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Entropy Demystified: The Second Law Reduced to Plain Common Sense

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After seeing nothing but five-star reviews for this book, I figured I'd pick it up despite having little feel for what its target audience was since none of it was actually viewable on Amazon.

In a nutshell, this is very much a book for laymen. If you want an intuitive grasp of what entropy's about in the context of everyday physics without getting bogged down in math, then this may be a great book for you. The book uses as little math as possible in its explanations, and effectively assumes you're unfamiliar with or have forgotten high-school-level math operations such as factorials and logarithms. It manages to pound its point home reasonably well using lots and lots of fairly simple thought experiments that only differ from each other by little incremental steps.

On the other hand, if you already know anything at all about the information-theoretic formulation of entropy, already have an appreciation for the Law of Large Numbers, and have heard the words "macrostates" and "microstates" before, then there's nothing in this book you aren't likely to understand already. If you've taken a course on statistical mechanics and finished it without being horrendously confused, but maybe were hoping for a useful refresher on how different formulations of entropy are related, you should pass on this book. If you were hoping for illumination about the aspects of entropy that are actually at all "interesting" to modern physicists, such as black hole entropy (or the bizarre theories it's spawned such as the holographic principle), this is definitely not the book you're looking for.

Also, the book has no index. This is less annoying than it would be in a book that had more meat to it, but still, any 200+ page nonfiction book with no index should be taken out and shot as a matter of principle.

Arieh Ben-Naim, professor at the Hebrew University of Jerusalem, taught
thermodynamics and statistical mechanics for many years and is well
aware that students learn the second law but do not understand it,
simply because it can not be explained in the framework of classical
thermodynamics, in which it was first formulated by Lord Kelvin (i.e.
William Thomson, 1824-1907) and Rudolf Julius Emanuel Clausius
(1822-1888). Hence, this law and the connected concept of entropy are
usually surrounded by some mysterious halo: there is something (the
entropy), defined as the ratio between heat and temperature, that is
always increasing. The students not only do not understand _why_ it is
always increasing (it is left as a principle in classical
thermodynamics), but also ask themselves what is the _source_ of such
ever increasing quantity.

We feel comfortable with the first law, that is the principle of energy conservation, because our experience always
suggests that if we use some resource (the energy) to perform any work,
then we are left with less available energy for further tasks. The
first law simply tells us that the heat is
another form of energy so that nothing is actually lost, something which
we can accept without pain. In addition, the second law says that,
though the total energy is constant, we can not always recycle 100% of
it because there is a limit on the efficiency of conversion of heat into
work (the highest efficiency being given by the Carnot cycle, named
after Nicolas Léonard Sadi Carnot, 1796-1832). Again, we can accept it
quite easily, because it sounds natural, i.e. in accordance with our
common sense: we do not know any perpetual engine. But our daily
experience is not sufficient to make us understand what entropy is, and
why it must always increase.

The author shows that, if we identify the entropy with the concept of
"missing information" of the system at equilibrium, following the work
done by Claude Elwood Shannon (1916-2001) in 1948, we obtain a well
defined and (at least in principle) measurable quantity. This quantity,
apart from a multiplicative constant, has the same behavior as the
entropy: for every spontaneous process of an isolated system, it must
increase until the equilibrium state is reached. The missing
information, rather than the disorder (not being a well defined
quantity), is the key concept to understand the second law.

I should say here that the identity of entropy and missing
information is not a widespread idea among physicists, so that many
people may not appreciate this point. However, the arguments of this
book are quite convincing, and different opinions are also taken into
account and commented.

In addition, Ben-Naim thinks that the entropy should be taught as an
dimensionless quantity, being defined as the ratio between heat, that is
energy, and temperature, that is a measure of the average kinetic energy
of the atoms and molecules. The only difference with the missing
information, again dimensionless, is the scale: because the missing
information can be defined as the number of binary questions (with
answer "yes" on "no" only) which are necessary to identify the
microscopic state of the system, this number comes out to be incredibly
large for ordinary physical systems, involving a number of constituents
of the order of the Avogadro's number. This numerical difference makes
me think about the difference between mass and energy, connected by the
Einstein's most famous equation E = m c^2: they could be measured using
the same units (as it is actually done in high-energy physics), the sole
difference being that even a very small mass amounts to a huge quantity
of energy.

