Universal restrictions to the conversion of heat into work derived from the analysis of the Nernst theorem as a uniform limit
Jos\'{e} Mar\'{\i}a \surname{Mart\'{\i}n-Olalla}
The relationship between the Nernst theorem and the Kelvin-Planck statement of the second law is revisited. We put forward the fact that the exchange of entropy is uniformly vanishing as the temperature goes to zero. The analysis of this assumption shows that is equivalent to fact that the compensation of a Carnot engine scales with the absorbed heat so that the Nernst theorem should be embedded in the statement of the second law.
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\newcommand{\mecanica}{\ensuremath{x}} %Symbol for mechanical properties
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\begin{document}

\title[Universal restrictions to...]{Universal restrictions to the conversion of heat into work derived from the analysis of the Nernst theorem as a uniform limit}
\author{Jos\'{e} Mar\'{\i}a \surname{Mart\'{\i}n-Olalla}} 
\email{olalla@us.es}
\author{Alfredo \surname{Rey de Luna}}
\affiliation{Departamento de F\'{\i}sica de la Materia Condensada. Universidad de Sevilla. \\ Ap. Correos 1065 E-41080Sevilla. Spain}
\preprint{J.  Phys. A: Math. Gen. \textbf{36}(29) (2003) 7909--7921}

\begin{abstract}
The relationship between the Nernst theorem and the Kelvin-Planck statement of the second law is revisited. We put forward the fact that the exchange of entropy is uniformly vanishing as the temperature goes to zero. The analysis of this assumption shows that is equivalent to fact that the compensation of a Carnot engine scales with the absorbed heat so that the Nernst theorem should be embedded in the statement of the second law.
\end{abstract}
\pacs{05.70.-a,05.70.Ce}%, 11.30.-j, 65.40}
\keywords{First law; Third law; Foundations of Thermodynamics; Second Law; Carnnot engine; Ideal Gases}


\maketitle


\section{Introduction}
\label{sec:intro}
The classical formulation of the Kelvin-Planck statement of the second law reads:\cite[page~89]{planck-1927}
\begin{quote}
``it is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir''
\end{quote}
Some other, essentially equivalent formulations of the law are possible ---Kelvin\cite{thomson-1853}, Clausius\cite{clausius-1854} and Carath\'eodory\cite{caratheodory-1909} statements--- but, for our purpose, we will refer to the formulation posed above. Essentially the statement requires  the presence of another  reservoir.

The development of the law needs the concept of  ``working fluid'', the substance that undergoes the cyclic process. The properties of the working fluid are usually discarded because the initial and final state of the fluid coincides and ``it has done  service  only as a transmitting agent in order to bring about the changes in the surroundings''\cite[page~68]{planck-1927}. Nonetheless, the fluid \emph{must} be able to do ``service'' in the way required.

We will show in this work that a general property of the matter will be restricting the ability of working fluids to perform cycles thus restricting what the  Kelvin-Planck statement allows. We will show that, in fact, such restriction follows from a comprehensive interpretation of the statement posed above.

The general property we are speaking about is nowadays known as the third law of thermodynamics. The necessity and character this law has been a matter of discussion from the early years of 1900's. Some chemical problems lead  Nernst\cite{nernst-1906,nernst-1924} to discover his \emph{heat theorem} which reads\cite[page~85]{nernst-1924}:
\begin{quote}
``in the neighborhood of the absolute zero \emph{all} processes proceed without alteration of entropy''
\end{quote}
The theorem ---which classically does not follow from the Kelvin-Planck statement\cite{einstein-1921,epstein-1937}--- is supported by a formidable array of experimental data. We choose this elder ---though very valuable--- version for the reasons that will be disclosed in \sref{sec:space} but we are seeing no particular reason for this statement to have been forgotten other than it refers to properties of processes rather than to properties of systems as it is nowadays stated\cite{kestin-1968ii,landau-lifshitz-1968,landsberg-78,baierlein-99,zemansky-1968}.

Nernst derived the theorem  from two quite general observations. The first one is the so-called \emph{principle of unattainability of the zero isotherm} which recalls the fact that no process can diminish the temperature of a system to the absolute zero. The second one is the fact that the specific heat of substances goes to zero as the temperature goes to zero.  

It should be acknowledge that Planck\cite{planck-1927} noticed that these observations should have lead to a ``more comprehensive'' conclusion: ``as the temperature diminishes indefinitely the entropy of a chemical homogeneous body of finite density approaches indefinitely to a definite value, which is independent of the pressure, the state of aggregation and of the special chemical modification.'' The Planck formulation avoids that $\Delta S\to0$ while $S\to-\infty$ as $T\to0$ and thus expresses that the absolute value of the entropy is bounded in the absolute zero.

Yet, our work will be just related to the analysis of $\Delta S$ since it just deals with the analysis of the conversion of heat into work. That problem is insensitive to a translation of the value of the entropy and thus, Planck's formulation lies out of our scope of interest. The same can be said about the vanishing of the specific heats.


