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To appear in the April 2000 issue of PHYSICS TODAY (together with 4
figures, not included here).
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Thermodynamics, Second law, Entropy
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\begin{document}
\title{\bf{
A Fresh Look at Entropy and the Second Law of Thermodynamics}}
\author{\vspace{5pt} Elliott H.~Lieb$^1$, and Jakob
Yngvason$^{2}$\\
\vspace{-4pt}\small{$1.$ Departments of Physics and Mathematics, Jadwin
Hall,} \\
\small{Princeton University, P.~O.~Box 708, Princeton, New Jersey
08544}\\
\vspace{-4pt}\small{$2.$ Institut f\"ur Theoretische Physik, Universit\"at
Wien}\\
\small{Boltzmanngasse 5, A 1090 Vienna, Austria}}
\date{}
\maketitle
\begin{abstract}The existence of entropy and its increase can be understood without
reference to either statistical mechanics or heat engines.
\end{abstract}
\renewcommand{\thefootnote} {}
\footnotetext{To be published in PHYSICS TODAY, April 2000
}
In days long gone, the second law of thermodynamics (which predated
the first law) was regarded as perhaps the most perfect and
unassailable law in physics. It was even supposed to have
philosophical import: It has been hailed for providing a proof of the
existence of God (who started the universe off in a state of low
entropy, from which it is constantly degenerating); conversely, it has
been rejected as being incompatible with dialectical materialism and
the perfectability of the human condition.
Alas, physicists themselves eventually demoted it to a lesser position
in the pantheon---because (or so it was declared) it is ``merely''
statistics applied to the mechanics of large numbers of atoms.
Willard Gibbs wrote: ``The laws of thermodynamics may easily be
obtained from the principles of statistical mechanics, of which they
are the incomplete expression'' [1]---and Ludwig Boltzmann expressed
similar sentiments.
Is this really so? Is it really true that the second law is merely an
``expression'' of microscopic models or could it exist in a world that
was featureless at the $10^{-8}$ cm level? We know that statistical
mechanics is a powerful tool for understanding physical phenomena and
calculating many quantities, especially in systems at or near
equilibrium. We use it to calculate entropy, specific and latent
heats, phase transition properties, transport coefficients and so on,
often with good accuracy. Important examples abound, such as Max
Planck's realization that by staring into a furnace he could find
Avogadro's number or Linus Pauling's highly accurate
back-of-the-envelope calculation of the residual entropy of ice. But
is statistical mechanics essential for the second law?
In any event, it is still beyond anyone's computational ability
(except in idealized situations) to account for a very precise,
essentially infinitely accurate law of physics from statistical
mechanical principles. No exception has ever been found to the second
law of thermodynamics---not even a tiny one. Like conservation of
energy (the ``first'' law) the existence of a law so precise and so
independent of details of models must have a logical foundation that
is independent of the fact that matter is composed of interacting
particles. Our aim here is to explore that foundation. The full
details can be found in [2].
As Albert Einstein put it, ``A theory is the more impressive the
greater the simplicity of its premises is, the more different kinds of
things it relates, and the more extended is its area of applicability.
Therefore the deep impression which classical thermodynamics made upon
me. It is the only physical theory of universal content concerning
which I am convinced that, within the framework of the applicability
of its basic concepts, it will never be overthrown'' [3].
In an attempt to reaffirm the Second Law as a pillar of physics in its
own right, we have returned to a little noticed movement that began in
the 1950's with the work of Peter Landsberg [4], Hans Buchdahl [5],
Gottfried Falk, Herbert Jung [6], and others (see [2] for references)
and culminated in the book of Robin Giles [7], which must be counted
one of the truly great, but unsung works in theoretical physics. It
is in these works that the concept of ``comparison'' (explained below)
emerges as one of the key underpinnings of the second law. The
approach of these authors is quite different from lines of thought in
the tradition of Sadi Carnot that base thermodynamics on the
efficiency of heat engines. (See [8], for example, for a modern exposition of
the latter.)
