The basics of Statistics, Information Theory and Statistical Mechanics in a nutshell

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	• A Farewell To Entropy Arieh Ben-Naim World Scientific Publishing Company, 2008 - 412 pages average customer review:based on 2 reviews view larger image
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A modern approach to statistical thermodynamics

In the Preface, the author states that there are many good books on statistical thermodynamics, and that this is not a textbook on this subject. However, though few other books exist that make use of the concept of "information" in statistical thermodynamics, this seems to be the first one that bases the full construction of the theory upon the information. For this reason, the approach by Arieh Ben-Naim is really modern and deserves a careful reading.

Personally, I don't think that this book can not be used as a textbook. Indeed, it is quite self consistent and builds step by step the core of the theory in such a way that any student is able to follow all arguments. Actually, it is true that it does not contain everything, but what is the textbook that really contains everything? May be, the only unpleasant thing for a student is the non negligible amount of time spent commenting the differences with other well known references, in particular the explanations of the probable reasons why Gibbs did not reach the very same results. But comments like these would be a valuable resource for teachers, on the other hand.

If I had to choose one thing in this book, I would recommend to enjoy the derivation of the Sackur-Tetrode equation (chapter 4): it is really beautiful and does not have the "shadows" that classical derivations suffer. For the very first time, I should say, I think I have understood it, thanks to this book.

However, the most important point of the whole book, the real starting point of the full construction, is the following. Shannon's measure H of the missing information (MI) is a more general concept than the entropy S, that is a thermodynamic quantity defined _only_ for equilibrium states: H can be defined for _any_ probability distribution and it comes out that H=S for the equilibrium state. Hence, though in thermodynamics, only changes of S between different equilibrium states are defined, one can make use of the properties of H to perform derivations in a more general context. The results, when applied to thermodynamic equilibrium states, will be also valid for the entropy S.

As the author explained in his introductory book Entropy Demystified: The Second Law Reduced to Plain Common Sense, the fact that S can only increase is an experimental observation, whose explanation is provided by a framework in which matter is discrete and composed by intrinsically indistinguishable particles, with the postulate of equally probable microscopic states and the postulate that the system will be found more often in macroscopic (or "dim") states with higher probability (the latter being the sum of the probabilities of all practically indistinguishable microscopic states, under the assumption that they are all independent).

What it is known as "thermodynamic equilibrium state" is really a set of "dim states" (or "macro-states", following Gibbs) for which the measurable quantities (that are inherently macroscopic) differ only by negligible amounts, so that they are practically (though not in principle) indistinguishable. In turn, these dim states are (in principle) different because they contain all "micro-states" (or "specific states", for Ben-Naim) that features the very same values of the observable quantities. In the assumption that all microscopic states are equally probable (dating back to Boltzmann and fully used by Gibbs), it turns out that the macro-states containing more micro-states are more probable, so that the system will spend more time on them. The family of macro-states around the macro-state with the maximum number of micro-states is what it is called thermodynamic equilibrium state.

The entropy S is defined _only_ for this set of macro-states, that is for the equilibrium state. However, the Shannon's (missing) information theory is defined for _each_ individual macro-state. That is why H is more general than S. By following these arguments, the full theory of statistical thermodynamics can be built, as you will find in the book by Arieh Ben-Naim.

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The basics of Statistics, Information Theory and Statistical Mechanics in a nutshell

As all previous books of Arieh Bem-Naim have already shown, he is an excellent writer. This is a very clearly written book, wherein the explanations are sufficiently detailed for non-experts to understand. The book is very complete and stands for its own. It starts by presenting all the statistical background needed for the ulterior chapters. Then, the author presents the core construction and the historical development of Information Theory and, in later chapters, applies it to the fundamentals of Statistical Mechanics. Furthermore, the whole book is full of interesting and engaging examples. Therefore, the book is a highly recommended reading for anyone interested in these topics.

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The principal message of this book is that thermodynamics and statistical mechanics will benefit from replacing the unfortunate, misleading and mysterious term entropy with a more familiar, meaningful and appropriate term such as information, missing information or uncertainty. This replacement would facilitate the interpretation of the driving force of many processes in terms of informational changes and dispel the mystery that has always enshrouded entropy.

It has been 140 years since Clausius coined the term entropy ; almost 50 years since Shannon developed the mathematical theory of information subsequently renamed entropy. In this book, the author advocates replacing entropy by information, a term that has become widely used in many branches of science.

The author also takes a new and bold approach to thermodynamics and statistical mechanics. Information is used not only as a tool for predicting distributions but as the fundamental cornerstone concept of thermodynamics, held until now by the term entropy.

The topics covered include the fundamentals of probability and information theory; the general concept of information as well as the particular concept of information as applied in thermodynamics; the re-derivation of the Sackur Tetrode equation for the entropy of an ideal gas from purely informational arguments; the fundamental formalism of statistical mechanics; and many examples of simple processes the driving force for which is analyzed in terms of information.

Contents: Elements of Probability Theory; Elements of Information Theory; Transition from the General MI to the Thermodynamic MI; The Structure of the Foundations of Statistical Thermodynamics; Some Simple Applications.

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