Modern approaches and presentations of the third law of thermodynamics usually relates it to the microscopic properties of systems under consideration\cite{balian-91,huang-87,mafe-ajp-98,roseinnes-ajp-99,wu-pre-98}. Some efforts also try to clarify its macroscopic meaning\cite{falk-pr-1959,yan-chen-jpa-88,landsberg-jpa-89,oppenheim-jpa-89,liboff-physicsessays-94,landsberg-ajp-97,belgiorno-pra-03a}.


The goal of this paper is a revision of the mathematical description of the statement of the Nernst theorem posed above as well as its physical consequences. We will study pure macroscopic observations in the field of low temperature physics. In so doing no hypothesis about the constitution of the systems under study will be considered. 

Also, it is a burden of this manuscript an energetic analysis of the consequences of the statement posed above. To put it shortly, the simplest, most efficient engine contains two processes in which entropy is altered. This kind of processes are restricted at the neighborhood of absolute zero by the Nernst theorem; we will proof that such restriction will lead to a further condition  that any engine must satisfy.



\section{Limitations to the description of the Nernst theorem}
\label{sec:limit}

The classical formulation of the Nernst theorem reads:\cite{kestin-1968ii} ``the change in entropy associated with any isothermal process between two states of a system in internal equilibrium vanishes in the limit of zero temperature,'' which is usually translated into the mathematical condition:
\begin{equation}
\label{eq:weak}
    \forall\mecanica_1\mecanica_2\in\mathcal{D}\quad\lim_{T\to0^+}[S(T,\mecanica_1)-S(T,\mecanica_2)]=0%\Longleftrightarrow
\end{equation}
where $\mecanica$ is any mechanical variable such as volume, pressure or magnetic field and $\mathcal{D}\subseteq\mathbb{R}$ is its domain of definition.\footnote{Nernst himself clothed \eref{eq:weak} through the statement posed in  \sref{sec:intro}.\cite[page~85]{nernst-1924}} 

Landau and Lifshitz\cite{landau-lifshitz-1968} showed the importance of keeping $\mecanica_1,\mecanica_2$ fixed in \eref{eq:weak}. Otherwise, they said, if, for instance, $\mecanica_1$ goes to infinity, the theorem may no be valid. Quite generally it could be said that that description works fine if the values of $\mecanica$ are indeed fixed but it comes into trouble in case one looks for double limits of the form $T\to0$ and $\mecanica\to\infty$.

An academic example of this problem is provided by the following naive model
\begin{equation}
  \label{eq:modelo}
S(T,\mecanica)=S_0+\chi T\mecanica^g \Longrightarrow S(T,\mecanica_1)-S(T,\mecanica_2)=\chi T(\mecanica_1^g-\mecanica_2^g)\quad
\end{equation}
with $\mecanica\in\mathbb{R}^+$. In this expression $\chi$ is a positive constant that fits the dimensions of the model. Notice that  \eref{eq:modelo} satisfies \eref{eq:weak} but the double limit $T\to0,\mecanica\to\infty$ depends on the path it is achieved. \Fref{fig:modelo} depicts a $T-S$ plot for a system satisfying \eref{eq:modelo}.

The model does not fulfill the unattainability statement. First, it true that for fixed $\mecanica_1$ and $\mecanica_2$ the zero isotherm is unattainable\cite{zemansky-1968}; this proposition is essentially equivalent to \eref{eq:weak}. Yet, there is no need for so doing if one wants to achieve the neighborhood of the zero isotherm and it is also true that any isentropic path is endless for that model as ---let $g>0$--- by increasing indefinitely the mechanical parameter, the temperature is indefinitely decreasing to zero in a single step through $T\mecanica^g=\mathrm{cte}$. The ``unattainability''  of the zero isotherm would be a matter of practical limitations ---how to get an infinite $\mecanica$--- rather than a  fundamental restriction posed by a law of nature ---strictly speaking the absolute zero is here attained asymptotically.---  

\begin{figure}[tb]
  \centering
  \includegraphics[bb=131 406 465 720,scale=0.5]{modelo.eps}
  \caption{$T-S$ plot for the model \eref{eq:modelo}. Iso-$\mecanica$ lines are depicted. The unattainability statement and \eref{eq:weak} are satisfied for fixed $\mecanica_1$ and $\mecanica_2$. Yet an isentropic process ---horizontal left arrow--- diminishes the temperature of the system arbitrarily. Also any alteration of entropy is possible in the neighborhood of $T=0$ ---vertical double arrow--- provided that $\mecanica$ would change appropriately.}
  \label{fig:modelo}
\end{figure}
%The model does not satisfy the vanishing of the specific heat $c_\mecanica(T,\mecanica)=T\mecanica^g$ since this goal is not a sole matter of $T$ but also of $\mecanica$ and, again, the limit $\mecanica\to\infty$ get into troubles.