\section*{The basic question}
The paradigmatic event that the second law deals with can be described
as follows. Take a macroscopic system in an equilibrium state
$X$ and place it in a room, together with a gorilla equipped with
arbitrarily complicated machinery (a metaphor for the rest of the
universe), and a weight---and close the door. As in the old
advertisement for indestructible luggage, the gorilla can do anything
to the system---including tearing it apart. At the end of the day,
however, the door is opened and the system is found in some other
equilibrium state, $Y$, the gorilla and machinery are found in their
original state, and the only other thing that has possibly changed is
that the weight has been raised or lowered. Let us emphasize that
although our focus is on equilibrium states, the processes that take
one such state into another can be arbitrarily violent. The gorilla
knows no limits. (See figure 1.)
The question that the second law answers is this: What distinguishes
those states $Y$ that can be reached from $X$ in this manner from
those that cannot? The answer: There is a function of the equilibrium
states, called entropy and denoted by $S$, that characterizes
the possible pairs of equilibrium states $X$ and $Y$ by the inequality
$S(X)\leq S(Y)$. The function can be chosen to be additive (in a
sense explained below), and with this requirement it is unique,
up to a change of scale. Our main point is that the existence of
entropy relies only on a few basic principles, independent of any
statistical model---or even of atoms.
What is exciting about this apparently innocuous statement is the
uniqueness of entropy, for it means that all the different methods for
measuring or computing entropy must give the same answer. The usual
textbook derivation of entropy as a state function, starting with some
version of ``the second law'', proceeds by considering certain slow,
almost reversible processes (along adiabats and isotherms). It is not
at all evident that a function obtained in this way can contain any
information about processes that are far from being slow or
reversible. The clever physicist might think that with the aid of
modern computers, sophisticated feedback mechanisms, unlimited amounts
of mechanical energy (represented by the weight) and lots of plain
common sense and funding, the system could be made to go from an
equilibrium state $X$ to $Y$ that could not be achieved by the
primitive quasistatic processes used to define entropy in the first
place. This cannot happen, however, no matter how clever the
experimentalist or how far from equilibrium one travels!
What logic lies behind this law? Why can't one gorilla undo what
another one has wrought? The atomistic foundation of this logic is
not as simple as is often suggested. It not only concerns things like
the enormous number of atoms involved ($10^{23}$), but also other
aspects of statistical mechanics that are beyond our present
mathematical abilities. In particular, the interaction of a system
with the external world (represented by the gorilla and machinery)
cannot be described in any obvious way by Hamiltonian mechanics.
Although irreversibility is an important open problem in statistical
mechanics, it is fortunate that the logic of thermodynamics itself is
independent of atoms and can be understood without knowing its source.
The founders of thermodynamics---Rudolf Clausius, Lord Kelvin, Max
Planck, Constantin Carath\'eodory, and so on---clearly had
transitions between equilibrium states in mind when they stated the
law in sentences such as ``No process is possible, the sole result of
which is that a body is cooled and work is done'' (Kelvin). Later it
became tacitly understood that the law implies a continuous increase
in some property called entropy, which was supposedly defined for
systems out of equilibrium. The ongoing, unsatisfactory debates (see
referce [9, for example) about the definition of this nonequilibrium
entropy and whether it increases shows, in fact, that what is
supposedly ``easily'' understood needs clarification. Once again, it
is a good idea to try to understand first the meaning of entropy for
equilibrium states---the quantity that our textbooks talk about when
they draw Carnot cycles. In this article we restrict our attention to
just those states; by ``state'' we always mean ``equilibrium state''.
Entropy, as the founders of thermodynamics understood the quantity is
subtle enough, and it is worthwhile to understand the ``second law''
in this restricted context. To do so it is not necessary to decide
whether Boltzmann or Gibbs had the right view on irreversibility.
(Their views are described in Joel L. Lebowitz's article ``Boltzmann's
Entropy and Time's Arrow'', Physics Today, September 1993, page 32.)
\section*{The basic concepts}
To begin at the beginning, we suppose we know what is meant by a
thermodynamic system and equilibrium states of such a system.