The mystery of the ever increasing entropy can be explained once (and
only if) we realize that the matter is not continue, but discrete. The
author basically follows the work of Josiah Willard Gibbs (1839-1903),
who developed the statistical mechanical theory of matter based on a
purely probabilistic approach. First, one has to accept the fact that
macroscopic measurements are not sensitive enough to distinguish
microscopic configurations when they differ for thousands or even
millions of atoms, just because the total number of particles is usually
very large (usually of the order of 10^23 at least). Then, under the
hypothesis that each microscopic state is equally probable, i.e. that
the system will spend almost the same time in each micro-state, one can
group indistinguishable micro-states into macro-states. The latter are
the only thing we can monitor with macroscopic measurements. Under the
commonly accepted hypothesis that all microscopic configurations are
equally probable, macro-states composed by larger numbers of
micro-states will be more probable, i.e. the system will spend more time
in such macro-states.

As a naive example, one could start with a system prepared in such a way
that all its constituents are in the same microscopic configuration.
One could think about a sample of N dices, all of them showing the same
face, say the first one. The questions could be: (1) "Are all dices
showing the same face?"--Yes--; (2) "Is the face value larger or equal
than 3?"--No--; (3) "Is the face value larger or equal than 2?"--No--;
at this point we know that the value is 1. In general, the number of
binary questions is proportional to the logarithm in base 2 of the
number C of possible configurations, that is O(log_2 C). Now imagine to
randomly mix the dice by throwing all of them. The answer to the first
question would be "No", so that a completely different series of
questions has to be asked to find the microscopic configurations.
First, one may procede by finding how many dice show the value 1, for
example, asking O(log_2 N) questions. Suppose that the answer is M<N:
then one should find exactly what dice are showing this face, by asking
O(N) questions. The next step is to find how many dice show the value
2, among the N-M remaining ones, and so on. When N is very large, the
number of questions increases rapidly. So far, we have being speaking
about "microscopic" configurations, describing the exact state of all
dice. Now, we can imagine to be interested only in the "macroscopic"
configuration defined by the sum of all values. It is very easy to
imagine that the "microscopic" configurations corresponding to sum
values around 3N (corresponging to a uniform distribution of values)
will be many more than those with sum near N or 6N (which need all dice
showing 1 or 6, respectively). If we repeatedly shake the box or throw
all dice, most of the time we will obtain a sum near to 3N, and larger
deviations will be rarer. Hence, such a system will soon approach the
"equilibrium" state in which the sum is very near to 3N.

As a matter of fact, when the number of possible microscopic
configurations increases, the probability distribution of macro-states
becomes narrower and narrower, so that for ordinary systems the
probability to have a fluctuation large enough to be measured is
incredibly small. Actually, as Ben-Naim clearly emphasizes, the
probabilistic formulation of the second law of thermodynamics allows us
to quantify its validity, in terms of the time one should wait to be
able to find a fluctuation large enough to be measured. It comes out
that, for ordinary systems, the probability to have any measurable
fluctuation away from the equilibrium state is so low that the universe
age is practically negligible compared to the time we should wait to
observe such fluctuation. From this point of view, the second law is
far more "absolute" than the other laws of physics, for which at best we
could state that they are valid since the beginning of the universe life.

The book is a very good reading for all students who approach the
thermodynamics and also for more advanced people who do or do not feel
comfortable with the fascinating concept of entropy. Ben-Naim is also
the author of a more technical book ("Statistical Thermodynamics Based
on Information. A Farewell to Entropy", World Scientific, A Farewell To Entropy) in
which these guidelines are the base for a more detailed treatment of
statistical mechanics. Because we usually learn things much better when
following a cyclical approach, I encourage the readers to start with the
book "Entropy Demystified" and then seriously consider to go deeper into
the details of statistical mechanics with the more technical book by
Ben-Naim, of which I was delighted to read the draft.

"... Arieh Ben-Naim invites the reader to experience the joy of appreciating something which has eluded understanding for many years -entropy and the second law of thermodynamics". This statement on the back cover for sure will reflect the experience of many who read this book. I highly recommend it to anyone who wants to understand or teach the mysterious concept "entropy". Just sit back, open this delightful book, and experience how your foggy ideas are cleared up within just a couple of enjoyable hours. You need no prior knowledge; if you have learned how to read and how to count numbers between one and ten you possess all qualifications needed to read and appreciate all of its contents. The author not only succeeds to brilliantly explain the meaning of entropy, its statistical interpretation and why common sense leads us to conclude entropy (most likely) is ever-increasing - he moreover provides compelling arguments to do away with the second law altogether: ".. because science will find it unnecessary to formulate a law of physics based on purely logical deduction". This concluding sentence by Ben-Naim will be further substantiated in a forthcoming book by the same author. In addition to the present book, which I highly recommend to everbody who wants to learn about entropy in general, I also want to recommend another recent book by Ben-Naim on molecular theory of solutions to students and scientists interested in the entropy of solvation processes. The scientific literature on this topic is huge and -above all - utterly confusing. Ben-Naim's clearly formulated ideas have helped me a lot in understanding the subject better.