Finally, the model does not accomplish for the words given by Nernst ---see \sref{sec:intro}--- either. No matter how close to zero the temperature can be \emph{any alteration} of entropy is possible provided that the mechanical variable increases sufficiently.

The ideal gas behaves similarly to that model. A classical description of the particles leads to an entropy with no lower bound and which does not accomplish for the Nernst theorem. On the contrary a quantum description of the problem leads to\cite{landau-lifshitz-1968,huang-87,wu-pre-98} \eref{eq:modelo} where $\mecanica$ is the volume and $g=2/3$ for fermions. A deep analysis of the model shows that the quantum or classical description is driven by the condition\cite{landau-lifshitz-1968,balian-91}:
$$
\frac{V}{N}\left(\frac{mkT}{2\pi\hbar^2}\right)^{3/2}\gg1%\frac{(2\pi m)^{3/2}}{N_ah^3}\frac{V}{N}T^{3/2}\gg1
$$
known as ``classical limit.'' Here $N$ is the number of particles, $\hbar$ is Dirac's constant, $m$ is the particle mass and $k$ is the Boltzmann's constant. Thus, the competition between $T\to0$ and $V\to\infty$ is again crucial. It is likely that these contradictions are due to the fact that interactions are unavoidable at the very limit $T\to0$ so that microscopically ideal models may not be reflecting the array of data on macroscopic systems\cite{balian-91}.

We have shown then that a model satisfying \eref{eq:weak} ---the classical description of the Nernst theorem--- does not lead to the unattainability statement. Even worse, it does not fit to the words given by Nernst. Thus, further assumptions are required to a comprehensive and accurate mathematical description of the empirical laws observed at very low temperatures.



\section{The Nernst theorem as a uniform limit}
\label{sec:space}

The mathematical description of the statement of the Nernst theorem posed in Sec.~\ref{sec:intro} is improved by considering the following \emph{hypothesis I}: \emph{the isothermal exchange of entropy is \emph{uniformly} vanishing as the temperature goes to zero}:
\begin{equation}
  \label{eq:uniform}
  \forall\epsilon>0\quad\exists\delta(\epsilon)>0:\: T<\delta\Rightarrow \left|S(T,\mecanica_1)-S(T,\mecanica_2)\right|<\epsilon
\end{equation}
The key question\cite{jones-82} of the ``uniform convergence'' is that the \emph{same} $\delta(\epsilon)$ fits \emph{for any} $\mecanica_1,\mecanica_2$ belonging to $\mathcal{D}$.  It is straightforward that \eref{eq:uniform} matches the statement posed in \sref{sec:intro}. We will argue that it is the best choice to express mathematically that proposition.


The Nernst theorem is classically supposed to be  restricting the functional dependence of isothermal exchange of entropy $\Delta S=S(T,\mecanica_2)-S(T,\mecanica_1)$  on $T$ so that it converges to zero at the absolute zero. The ``uniform'' condition, here presented, essentially means that no value of $\mecanica$ can challenge this convergence. That is, no accidental divergence can possibly occur for a given value of $\mecanica$ in the neighborhood of $T=0$. In this way, the Nernst theorem would be also restricting the functional dependence of $\Delta S(T,\mecanica)$ on $\mecanica$. 


\paragraph{R\^ole of $x$:}
\label{con:x}
If hypothesis I is taken into account, the mechanical variable plays no r\^ole in the description of the problem. That is the primary consequence of the uniform convergence since given $\epsilon$ then $\delta$ is just a property of the system under consideration regardless the value of $x$. The reader should notice that this is a burden in the formulation of the Nernst theorem as a law of nature which does not depends on the configuration of the system under consideration. 

In the classical description of the theorem \eref{eq:weak} $\delta$ is a function of $\epsilon,\mecanica_1,\mecanica_2$ so the mechanical variable does play a r\^ole in the description of the problem. Although this r\^ole is usually discarded it is of the most importance when considering, for instance, double limits. 



\paragraph{Existence of inaccessible regions in a $T-S$ plot:}
\label{con:inaccessible}
\Eref{eq:uniform} ensures that $S(T,\mecanica_1)-S(T,\mecanica_2)$ is  bounded in the neighborhood of $T=0$ so that it has a supremum:
\begin{equation}
  \label{eq:sigma}
  \sigma(T)=\underset{\mecanica\in\mathcal{D}}{\sup}\big\{S(T,\mecanica_1)-S(T,\mecanica_2)\big\}
\end{equation}
The function $\sigma$ exists and is a monotonically increasing function at least in the neighbourhood of $T=0$. The function depends on the thermophysical properties of the system under consideration. What follows describe the relevant properties of the function in that neighbourhood. 