Admittedly these are not always easy to define, and there are
certainly systems, such as a mixture of hydrogen and oxygen or an
interstellar ionized gas, that can behave as if they are in
equilibrium even if it is not truly so. The prototypical system is a
``simple system'', consisting of a substance in a container with a
piston. But a simple system can be much more complicated than that.
Besides its volume it can have have other coordinates, which can be
changed by mechanical or electrical means---shear in a solid, or
magnetization, for example. In any event, a state of a simple system
is described by a special coordinate $U$, which is its energy, and one
or more other coordinates (such as the volume $V$) called work
coordinates. An essential point is that the concept of energy, which
we know about from the moving weight and Newtonian mechanics, can be
defined for thermodynamic systems. This fact is the content of the
first law of thermodynamics.
Another type of system is a ``compound system'', which consists of
several different or identical independent, simple systems. By means
of mixing or chemical reactions, systems can be created or destroyed.
Let us briefly discuss some concepts that are relevant for systems and
their states, which are denoted by capital letters such as
$X,X',Y,\ldots$. Operationally, the composition, denoted $(X,X')$, of
two states $X$ and $X'$ is obtained simply by putting one system in a
state $X$ and one in a state $X'$ side by side on the experimental
table and regarding them jointly as a state of a new, compound system.
For instance, $X$ could be a glass containing 100 g of whiskey at
standard pressure and $20^\circ$ C, and $X'$ a glass containing 50 g
of ice at standard pressure and $0^\circ$ C. To picture $(X,X')$ one
should think the two glasses standing on a table without touching each
other. (See figure 2.)
Another operation is the ``scaling'' of a state $X$ by a factor
$\lambda>0$, leading to a state denoted $\lambda X$. Extensive
properties like mass, energy and volume are multiplied by $\lambda$,
while intensive properties such as pressure stay intact. For the
states $X$ and $X'$ as in the example above the example above ${1\over
2} X$ is 50 g of whiskey at standard pressure and $20^\circ$ C, and
${1\over 5}X'$ is 10 g of ice at standard pressure and $0^\circ$ C.
Compound systems scale in the same way: ${1\over 5}(X,X')$ is 20 g of
whiskey and 10 g of ice in separate glasses with pressure and
temperatures as before.
A central notion is adiabatic accessibility. If our gorilla can take
a system from $X$ to $Y$, as described above---that is, if the only
net effect of the action, besides the state change of the system, is
that a weight has possibly been raised or lowered, we say that $Y$ is
adiabatically accessible from $X$ and write $X\prec Y$ (the symbol
$\prec$ is pronounced ``precedes''). It has to be emphasized that for
macroscopic systems this relation is an absolute one: If a transition
from $X$ to $Y$ is possible at one time, then it is {\it always}
possible (that is, it is reproducible), and if it is impossible at one
time it {\it never} happens. This absolutism is guaranteed by the
large powers of 10 involved---the impossibility of a chair's
spontaneous jumping up from the floor is an example.
\section*{ The role of entropy}
Now imagine that we are given a list of all possible pairs of states
$X,Y$ such that $X\prec Y$. The foundation on which thermodynamics
rests, and the essence of the second law, is that this list can be
simply encoded in an {entropy function $S$ on the set of all states of
all systems (including compound systems) so that when $X$ and $Y$ are
related at all, then
$$
X\prec Y \ \ \hbox{\rm if and only if}\ \ S(X) \leq S(Y) \ .
$$
Moreover, the entropy function can be chosen in such a way that
if $X$ and $X'$ are states of two (different or identical)
systems, then the entropy of the compound system in this pair of states
is given by
$$ S(X,X') = S(X) + S(X').
$$
This additivity of
entropy is a highly nontrivial assertion. Indeed, it is one of the
most far reaching properties of the second law. In compound systems
such as the whiskey/ice example above, all states $(Y,Y')$ such that
$X\prec Y$ and $X'\prec Y'$ are adiabatically accessible from $(X,X')$.
