wiley & Sons, New York, second, 1997

In the back cover, it reads “All the tools necessary to understand the concepts underlying today’s statistical physics”. Admittedly, the book really covers huge contents and many advanced topics which are usually under the special sections. I don’t have the necessary background to judge whether it is a good textbook in statistical physics. It is not too difficult to read when skipping many mathematical derivations, which does not mean I hate equations, but I wanted to read through the book as fast as possible and had no time to linger on the equations. However, I read the contents about homogeneous functions carefully and grasped the central idea behind homogeneous functions easily for the first time, although I did have some vague illusion about homogeneous functions and its relation with renormalization group theory. 

I was attracted by the Einstein fluctuation theory and thought it could be applied to the investigate the Prandtl number effect on the nucleation of aerosol which I have been considering for months. To cite Einstein’s work (A. Einstein, Investigation on the Theory of Brownian Movement, Methuen and Co., London, 1926) in a paper is an appealing idea.

Errata:

There is a misspelling in the book, but I forgot the right place. The word “proportioality” should be “proportionality”, but when I first encountered it, I just though it was a new word and did not note down the place it occurred. Then I read the right spelling word and noticed the former one was misspelled. This is a very scare erratum.

On the seventh line in page 711, should be .

Innumeracy: Mathematical Illiteracy and Its Consequences

John Allen Paulos, Viking, London, 1989

At first sight, I though the book is too simple and is not worth reading. Soon, I changed my mind after reading the following joke:

A man who travels a lot was concerned about the possibility of a bomb on board his plane. He determined the probability of this, found it to be low but not low enough for him, so now he always travels with a bomb in his suitcase. He reasons that the probability of two bombs being on board would be infinitesimal.

Equilibrium Statistical Physics: Phases of Matter and Phase Transitions

Marc Baus and Carlos F. Tejero, Springer, Berlin, 2008

This is the first book in statistical physics that I have almost read through. It’s very easy to read, both contents and language. Only after reading the book, I can distinguish various Ensembles: microcanonical, canonical and grand canonical.

Errata:

The double integral with respect to p and q was missing in equation (4.44).

Kinetic Theory: Classical, Quantum, and Relativistic Descriptions

Richard L. Liboff, Springer-Verlag, New York, third, 2003

I first encountered this book several years ago. The contents were so new to me and I was impressed. But it was hard to grasp the contents also. I have been hoping to read through the book one day.

In order to figure out the clear relation between viscosity and conductivity of fluid and the particle kinetics, I pick up the book again.

“A distribution function is worth a thousand macroscopic variables.” The quip is excepted from the footnote 5 of page 162.

Errata:

It seems from page 29 on all the references to (4.20) should be (4.23).

The subsection 3.3.4 title “Temperance: Variance of the Velocity Distribution” is so weird, perhaps “Temperance” should be “Temperature”. But this kind of error is unimaginable, except the word was automatically chosen by the word processor through the auto-correction function, which I encountered several times.

Page 169, the second sentence was repeated.

Page 181, “+” in equation (4.18) should be “=”.

Probability And Related Topics in Physical Sciences

Mark Kac with special lectures by G. E. Uhlenbeck, A. R. Hibbs, and Balth. van der Pol, American Mathematical Society, Providence, Rhode Island, 1959

It is too hard to follow the author’s derivation owing to very subtle math, and there was any subsections at all which made the context more difficult to grasp. In conclusion, I learned almost nothing new. However, I do respect Kac’s math ability, and appreciate the style to solve a concrete physical problem by presenting a math model and then analyzing the model. The style is not unfamiliar, but the author’s ability is admirable. There are special lectures appended in the book by Uhlenbeck and van der Pol (I have not heard about Hibbs.), but I did not read them. So many famous names in a single title page reminds me of the saying birds of a feather flock together!

The book contains too much on the topic of irreversibility from the Boltzmann standpoint to be of much interest today, excepting one chapter: the one by Kac on functional integration and partial differential equations.

Above is the review by professor Joseph L. McCauley in Amazon.com. I cannot judge professor McCauley’s review either, since I do not know the popular point of view on the paradox of Boltzmann’s irreversibility.