Now, consider a system compliant with hypothesis I whose equilibrium state is defined by a given temperature and a given mechanical configuration. The entropy of this state equals $S(T,\mecanica)$. Let us suppose that entropy is isothermally increased, the existence of $\sigma$ ensures that the final entropy cannot exceed $S(T,\mecanica)+\sigma$. The same argument applies for a process that decrease entropy. Hence, $S(T,\mecanica)$ is an upper and lower bounded function of $\mecanica$ for a given temperature and exists the following functions:
\begin{equation}
  \label{eq:sigma2}
  \umax{S}(T)=\underset{\mecanica\in\mathcal{D}}{\sup}\{S(T,\mecanica)\};\quad\umin{S}(T)=\underset{\mecanica\in\mathcal{D}}{\inf}\{S(T,\mecanica)\}
\end{equation}

Since the stability condition $(\partial S/\partial T)_\mecanica>0$ holds in the neighborhood of $T=0$ ---except perhaps at $T=0$--- the preceding functions are increasing functions of $T$ so that states of the type $\{T,S>\umax{S}(T)\}$ and $\{T,S<\umin{S}(T)\}$ cannot exist.  Hence, equilibrium states do not fill the plane $T-S$ and two boundaries arises from the fulfillment of the Nernst theorem as an uniform condition. 

In more detail it could be said that one of the goals of the third law is to ensure that the entropy $S$ has a single value   in the neighborhood of zero isotherm\cite{kestin-1968ii}.  In the classical formulation of the theorem, points of the type $\{T=0,S\neq S_0\}$ are excluded\cite[Figure~23.5]{kestin-1968ii} in a $T-S$ plot (these points are sketched by the symbol $\times$ in \fref{fig:f}). From a physical point of view ``when a certain point is excluded, we must demand that the same must be true about a small region surrounding the point''\cite{caratheodory-1909}\cite[page~236]{kestin-1976}.\footnote{Carath\'edory is here speaking about the adiabatic inaccessibility and he is preparing his celebrated Axiom~II. It is noteworthy that the Axiom literally states that ``(\ldots) there exist states that cannot be approached \emph{arbitrarily close} by adiabatic process'' instead of bare phrase ``states that are \emph{inaccessible} by adiabatic process.'' The concept ``arbitrarily close'' or ``neighborhood'' will play a leading r\^ole in the following discussion.} On the contrary, if a neighborhood of $\{T=0,S\neq S_0\}$ could be reached,  the exclusion of those ---isolated--- points would be fictitious. 

Thus, the plot $T-S$ (see \fref{fig:f}) consists in the region~I of allowed values of $\{T,S\}$, and  the forbidden region~II. The existence of region II is a goal of hypothesis I. The boundaries  ---which may or may not be physically accessible--- does not coincide with the axis $T=0$. In \Fref{fig:f} and in the preceding discussion we have make use of the Planck hypothesis for the purpose of clarity. The same argument would apply if $S_0$ comes down to $-\infty$ and, simultaneously, $\Delta S$ is vanishing.


\begin{figure}[tbp]
  \centering
%\includegraphics[bb=0 0 350 350,scale=0.7]{figuras/nernst.eps}
\includegraphics[bb=143 381 492 726,scale=0.6]{nernst.eps}
%\includegraphics[angle=270,scale=0.8]{figuras/nernst.pdf}
\caption{The $T-S$ plot and the Nernst theorem. The symbol $\times$ represents the points classically excluded by the theorem\cite[Figure~23.5]{kestin-1968ii}. A more comprehensive analysis of the theorem reveals that there exist a region~I whose points  represents equilibrium states, and a region~II which does not do so. The cycle 1-2-3-4-1 is an engine consisting in two isotherms 1-2 and 3-4, and two processes, 2-3 and 4-1, which differ just in a shift of entropy $S'=S+\Lambda$. Area 1-2-3-4-1 is equal to $\Lambda[T-\sigma^{-1}(\Lambda)]$ as in a Carnot engine. The essential of this picture is valid even if $S_0\to-\infty$ while $\Delta S\to0$}
\label{fig:f}
\end{figure}




The following issue is known, however hypothesis I enriches and clarifies its meaning. 

\paragraph{Processes that come to an end (unattainability statement):}
\label{con:unattainability}


Let us consider the isentropic process $S=\Sigma_0$ starting at some temperature so that $\umin{S}<\Sigma_0<\umax{S}$. The process will go on until the temperature $T_1$ defined by $\umax{S}(T_1)=\Sigma_0$ is attained. That temperature is non zero.

Here we may decrease the entropy of the system isothermally until the condition $\Sigma_1=\Sigma_0-\sigma(T_1)=\umin{S}(T_1)$ is achieved. At that point, an isentropic process will cool the system down to the temperature $T_2$ defined by $\umax{S}(T_2)=\Sigma_1$.

The endless staircase process that goes to the absolute zero is then defined.