For instance, by letting a falling weight run an electric generator one
can stir the whiskey and also melt some ice. But it is important to
note that $(Y,Y')$ can be adiabatically accessible from $(X,X')$
without $Y$ being adiabatically accessible from $X$. Bringing the two
glasses into contact and separating them again is adiabatic for the
compound system but the resulting cooling of the whiskey is not
adiabatic for the whiskey alone. The fact that the inequality
$S(X)+S(X')\leq S(Y)+S(Y')$ {\it exactly} characterizes the possible
adiabatic transitions for the compound system, even when $S(X)\geq
S(Y)$, is quite remarkable. It means that it is sufficient to know the
entropy of each part of a compound system in order to decide which
transitions due to interactions between these parts (brought about
by the gorilla) are possible.
Closely related to additivity is extensivity, or scaling of entropy,
$$S(\lambda X)=\lambda S(X),$$ which means that the entropy of an
arbitrary mass of a substance is determined by the entropy of some
standard reference mass, such as 1 kg of the substance. Without this
property engineers would have to use different steam tables each time
they designed a new engine.
In traditional presentations of thermodynamics, based for example on
Kelvin's principle given above, entropy is arrived at in a rather
roundabout way which tends to obscure its connection with the relation
$\prec$. The basic message we wish to convey is that existence and
uniqueness of entropy are equivalent to certain simple properties of
the relation $\prec$. This equivalence is the concern of [2].
An analogy leaps to mind: When can a vector-field, ${\bf E}(x)$, be
encoded in an ordinary function (potential), $\phi(x)$, whose gradient
is ${\bf E}$? The well-known answer is that a necessary and
sufficient condition is that ${\rm curl}\, {\bf E} =0$. The
importance of this encoding does not have to be emphasized to
physicists; entropy's role is similar to the potential's role and the
existence and meaning of entropy are not based on any formula such as
$S=-\Sigma_i p_i\ln p_i$, involving probabilities $p_i$ of
``microstates''. Entropy is derived (uniquely, we hope) from the list
of pairs $X\prec Y$; our aim is to figure out what properties of this
list (analogous to the curl-free condition) will allow it to be
described by an entropy. That entropy will then be endowed with an
unambiguous physical meaning independent of anyone's assumptions about
``the arrow of time'', ``coarse graining'' and so on. Only the list,
which is given by physics, is important for us now.
The required properties of $\prec$ do {\it not} involve concepts like
``heat'' or ``reversible engines'', not even ``hot'' and ``cold'' are
needed. Besides the ``obvious'' conditions ``$X\prec X$ for all $X$''
(reflexivity) and ``$X\prec Y$ and $Y\prec Z$ implies $X\prec Z$''
(transitivity) one needs to know that the relation behaves reasonably
with respect to the composition and scaling of states. By this we
mean the following:
\begin{itemize} \item[{$\bullet$}] Adiabatic accessibility is
consistent with the composition of states: $X\prec Y$ and $Z\prec W$
implies $(X,Z)\prec (Y,W)$. \item[{$\bullet$}] Scaling of states does not
affect adiabatic accessibility: If $X\prec Y$, then $\lambda X\prec
\lambda Y$. \item[{$\bullet$}] Systems can be cut adiabatically into two
parts: If $0<\lambda<1$, then $X\prec ((1-\lambda)X,\lambda X)$, and
the recombination of the parts is also adiabatic:
$((1-\lambda)X,\lambda X)\prec X$. \item[{$\bullet$}] Adiabatic accessibility
is stable with respect to small perturbations: If $(X,\varepsilon
Z)\prec (Y, \varepsilon W)$ for arbitrarily small $\varepsilon>0$,
then $X\prec Y$. \end{itemize}
These requirements are all very natural. In fact, in traditional
approaches they are usually taken for granted, without mention. They
are not quite sufficient, however, to define entropy. A crucial
additional ingredient is the comparison hypothesis for the relation
$\prec$. In essence, this is the hypothesis that all equilibrium
states, simple or compound, can be grouped into classes, such that if
$X$ and $Y$ are in the same class, then either $X\prec Y$ or $Y\prec
X$. In nature, a class consists of all states with the same mass and
chemical composition---that is, with the same amount of each of the
chemical elements. If chemical reactions and mixing processes are
excluded, the classes are smaller and may be identified with the
``systems'' in the usual parlance. But it should be noted that
systems can be compound, or consist of two or more vessels of
different substances. In any case, the role of the comparison
hypothesis is to insure that the list of pairs $X\prec Y$ is
sufficiently long. Indeed, we shall give an example later where the
list of pairs satisfies all the other axioms, but which is {\it not}
describable by an entropy function.