Adam Smith's "Invisible Hand" leads many people to think, that markets have the power to repair "themselves". But even in markets as open systems, there are irreversible processes, as the openness of real systems always is limited. Adam Smith, still in a Newtonian world, didn't know anything about the "second 'law' of thermodynamics" and "entropy". But at least today we should know better. Unfortunately entropy still seems to be some mystic thing to many, which to deal with should be avoided. (Knowing about entropy also increases responsibility. Some like to avoid that as well.)

You can't "avoid" entropy. Entropy is something very real: E.g. in broadband transmission the cost (e.g. chip size, power dissipation, heat generation) of managing entropy is almost proportional to the amount of entropy, which is to be managed. And climate change also can be explained by the entropy accounting (entropy generation, import, export) of the biosphere and the clogging of the interfaces of the biosphere, which are required to get rid of the entropy generated within the biosphere.

Therefore we need comprehensible explanations for entropy. My personal interest is not so much in entropy itself, but in how teachers and authors manage to explain entropy. Arieh Ben-Naim manages to get rid of all the fuzz which comes with so many publications related to entropy. He really manages to demystify entropy. I think, there are two paths which one could select to explain entropy. One is within information theory, the other one uses statistical physics. Ben-Naim chose the second one and thus not only managed to demystify entropy, but also demystified statistical physics: From my point of view, you just need a high school degree in order to be able to comprehend his book. Or you even may be lucky to have a teacher, who uses this book in the final high school year.

Economists and social scientists could get some help from the book too in understanding, what entropy really means. Indicators like the inequality measures of Theil and Kolm are entropy measures. And Nicholas Georgescu Roegen will be easier to understand. (The book would have been helpful to him too.)

Besides its content, I also like the making of the little book from Arieh Ben-Naim. It got very nice illustrations. And they are not just nice, they also are helpful. Here scientific thinking comes together with simple love to make things beautiful. It seems, that good science also leads to good aesthetics.

Related to this book, I also recommend the publications of M.V.Volkenstein (like Physics and Biology), although they are mostly out of print.

In this book, Arieh Ben-Naim gives a very clear exposition of the Second Law of Thermodynamics--one of the most important principles of modern science. Moreover, the author lays out a compelling argument for why this "law" is not deeply mysterious, as is commonly believed, but natural and intuitive. The argument is based on an analogy with a class of easily understood dice games, which are analyzed in great detail. The essential features of the dice-game model--that is, the features which exhibit the exact behavior predicted by the Second Law--are then extrapolated to various real-world thermodynamic systems in a very lucid way. The book includes nice introductions to probability theory and information theory, although only the bare rudiments of these theories are needed to understand Ben-Naim's arguments. The author concludes the book with a critique of some commonly held views among the scientific community regarding the Second Law. In particular, it is argued that if the atomic theory of matter had preceded the formulation of the Second Law, rather than visa-versa, the Second Law would have been understood in a much more intuitive way from its inception, and thus would never have gained the near mythical status it has commanded over the past 150 years.

I consider this book to be of great value to be read by any scientist (I am one, hence I will not speak for non-scientists).

Anyone who learns entropy in terms of thermodynamics (that is, heat cycles) is done a horrible disservice. The microscopic view of entropy makes intuitive sense, and most people do get it. I, however, was stuck for the longest time in that I could not understand why the thermodynamic definitions of entropy were intuitively obvious, why they had to exist, from purely macroscopic reasoning. Ben-Naim clarifies, as most professors of this subject do not or probably do not know, that it is not possible to justify such equations based purely on reasoning.

Additionally, Ben-Naim describes entropy itself in terms of information theory. This is invaluable; it is far more rigorous than the naive "disorder" analogy. Anyone who has done more than just basic qualitative questions recognizes that the notion of "ordered" vs "disordered" is inherently fuzzy in examples of solvation. The value of using information theory to then discuss tempergy, or temperature in units of energy, is intuitively valuable.

Ben-Naim also discusses entropy as a generalized property from several different common views, which are equivalent. The argument of showing how a certain quantity, seemingly different because of the dependent variable, is actually logically the same is an argument familiar to physicists, helping to put the macroscopic notion of entropy put on firm footing, and not the "Where did this come from" basis of saying it is the contour integral of the change in heat per temperature.

If you are a scientist, you will fly through this book, and reap quick rewards. Chemists/Physicists will be already quite used to much of the material in the book, but the analysis of certain chapters (for me, 2, 6, and his epilogue 8) are invaluable, clearly spoken insights.