\paragraph{Vanishing of the thermal expansions coefficients}
\label{con:expansion}
 
The thermal coefficients are related to the derivative $(\partial S/\partial\mecanica)_T$ through Maxwell's relations\cite{kestin-1968ii}. From \eref{eq:uniform} it is derived the vanishing of such derivative since:
$$
\lim_{T\to0}\left(\frac{\partial S}{\partial\mecanica}\right)_T=\lim_{T\to0}\lim_{\mecanica'\to\mecanica}\frac{S(T,\mecanica')-S(T,\mecanica)}{\mecanica'-\mecanica}\quad\quad\forall\mecanica\in\mathcal{D}
$$
If this double limit exists, it can be computed in whichever order. By taking first $T\to0$ and invoking \eref{eq:uniform} one has, of necessity\cite{kestin-1968ii}:
\begin{equation}
  \label{eq:alfa}
  \lim_{T\to0}\left(\frac{\partial S}{\partial\mecanica}\right)_T=0\quad\quad\forall\mecanica\in\mathcal{D}
\end{equation}


Unlike properties a and b ---being mathematical propositions,--- it should be pointed out that properties c and d are trends\cite{nernst-1924,kestin-1968ii} amply confirmed by experiment providing a support for the hypothesis. Yet, the most important consequence of the hypothesis relates it to the problem of the conversion of heat into work and will be considered in detail in the following section.


\section{The Nernst theorem and the continuous production of work}
\label{sec:work}

In the preceding section we have shown that  the Nernst theorem forces the existence of forbidden regions in a $T-S$ plot. We now derive  consequences taking in mind that the uniformity condition introduced in the preceding section makes $\mecanica$ play no r\^ole in the problem.

Let us now consider the following question: we wish to build up an engine which produces mechanical work $W$ by using a given working fluid that draws a given  amount of heat $Q$ from a reservoir of a given temperature $T$, \emph{which is the minimal amount of heat $\umin{Q'}$ that is to be taken up to the cold reservoir?}\footnote{This question inspects the behaviour of $W$ ---or $Q'$--- once the hot reservoir and the absorbed heat are fixed. It is also customary to inspect the behaviour of $W$ once the hot and cold reservoirs are fixed. This problem is related to the concept of irreversibility\cite{landau-lifshitz-1968} and lies out of the scope of the following discussion.}

The classical reading of the Kelvin-Planck statement would just say that the heat taken up at the cold reservoir ---hereafter called ``compensation''--- must be non-zero $Q'\neq0$. It then seems that so long as this statement is concerned, a negligible $Q'$ would suffice. Thus, $\umin{Q'}=0^+$ ---ie arbitrarily close to zero but non-zero---. The answer is independent from $Q$, $T$ and the working fluid and comes from the fact we feel free to place a two-reservoir engine in a $T-S$ plot since no other restriction happens to be.

A machine having $\umin{Q'}=0^+$ would result in an efficiency $\eta=W/Q$ as close to unity as desired. This most efficient engine has never been built up; we will now put forward the fact that this is due to fundamental properties of matter despite of practical limitations to achieve such engine.


As a general rule, $\sigma$ is nonzero for nonzero temperatures and from \eref{eq:uniform} and \eref{eq:sigma} one gets:
\begin{equation}
  \label{eq:limsigma}
  \forall\epsilon>0\quad\exists\delta(\epsilon)>0:\quad T<\delta\Longrightarrow \sigma(T)<\epsilon
\end{equation}
That is, $\lim_{T\to0^+}\sigma(T)=0$.

In \eref{eq:limsigma} there is no need to call for absolute value delimiters since both $T$ and $\sigma$ are positive magnitudes. From \eref{eq:limsigma} and the preceding argument  the existence of the  inverse function $\sigma^{-1}(\Lambda)$ is straightforward. In fact, the inverse function is nothing else but a suitable representation for the parameter $\delta(\epsilon)$. The inverse function gives the temperature at which the width in entropy of the accessible states equals $\Lambda$. That temperature also depends on the thermophysical properties of the system under consideration.


Now, let us consider again the question posed at the beginning of this section but now consider that the Kelvin-Planck statement and the Nernst theorem, as stated in \sref{sec:intro}, apply. Hence, the restrictions posed in \sref{sec:space}, shown in \fref{fig:f}, are valid.  The working fluid is undergoing a cycle that is extracting an amount of entropy  $\Lambda=Q/T$ from the hot reservoir. For so doing, it is necessary that $\Lambda<\sigma(T)$.