\section*{The construction of entropy}
Our main conclusion (which we do not claim isobvious, but whose proof
can be found in reference [2]) is that the existence and uniqueness of
entropy is a consequence of the comparison hypothesis and the
assumptions about adiabatic accessibility stated above. In fact, if
$X_0$, $X$ and $X_1$ are three states of a system and $\lambda$ is any
scaling factor between 0 and 1, then either $X\prec ((1-\lambda)
X_0,\lambda X_1)$ or $((1-\lambda) X_0,\lambda X_1)\prec X$ must be
true, by the comparison hypothesis. If {\it both} alternatives hold,
then the properties of entropy demand that
$$
S(X)=(1-\lambda) S(X_0)+\lambda S(X_1).
$$
If $S(X_0)\neq S(X_1)$ this equality can hold for at most one
$\lambda$. With $X_0$ and $X_1$ as reference states, the entropy is
therefore {\it fixed}, apart from two free constants, namely the values
$S(X_0)$ and $S(X_1)$.
{}From the properties of the relation $\prec$ listed above, one can
show that there is, indeed, always a $0\leq \lambda\leq 1$ with the
required properties, provided that $X_0\prec X\prec X_1$. It is the
{\it largest} $\lambda$, denoted $\lambda_{\rm max}$, such that
$((1-\lambda) X_0,\lambda X_1)\prec X$. Defining the entropies of the
reference states arbitrarily as $S(X_0)=0$ and $S(X_1)=1$ unit, we
obtain the following simple {\it formula for entropy}:
$$
S(X)=\lambda_{\rm max}\ \hbox{\rm units}.
$$
The scaling factors $(1-\lambda)$ and $\lambda$ measure the amount of
substance in the states $X_0$ and $X_1$ respectively. The formula for
entropy can therefore be stated in the following words: $S(X)$ is the
maximal fraction of substance in the state $X_{1}$ that can be
transformed adiabatically} (that is, in the sense of $\prec$) into the
state $X$ with the aid of a complementary fraction of substance in the
state $X_{0}$. This way of measuring $S$ in terms of substance is
reminiscent of an old idea, suggested by Pierre Laplace and Antoine
Lavoisier, that heat be measured in terms of the amount of ice melted
in a process. As a concrete example, let us assume that $X$ is a
state of liquid water, $X_{0}$ of ice and $X_{1}$ of vapor. Then
$S(X)$ for a kilogram of liquid, measured with the entropy of a
kilogram of water vapor as a unit, is the maximal fraction of a
kilogram of vapor that can be transformed adiabatically into liquid in
state $X$ with the aid of a complementary fraction of a kilogram of
ice. (See figure 3.)
In this example the maximal fraction $\lambda_{\rm max}$ cannot be
achieved by simply exposing the ice to the vapor, causing the former
to melt and the latter to condense. This would be an irreversible
process---that is, it would not be possible to reproduce the initial
amounts of vapor of ice adiabatically (in the sense of the definition
given earlier) from the liquid. By contrast, $\lambda_{\rm max}$ is
uniquely determined by the requirement that one can pass adiabatically
from $X$ to $((1-\lambda_{\rm max})X_{0}, \lambda_{\rm max}X_{1})$
{\it and} vice versa. For this transformation it is necessary to
extract or add energy in the form of work---for example by running a
little reversible Carnot machine that transfers energy between the
high-temperature and low-temperature parts of the system (see figure
3). We stress, however, that neither the concept of a ``reversible
Carnot machine" nor that of ``temperature" is needed for the logic
behind the formula for entropy given above. We mention these concepts
only to relate our definition of entropy to concepts for which the
reader may have an intuitive feeling.
