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# Statistical Mechanics

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Here are sections that I have now:

- Fundamental postulate. Entropy
- Principle of the conservation of energy
- Empirical observations of temperature
- Energy and fundamental postulate
- Carnot cycle
- Free energy
- The Boltzmann distribution
- Freely jointed chain
- Pressure
- Electric potential
- Chemical potential

Appendix

For notes on thermodynamics of inhomogeneous fields and non-equilibrium processes, see notes on Advanced Elasticity.

I am posting notes on statistical mechanics. Although parts of the notes have been used in courses that I have taught, I have written these notes mainly to help myself learn the subject. I have never taught a full course on the subject, and have never taken a full course on the subject, either.

This is a work in progress. I will update them and add new sections. I have tried to make each section modular, with as little coupling between sections as I can manage. Each section will be in a separate post, which will also serve as the home page for the section, for future updates. I will refrain from deleting any old versions, no matter how wrong and naïve they are.

The notes will be in the form of Word and PDF files. If you also like to learn a subject by writhing things down, please feel free to download and modify them to suit your own need. If you have improved a section of the notes, please upload a Word file in your own blog, and add a comment beneath my post of the section, with a link pointing to your blog entry. I'll try to incorporate your improvements in the next update of the section. Of course, you are always welcome to add comments. Together and over some time, perhaps we can turn these notes into a form satisfactory to both of us, and possibly to other people who think just like us.

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## Comments

## suggestion for a book

Zhigang,

There is a recently released book by JP Sethna on Statistical Mechanics. I found it to be quite good especially since it unifies the relevance of statmech to diverse subjects (black holes to econonics). It has a chapter on (after paraphrasing) "how to construct a landau model for anything" which is certainly worth reading-- especially for a student. For a long time, this book was available online on the authors website for free. The link appears here.

## I have just downloaded Sethna's book

Thank you Pradeep. I have just downloaded Sethna's book from the website you pointed out, using this link. It looks like a fun but demanding book to read.

## Very interesting

Thanks to both. Statistics is widely used in the field of durability. Does anyone has recommendation for a good book in that field?

Zhigang, any thought to include durability as a chapter in the book?

Xiao-Yan Gong, PhD

## A Dover book on statistics

Xiao-Yan:

I have not studied statistics systematically, and cannot recommend a good book. Here is a Dover book that I studied: Principles of Statistics, by M.G. Bulmer. I just noticed that readers reviews on Amazon are excellent. Apparently other people have also benefited from reading it.

If you'd like to have some fun, Cartoon Guide to Statistics may give you a quick overview of the subject. As a cartoon book goes, this one covers serious stuff.

Maybe some other iMechanicians will give you more advice.

## Statistics reference (Bayesian approach)

Xiao-Yan, I'd recommend Sivia's book on Bayesian data analysis for an intuitive look into data/statistical analysis. It uses the Bayesian approach (probability is a degree-of-belief and is always conditional) as opposed to the classical (frequentist) one. This book is uniformly recommended by all who subscribe to the Bayesian school of statistics, and on reading parts of the book I came away with the same conclusion. The book is concise (~200 pages), very well-written, and proceeds in a very systematic manner to de-mystify standard statistics as is taught in school. The above link is the 1996 edition; I noticed that a new edition (with Skilling as co-author) has come out in 2006, so that might be preferable if one wants to buy the book. Reviewers' comments are available on the 1996 edition, and hence I have provided a link to that. For one, I think a tool such as this might be relevant to estimate material parameters (e.g., for constitutive theories) from available experimental data. Given the fact that there exists inherent variabilities in real materials (structure, properties and ergo response), such a tool would seem to be useful. If not anything, it'll give you new perspectives and indicate that there are other options than classical statistics (and the many ad-hoc tests therein) to deal with data and uncertainty. In spite of my obvious bias, I hope this helps?

## Statistical mechanics and thermodyanmics made easy

I wish I did. People have given me warm feedback about these study notes, which I wrote to teach myself. For a college student in engineering and science, perhaps the right place to start is the section "Isolated Systems" The section reminds you of what a quantum state is, and tells you the fundamental postulate of statistical mechanics.

The whole statistical mechanics is then a collection of elaborate applications of the fundamental postulate. A first prototypical application is the definition of Temperature, the section you might read as the second section. The section relates the fundamental postulate to how we measure temperature and basic quentities like heat and the number of quantum states. The section also reminds people that temperature should really have the same unit as energy, and Boltzmann's constant is just a conversion factor between two units of temperature, rather like 1.6 is the conversion factor between miles and kilometers. These units are made up by humans, and have no fundamental significance. To think Boltzmann constant as a fundamental constant of nature is an abuse of nature and of the man.