Now, the entropy should be being deposited into  the cold reservoir which would receive an amount of heat $Q'$. To achieve the maximum efficiency, the temperature of the cold reservoir must be the coldest temperature able to exchange with the working fluid that  amount of entropy. Following the preceding paragraphs, that minimal temperature is given by $\sigma^{-1}(\Lambda)$ which is a property of the fluid under consideration. Thus,
%\nopagebreak[5]
\begin{eqnarray}
  \label{eq:eta}
  \forall Q\neq0,T\neq0:\, \frac{Q}{T}=\Lambda<\sigma(T)\nonumber\\
\Longrightarrow\exists \sigma^{-1}(\Lambda):\, Q'\geq\Lambda\times\sigma^{-1}(\Lambda)=\umin{Q'}
\end{eqnarray}
This most efficient engine is depicted in \fref{fig:f} by the cycle $1-2-3-4-1$ which consists of two isotherm and two processes that differ in a shift of entropy; exchanges of energy and entropy in $2-3$ cancel with those of $4-1$ so the reservoirs needed for these two processes play no r\^ole in the problem.\footnote{It would be possible to decrease $Q'$  by considering the cycle $1-2-3-5-4-1$ ---see \fref{fig:f}--- because entropy will be being deposited to colder reservoirs, but the condition of two reservoirs is broken. However, in that case, the compensation is equally expressed by $\sigma''\Lambda$ where $\sigma''$ is an unknown temperature ranging between $T_5$ and $T_4=\sigma^{-1}(Q/T)$. The essential of the following discussion also applies to this compensation.} The performance of work equals to  $\umax{W}=\Lambda\times\left[T-\sigma^{-1}(\Lambda)\right]$.


It is very noticeable that the value of $\umin{Q'}$ is now a function of the parameters of the problem: $Q$, $T$ and the working fluid which enters through $\sigma^{-1}$. Moreover, the minimal compensation is a function of the exchange of entropy $\Lambda$.

Though the particular value of $\umin{Q'}$ depends on the thermophysical properties of the working fluid it is a quite noteworthy fact that, as a general rule, $\umin{Q'}$ is never arbitrarily close to zero for a given $\Lambda$ since, according to the Nernst theorem, $\sigma^{-1}$ is not arbitrarily close to zero either (see \fref{fig:f}).

\begin{figure}[tbp]
  \centering 
% \includegraphics[angle=270,scale=0.8]{figuras/qfrio.pdf}
% \includegraphics[scale=0.7]{figuras/qfrio.eps}
 \includegraphics[bb=149 444 457 721,scale=0.6]{qfrio.eps}
  \caption{Plot of $\Lambda-Q'$ for a given working fluid. A point of region~I, say $a$, represents a set of engines each having the same $Q'$ and $\Lambda$ and the same temperature of the cold reservoir: the slope of the straight line.  The thick line ---which depends on the working fluid--- represents \eref{eq:eta} and comes to $\{0,0\}$ with zero slope. For the working fluid under consideration, it is impossible to built up an engine that enters in region~II.  The symbol $\times$ represents  the restriction posed by the classical reading of  the Kelvin-Planck. }
  \label{fig:etamax}
\end{figure}
The universal resemblance of \eref{eq:eta} allow to outline the plot $\Lambda-Q'$ (see \fref{fig:etamax}). Notice that in the context of the classical reading of the Kelvin-Planck statement that plot would have no restriction other than the exclusion of the points of the type $\{\Lambda\neq0,Q'=0\}$, those points are shown in that figure by the symbol $\times$. Now, if we take into account the Nernst theorem it is clear that points of the type $\{\Lambda,Q'<\umin{Q'}(\Lambda)\}$ should be excluded as well. That points define a region whose boundary is given by \eref{eq:eta}; the analogy between the regions I and II, and the boundary of \fref{fig:f} and those of \fref{fig:etamax} is immediate. From the Nernst theorem it is easily probed that $\umin{Q'}$ goes to zero with zero slope as $\Lambda$ goes to zero. Summarizing the meaning of \fref{fig:etamax} it should be stressed that  the Nernst theorem is excluding a region around those points actually excluded by the classical reading of the Kelvin-Planck statement. That is in agreement with Carath\'eodory's argument posed in \sref{sec:space}.


The limit $\Lambda\to0^+$ is the unique possibility to get $\umin{Q'}\to0^+$. In that case, taking $T$ as a bounded, constant parameter, it is clear that $Q$ will be vanishing as well and, as an ultimate consequence $W$ will be also vanishing. Hence:
  \begin{equation}
    \label{eq:qmin}
    Q'\to0^+\Longrightarrow W\to0^+
  \end{equation}
This proposition contains the essence of the Nernst theorem. Its dramatic meaning is best viewed noticing that $Q'\to0^+$ and $\eta\to1^-$ are equivalent. It then means that \emph{as the most efficient engine is achieved, the delivery of work is decreasing to zero}. 