By interchanging the roles of the three states, the definition of
entropy is easily extended to situations where $X\prec X_{0}$ or
$X_{1}\prec X$. Moreover, the reference points $X_{0}$ and $X_{1}$,
where the entropy is defined to be 0 and 1 unit respectively, can be
picked consistently for different systems such that the entropy will
satisfy the crucial additivity and extensivity conditions
$$S(X,X')=S(X)+S(X')\qquad {\rm and}\qquad S(\lambda X)=\lambda S(X).$$
It is important to understand that once the existence and uniqueness
of entropy has been established one need not rely on the $\lambda_{\rm
max}$ formula displayed above to determine it in practice. There are
various experimental means to determine entropy that are usually much
more practical. The standard method consists of measuring pressures,
volumes and temperatures (on some empirical scale), as well as
specific and latent heats. The empirical temperatures are converted
into absolute temperatures $T$ (by means of formulas that follow from
the mere existence of entropy but do not involve $S$ directly), and
the entropy is computed by means of formulas like $\Delta S=\int
(dU+PdV)/T$, with $P$ the pressure. The existence and uniqueness of
entropy implies that this formula is independent of the path of
integration.
\section*{Comparability of states}
The possibility of defining entropy entirely in terms of the relation
$\prec$ was first clearly stated by Giles [7]. (Giles's definition is
different from ours, albeit similar in spirit.) The importance of the
comparison hypothesis had been realized earlier, however [4, 5, 6].
All these authors take the comparison hypothesis as a
postulate---that is, they do not attempt to justify it from other simpler premises.
However, it is in fact possible to {\it derive} comparability for any
pair of states of the same system from some natural and directly
accessible properties of the relation $\prec$ [2]. In this derivation
of comparison the customary parametrization of states in terms of
energy and work coordinates is used, but it has to be stressed that
such parametrizations are irrelevant, and therefore not used, for our
definition of entropy---once the comparison
hypothesis is established.
To appreciate the significance of the comparison hypothesis it may be
helpful to consider the following example. Imagine a world whose
thermodynamical sytems consist exclusively of incompressible solid
bodies. Moreover, all adiabatic state changes in this world are
supposed to be obtained by means of the following elementary
operations:
\begin{itemize}\item[{$\bullet$}]Mechanical rubbing of
the individual systems, increasing their energy.
\item[{$\bullet$}]Thermal equilibration in the conventional sense (by
bringing the systems into contact.)\end{itemize}
The state space of the compound system consisting of two identical
bodies, 1 and 2, can be paramertized by their energies, $U_{1}$ and
$U_{2}$. Figure 4 shows two states, $X$ and $Y$ of this compound
system, and the states that are adiabatically accessible from eachof
these states. It is evident from the picture that neither $X\prec Y$
nor $Y\prec X$ holds. The comparison hypothesis is therefore violated
in this hypothetical example, and it is not possible to characterize
adiabatic accessibility by means of an additive entropy function. A
major part of our work consists of understanding why such situations
do not happen---why the comparison hypothesis appears to be true in
the real world.
The derivation of the comparison hypothesis is based on an analysis of
simple systems, which are the building blocks of thermodynamics. As
already mentioned the states of such systems are described by one
energy coordinate $U$ and at least one work coordinate, like the
volume $V$. The following concepts play a key role in this analysis:
\begin{itemize} \item[{$\bullet$}] The possibility of
forming ``convex combinations'' of states of simple systems with
respect to the energy $U$ and volume $V$ (or other work coordinates).