He also offers to send him an email if you are confused!

This is what science should be: writing done without pretension (in contrast to Atkins).

Entropy Demystified, by Prof. Arieh Ben-Naim, is an absolute gem! Using very simple, easily followed, "games" with dice, Ben-Naim delivers handsomely on his promise, made at the beginning of the book, to remove all mystery from the concept of entropy and to make the reader appreciate that entropy is not only not a mystery but it is nothing more than a consequence of common sense. No advanced mathematics is required, only some very basic concepts in probability and a feeling for "large" numbers, both of which are developed for the reader so that no advanced preparation is required.

As a physicist, I am well aware that many more of my colleagues than might care to admit it are not altogether comfortable with the notion of entropy and, unfortunately, share, and even perpetuate, some of the inappropriate interpretations that have become fashionable, such as that entropy is a measure of the disorder of a system. Putting aside the fact that "disorder" is an ill-defined concept, entropy is not always synonymous with what one might characterize as disorder, as Ben-Naim well illustrates in the last chapter of the book.

If you would really like to know, once and for all, what entropy really is, and to be certain beyond any doubt, this is the book for you. What I especially like about the book is that Ben-Naim has also lived up to the first law of good technical writing, which is that it is the author's duty and responsibility to consider the reader first and foremost. At every step of the way, Ben-Naim anticipates the next question likely to be in the mind of the reader and provides immediate clarification. It is almost as though Ben-Naim is there in the room with you providing immediate feedback on every detail. His ability to anticipate and respond to the needs of the reader in this way is a rare talent indeed that makes this book a sheer delight to read and assures that the promise to remove all mystery concerning entropy will be fulfilled by the time you reach the last chapter. Actually, by the time you reach the next-to-last chapter. The last chapter is reserved for some of Ben-Naim's personal reflections on entropy, itself fascinating reading, enhanced immeasurably by the understanding provided by the preceding chapters.

This book is not going to teach you thermodynamics or statistical mechanics, and is not intended to do so. It's sole purpose is to give you a clear and unambiguous understanding of what entropy really is. Prof. Ben-Naim has succeeded in spades.

Those who may be interested in a more "in depth" discussion of statistical mechanics based on information, perhaps as the next step after Entropy Demystified, can take a look at Ben-Naim's recent book "A farewell to Entropy; Statistical Thermodynamics based on Information."

No matter what your background, there is promise in a book that contains "An Introduction to Probability Theory, Information Theory, and All the Rest." And Arieh Ben-Naim delivers.

Call something The Second Law of Thermodynamics and it's bound to have a forbidding quality. Partly this is due to the use of the word "Law", and partly it's because scientists have been challenged by the Second Law since it was first formulated 150 years ago. But despite this quality even the nonscientist needs a passing familiarity with the law's basic principles to understand some of nature's greatest puzzlements: Why do whole eggs break and broken eggs never again become whole? Why does a drop of red food coloring loosed in a bowl of water always disperse but the dye in a pool of pink water never coalesces to form an isolated spot of pure red? And why do teenagers' rooms only get messier? Ben-Naim can't help you with the deepest of these mysteries -- you just have to accept the room situation -- but he does shed considerable light on the hows and whys of the Second Law and on the scientific debates that have long surrounded it.

Understanding the Second Law means understanding entropy and the counterintuitive rule that, left alone, the entropy in a system always increases. Counterintuitive because what else in the universe always increases? In a clearly argued presentation, Ben-Naim makes the case that entropy is best thought of as information and that rather than some of the more typical expressions (e.g., an untended system always leads to greater disorder), what actually increases in a system left to itself is the amount of information needed to fully and correctly describe the whereabouts and behavior of the particles atoms and molecules therein.

It would be silly for a layperson to say much more about what is obviously a nuanced subject, and Ben-Naim plainly states that the nature of entropy has produced diametrically opposing opinions even among Nobel Prize winning physicists. But Ben-Naim does nonetheless provide even the lay reader with invaluable tools for better appreciating aspects of the Second Law. Among these tools are discussions and illustrations of the truly BIG numbers involved in the workings of the Second Law -- numbers so big that without scientific shorthand they could not be written in their entirety in all the time available since time began (numbers of the 1,000,000,000,000,000,... variety).