It should be pointed out once again that in the preceding discussion $\mecanica$ does not play any r\^ole and it is in this sense that the restrictions \fref{fig:etamax} and \eref{eq:qmin} are \emph{universal}. That feature comes from the property of uniformity. On the contrary, if the Nernst theorem is just considered as a limit ---without the requirement of uniformity,--- then \eref{eq:qmin} and \fref{fig:etamax} would be just valid  for transitions between two given values of $\mecanica$ ---see \cite[Figure~23.9]{kestin-1968ii}\cite{liboff-physicsessays-94}.--- Therefore, \eref{eq:weak} does not lead to any proper restriction, as the boundary of \fref{fig:etamax} will come arbitrarily close to the restriction posed by the classical reading of the Kelvin-Planck statement provided that we consider the appropriate values for $\mecanica_1$ and $\mecanica_2$. Hence, the improvement of an engine would be a matter of practical limitations if \eref{eq:weak} were valid.\footnote{If the working fluid behaves like \eref{eq:modelo}, the engine could always be improved by getting arbitrarily large values of $\mecanica$.}

\section{The Nernst theorem and the statement of the second law}
\label{sec:second}
 
\Sref{sec:work} clearly shows that the Nernst theorem is restricting the conversion of heat into work in a way which is independent from the mechanical configuration of the system. It should be desirable to link it to the Kelvin-Planck statement of the second law.

One of the most important results of \sref{sec:work} is the leading r\^ole played by $Q'$, the compensation, in the problem of the conversion of heat into work. That importance comes from the fact that its minimum value is universally expressed by \eref{eq:eta}.  

From an historical point of view, the r\^ole of the cold reservoir was  the key of the second law. The first known statement of the law, due to Kelvin\cite{thomson-1853}, stated that it is impossible to built up an engine that both produces work and  cools the coldest of the available reservoirs, no matter what happens to hotter reservoirs (see \fref{fig:staircase}).

Planck simplified the statement by noticing that it is impossible to built up an engine that performs work by cooling just one reservoir\cite{planck-1927}. In that sense, a cold reservoir is to be heated ---compensation--- in some amount (see \fref{fig:staircase}).  Yet he put forward no word about the ``size'' of the compensation so that one hopes it might  be negligible: that is the  ``classical'' reading of the statement that has prevailed from the beginning of the statement.

Furthermore, if the  Nernst theorem is taken into account through hypothesis I, we get the ultimate restriction: a minimal compensation is given by the properties of the working fluid and the exchange of entropy (see \eref{eq:eta} and \fref{fig:staircase}): \emph{wish you to transform a finite amount of heat into work, you must \emph{certainly} pay a  tax ---a compensation---, namely  the tax is not becoming finiteless at your willing}. The emphasized proposition is an informal statement for \eref{eq:qmin}.

\begin{figure}[tb]
  \centering  
  \includegraphics[bb=137 446 592 726,scale=0.5]{staircase.eps}
%  \includegraphics[scale=0.4]{figuras/staircase.eps}
%  \includegraphics[angle=270,scale=0.5]{figuras/staircase.pdf}
  \caption{The staircase of the statements of the second law. The reader should notice the restrictions on the compensation that arises from the violations of the statements. From left to right a violation of the Kelvin statement, of the classical reading of the  Kelvin-Planck statement, and of a ``comprehensive'' reading of the Kelvin-Planck statement. In this last case, $\umin{Q'}$ is given by \eref{eq:eta}.}
  \label{fig:staircase}
\end{figure}

The reader should notice that, in fact, \eref{eq:qmin}  is something else than the bare statement of the Nernst theorem since it also recalls the Kelvin-Planck statement. It is the dome that crowns the leitmotif of the principles of thermodynamics by putting forward the ultimate restriction on $Q'$. In fact, if the relation were clothed in a negative way the reader would have found a statement very much like any of the second law: \emph{it precludes delivering a \emph{finite} work without a \emph{finite} compensation}.



\section{From the Kelvin-Planck statement to the Nernst theorem}
\label{sec:inverse}


The preceding sections analyzed the Nernst theorem through the hypothesis I putting forward its close relation to the problem of the conversion of heat into work. Here we want to do the reverse trip: starting at the Kelvin-Planck statement, upon which assumptions is hypothesis I derived? 

In our opinion the classical reading of the Kelvin-Planck statement could be said to be ``crude'' in the sense that the cause ---heat absorbed from the hot reservoir--- and the unavoidable effect ---the minimal compensation--- has been being taking as uncoupled since the early stage of thermodynamics. A more comprehensive reading of the statement of the second law would have lead to some coupling between the $\umax{W}$ or $\umin{Q'}$ and $Q$ since a finiteless absorbed heat is necessary to get a finiteless compensation ---see \eref{eq:eta} and \fref{fig:f}.--- We put now forward the fact that this hypothesis ---labeled \emph{hypothesis II}--- suffice for the Nernst theorem:
\begin{quote}
  \emph{the compensation of a Carnot engine approaches indefinitely zero  \emph{only if} the heat absorbed from the reservoir is vanishing}
\end{quote}

The significance of the hypothesis would be revealed by the conclusions that it draws, however the reader should not conclude that hypothesis II is additional to the Kelvin-Planck statement since it is embedded in its words. The point is the meaning of the word ``effect'' that appears in the statement. One can get explicitly hypothesis II by changing ``effect'' by ``finite effect'' in the statement so that any ``finite'' absorbed heat necessarily leads to a ``finite'' compensation. But, in fact,  any effect is actually finite so that the modification would be a pleonasm.