Then you are ready to read the section on the Boltzmann distribution.

The section on Probability is inserted for people who want to have a quick review of math, the kind of necessary evil as we often talk about tensors at the beginning of a solid mechanics course. Of course, both the theory of probability and the theory of tensors are beautiful subjects in their own rights. But there is no point to let them overshadow mechanics, which is even more beautiful. Perhaps I should list the section as an appendix.

The section on Entropy, however, is perhaps an unnecessary evil. You certainly don't need it to read the rest of the notes. I inserted it right after Probability simply to remind people that Entropy is a quantity associated with any probability distribution. I make no distinction between information entropy and thermodynamic entropy and entropy associated with rolling a die. Because there is no distinction. Entropy is just a number you calculate when you have a probability distribution. No more, no less. You may find many things about entropy to impress other people at a dinner table.

The situation is rather like taking differentiation of a function. This is a mathematical operation: you can take a differentiation if you have a smooth function. You may make interesting applications in engineering and economics, but there is no point to make a distinction about an engineer's differentiation and an economist's differentiation. The distinction is in the applications, not the concept of differentiation itself.

To sum up, an easy reading sequence is

With a glance at Probability to refresh your memory, and at Entropy as an aside. Let me know if these notes help.

## PDFs?

Is it convenient for you to also post your notes in some format that is more portable than Doc, for instance PDF? My Doc reader messes up the symbols quite badly.

Thanks very much,

Kaushik

## PDFs of notes on Statistical Mechanics

Dear Kaushik: Thank you very much for your interest. I've just uploaded the pdf files.

## Thank you very much

Dear Zhigang,

Thank you very much for taking the time to convert your notes to PDF. I'm interested in learning some statistical mechanics, but I've been confused by the usual texts. I'm hoping that your mechanics perspective will be more accessible to me.

Kaushik

## Focus on the sections on isolated system and temperature

Dear Kaushik:

I've just finished teaching these sections in my class on Advanced Elasticity. Each section roughly corresponded to one 90-minute lecture. I did not include the sections on probability and entropy in my lectures.

Please go very slow with the section on isolated system and the section on temperature. Once you see through these two sections, perhaps the myth of thermodynamics will disappear. The rest is rather charming and technical applications of the same ideas.

Please don't be too disappointed if my notes don't help you. They are written for myself to teach the subject in class. I'm not sure that they can be used for self study.

## Re: Focus on the sections on isolated system and temperature

Dear Zhigang,

Thanks for the advice and warning. I'll probably supplement my reading of your notes with a regular text that's meant for self-study.

Kaushik

## Stat Mech for Elasticity

Kaushik, in case you are interested in a viewpoint that is anchored to continuum mechanics, you may wish to check out the not-quite-widely-known book by Weiner, "Statistical Mechanics of Elasticity". It is now available in Dover ($ 25) and is in my opinion a true gem.

The first few chapters review classical continuum mechanics and thermodynamics which provide a nice warm-fuzzy feeling to mechanicians who are studying statistical mechanics for the first time. Some of my students, who have taken stat mech courses both in physics as well as gone through this book, tend to favor Weiner.

## Re: Stat Mech for Elasticity

Thanks for the reference, Pradeep.

## Statistical mechanics books

It is quite interesting in that now the statistical mechanics course is provided in Engineering. As far as I know, statistical mechanics includes fundamental topics in physics - thermodynamics, phase transition, renormalization theory, hydrodynamics, etc. However, in recent years, since applied mechanics people have intrigued by nanomechanics, so that statistical mechanics has been regarded as a basic background for understanding machanics of nano-scale objects.

In my case, when I was in UT as a PhD student, I took the statistical mechanics provided in Physics department. During that time, the course used the text book "A Modern Course in Statistical Physics (ISBN 0-471-59520-9)" written by Linda E. Reichl, who lectured the course at that time. Reichl's book is quite readable to not only physics students but also engineering students.

Also, I like the book "Statistical Mechanics of Elasticity (ISBN 0-486-42260-7)" written by J. H. Weiner. This book includes the topics of thermoelasticity, rubber elasticity, crystal lattices, basic quantum mechanics, etc. This book is quite understandable, without any physics background, to engineering students.