When the effects of probability are then unleashed in the realm of such big numbers, Ben-Naim shows how big systems "always" stabilize around their most probable states (red dye diffusing to pink in a pool of water) and how rare will be the exceptions: Turn ten thousand coins all to show "heads" then give the whole lot a random toss. While it is possible that all ten thousand will fall so that each coin again shows heads, don't bet on it. The chance is so low, says Ben-Naim, that you probably wouldn't get them to show that one unique result even if you could flip the coins at the rate of a million times a second and were able to do this for the entire 15 billion years the universe has existed. Instead, what you're almost always likely to get is close to half the coins showing heads and close to half showing tails. Which, says Ben-Naim, is why the randomly moving molecules of red dye will "always" spread evenly throughout the pool and "never" again come together in their original single drop. And why -- because it takes more information to describe the location of the particles in the dispersed rather than the concentrated dye -- the entropy of the red-diffused-to-pink system has increased.

This coupling of clear explanation and vivid example goes a long way toward making the concepts Ben-Naim presents accessible. And while the lay reader is not apt to come away with a thorough understanding of why "the Boltzmann constant (k) should be expunged from the vocabulary of physics," he or she will undoubtedly gain a deeper insight into the way the world around us works and why we see it the way we do. And which is why everyone can benefit from this book.

Fifty years ago, Arieh Ben-Naim, as every student in a physics or chemistry class of that era, was mystified by his introduction to entropy and the second law of thermodynamics. Although he was a professor of chemistry before retiring 15 years ago, Ben-Naim has evidently not kept up with the teaching of those topics in current chemistry texts. Thus, he seems unaware that most general chemistry texts currently published in the US (16) and three in physical chemistry - most available from Amazon.com - now clearly and simply present entropy and the second law.

Therefore, his 217 pages of "Entropy Demystified" that are necessary to develop his personal viewpoint (an information theory variant, not present in a US undergraduate chemistry textbook) can be clarified by 3-4 pages in any of the chemistry texts listed with their ISBN numbers (for exact Amazon.com identification) in [...] at "May 2009". In fact, a conceptual summary of the second law and entropy for most interested readers can be abstracted from these texts in two sentences: "Energy of all types changes from being localized to become more dispersed, spread out, distributed in space (and abstractly, in more energy quantum states, microstates) if that energy is not constrained." Then, "entropy change is the quantitative measure of how much more widely distributed the initial energy becomes in a spontaneous process." Thus, in real processes, energy spreads out spatially. Probabilistic methods are one way of quantifying thermodynamic entropy.

Unfortunately, Professor Ben-Naim's fundamental error, summarized on page 204 but weakening all previous pages, is his misinterpretation of what occurs in real systems of molecules, especially in the simple isothermal expansion of ideal gases or in their "mixing"/expansion. These cases have misled him to focus on their particular lack of change in the total energy of the system, rather than on what is the fundamental cause of all thermodynamic entropy change: the increased spreading of the initial energy of actual molecules in space when constraints are removed - e.g., their spontaneously moving into a greater volume from a smaller volume (with unchanged energy) in a process such as expansion/mixing. This is what traditional thermodynamic entropy readily measures and, as just stated, can be readily understood.

The disconnect between information and the second law is stressed on page 203 by "a measure of information cannot be used to explain the Second Law of Thermodynamics." This is true, indeed. The connection between the second law and information is tenuous. Contrast this with the modern view in beginning collegiate chemistry texts, e.g. "whenever a product-favored chemical or physical process occurs, energy becomes more dispersed...This is summarized in the second law of thermodynamics, which states that the total entropy of the universe ... is continually increasing." (Moore, Stanitski, and Jurs; 3rd edition.) A physical chemistry text that is used world-wide states "...the Second Law of thermodynamics, may also be expressed in terms of another state function, the entropy, S. ...entropy...is a measure of the energy dispersed in a process..." (Atkins and de Paula, 8th edition.)

The connection between spontaneous chemical reactions or physical processes, dispersal of energy, and entropy is integral, tight, and generally accepted. It does not require 200 pages to justify.

Arieh Ben-Naim, a distinguished scientist from the field of solution physical chemistry, guides readers to grasp basic principles of the second law of thermodynamics. In the book the author provides clear examples of how widely prevailed use of 'increase in disorder' to explain the underlying microscopic mechanism of the second law can be subjective and misleading. The statements made in this regard are reinforced by his decades of incomparable contribution to the understanding of hydrophobicity. One may recognize longstanding controversial aspects in the interpretation of the second law from the author's conclusion of the necessity of changing the unit of the absolute temperature. The book is written in a lucid manner, which can be done only by individuals who understand the essence of the subject in depth. It is telling we may need to go back at least once to the simplest question after having worked on the every possible detail. Thorough repetition of examples as introduced in the book may be a necessary attitude to tackle on any most difficult subject. The book is recommended not only to readers in fundamental physics and chemistry but to ones in biologically related science.