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.