This means that given any two states $X$ and $Z$ of one kilogram of
our system one can pick any state $Y$ on the line between them in
$U$, $V$ space and, by taking appropriate fractions $\lambda$ and
$1-\lambda$ in states $X$ and $Z$, respectively, there will be an
adiabatic process taking this pair of states into the state $Y$. This
process is usually quite elementary. For example, for gases and
liquids one need only remove the barrier that separates the two
fractions of the system. The fundamental property of entropy increase
will then tell us that $S(Y)\geq \lambda S(X)+(1-\lambda)S(Z)$. As
Gibbs emphasized, this ``concavity'' is the basis for thermodynamical
stability---namely positivity of specific heats and
compressibilities. \item[{$\bullet$}] The existence of
at least one irreversible adiabatic state change, starting from
any given state. In conjuction with concavity of $S$ this seemingly
weak requirement excludes the possibility that the entropy is constant
in a whole neighborhood of some state. The classical formulations of
the second law follow from this. \item[{$\bullet$}] The concept of
thermal equilibrium between simple systems, which means,
operationally, that no state changes takes place when the systems are
allowed to exchange energy with each other at fixed work coordinates.
The zeroth law of thermodynamic says that if two systems are in
thermal equilibrium with a third, then they are in thermal
equilibrium with one another. This property is essential for the
additivity of entropy, because it allows a consistent adjustment of
the entropy unit for different systems. It leads to a definition of
temperature by the usual formula $1/T=(\partial S/\partial
U)_{V}$.\end{itemize}
Using these notions (and a few others of a more technical nature) the
comparison hypothesis can be established for all simple systems and
their compounds.
It is more difficult to justify the comparability of states if mixing
processes or chemical reactions are taken into account. In fact,
although a mixture of whiskey and water at $0^\circ$ C is obviously
adiabatically accessible from separate whiskey and ice by pouring
whiskey from one glass onto the rocks in the other glass, it is not
possible to reverse this process adiabatically. Hence it is not clear
that a block of a frozen whiskey/water mixture at $-10^\circ$ C, say,
is at all related in the sense of $\prec$ to a state in which whiskey and
water are in separate glasses. Textbooks usually appeal here to {\it
gedanken} experiments with ``semipermeable
membrane'' that let only water molecules through and withhold the
whiskey molecules, but such membranes really exist only in the mind
[10].
However, without invoking any such device, it turns out to be possible
to shift the entropy scales of the various substances in such a way
that $X\prec Y$ always implies $S(X)\leq S(Y)$. The converse assertion,
namely, $S(X)\leq S(Y)$ implies $X\prec Y$ provided $X$ and $Y$
have the same chemical composition, cannot be guaranteed {\it a priori}
for mixing and chemical reactions, but
it is empirically testable and appears to be true in the real world.
This aspect of the second law, comparability, is not usually
stressed, but it is important; it is challenging to figure out how to
turn the frozen whiskey/water block into a glass of whiskey and a
glass of water without otherwise changing the universe, except for
moving a weight, but such an adiabatic process is possible.
\section*{What has been gained?}
The line of thought that started more than forty years ago has led to
an axiomatic foundation for thermodynamics. It is appropriate to ask
what if anything has been gained compared to the usual approaches
involving quasi-static processes and Carnot machines on the one hand
and statistical mechanics on the other hand. There are several
points. One is the elimination of intuitive, but hard-to-define
concepts like ``hot'', ``cold'' and ``heat''. Another is the
recognition of entropy as a codification of possible state changes,
$X\prec Y$, that can be accomplished without changing the rest of the
universe in any way except for moving a weight. Temperature is
eliminated as an {\it a priori} concept and appears in its natural
place as a quantity derived from entropy and whose consistent
definition really depends on the existence of entropy, rather than the
other way around. To define enetropy, there is no need for special
machines and processes on the empirical side, and there is no need for
assumptions about models on the statistical mechanical side. Just as
energy conservation was eventually seen to be a consequence of time
translation invariance, in like manner entropy can be seen to be a
consequence of some simple properties of the list of state pairs
related by adiabatic accessibility.
If the second law can be demystified, so much the better. If it can be
seen to be a consequence of simple, plausible notions then, as Einstein
said, it cannot be overthrown.
\section*{Acknowledgements} We are grateful to Shivaji Sondhi and Roderich Moessner for helpful
suggestions. Lieb's work was supported by NSF grant PHY 9820650.
Yngvason's work was supported by
the Adalsteinn Kristj\'ansson Foundation and the University of Iceland.
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