The fact that the hypothesis leads to the Nernst theorem is surprisingly straightforward. Notice that the compensation of an engine equals $Q'=T'\times\Lambda$ where $T'$ is the temperature of the cold reservoir and $\Lambda$ the exchange of entropy.  That magnitude becomes zero (1) if $T'$ goes to zero regardless $\Lambda$ or (2) if $\Lambda$ goes to zero regardless $T'$. The hypothesis excludes option (1) that is, it excludes any alteration of entropy in the neighbourhood of the absolute zero. Hence, the Nernst theorem as stated in \sref{sec:intro} and as analyzed in \sref{sec:space} comes of necessity.


It is then concluded that hypothesis I and II are equivalent propositions so that the Nernst theorem ultimately follows from a comprehensive reading of  the Kelvin-Planck statement through hypothesis II.\footnote{Thus the second law ensures that the entropy goes to a value which does not depend on $\mecanica$ at the absolute zero. Its precise value, or whether this value is finite or infinite  ---Planck's formulation--- is alien to this description since the second law just concerns variations of entropy. } This proposition is valid  as far as  systems suitable to be working fluids of engines are being considered.





\section{Discussion and conclusion}
\label{sec:discu}



In 1909 Carath\'eodory\cite{caratheodory-1909}, following a suggestion by Born, successfully translated the classical statements of the second law ---which deal with the problem of production of work--- into a statement which refers to physical properties of an isolated system. This work introduces the reverse trip for the Nernst theorem. The theorem has been supposed to deal with the properties of systems in the neighborhood of $T\to0$; the study here  presented (see \sref{sec:work}, specially \eref{eq:eta} and \eref{eq:qmin})  relates the theorem back to the problem of production of work. The theorem will be expressing an universal property of the continuous production of work. 

The reader may ask which assumptions makes the Nernst theorem independent from the Kelvin-Planck and which does not do so. The goal of the second law of thermodynamics is to restrict the  continuous production of work putting forward the existence of a fundamental asymmetry: work is dissipated into heat but the reverse is not true. The goal enters by a statement which expresses, in words, a restriction.  We have shown in this paper that the degree of restriction matters and gets influence in the properties of systems. The classical reading of the Kelvin-Planck statement  assumes just that the compensation is nonzero. Upon this assumption the general properties of systems in the neighborhood of $T=0$ needs to be summarized as an independent law. Yet, a comprehensive reading of the statement through the cautious hypothesis II presented in \sref{sec:inverse} leads to some of these general properties: the unattainability statement and the vanishing of the expansion coefficients. We should again recall that the formulation here presented is insensitive to whether or not specific heats come to zero as the temperature comes to zero.

The new reading of the statement overcomes the embarrassing fact that $W$ must differ from $Q$ ---to what extend?--- by stating that they must do so in a measurable quantity which also depends on the working fluid which does play a r\^ole in the problem. The speech of Carath\'eodory quoted in \sref{sec:space} again makes the sense in this discussion: if the condition $W=Q$ is excluded, $W=Q^-$ should have been excluded as well.\footnote{We are presenting an analogy between the words by Carath\'eodory and the results of this work. It is not our willing to state that the former derives from the latter or conversely. It is a sort of coincidence that both problems speak about the same law of nature.} 

Any exception to the hypothesis would result in a failure of the consequences \emph{iii} and \emph{iv} quoted in \sref{sec:space} that is: (a) an experiment that would allow to increase $1/T$ indefinitely, (b) an experiment that would allow to decrease entropy indefinitely, or (c) a substance that would have non zero thermal expansion coefficients.\footnote{The reader should notice that the unattainability of the zero isotherm does not guarantee the Nernst theorem. The model $S(T,\mecanica)=\chi\log T\times (\mecanica+a)/(\mecanica+b)$ with $\chi,a\neq b>0,\mecanica\in\mathbb{R}^+$ provides a mathematical example of system which precludes experiments (a) and (b) but fails to obey the Nernst theorem.} 



It is out of the scope of this paper to describe the microscopic relevance of the hypothesis, that is to determine which kind of Hamiltonians ---interactions--- would lead to uniformly vanishing $\Delta S$  as well as their symmetry properties. Well-known models ---specially ideal gases--- do not satisfy the hypothesis here presented. However the reader should not consider them as an ``exception'' to the hypothesis  since the hypothesis are grounded on experimental macroscopic basis and not on the analysis of microscopic models.

The relation between interactionless models and the Nernst theorem has been recently suggested\cite{balian-91,roseinnes-ajp-99} and it seems that the r\^ole of interactions can not be neglected in real systems at sufficiently low temperatures. The suggestion comes from the analysis of the independent spin system. That model does not satisfies the Nernst theorem since the ground state is degenerate. Yet, independent spin systems does not happen in nature since \emph{real} solids always exhibits magnetic correlation and ordering at sufficiently low temperatures. The resulting ordering ---usually a macroscopic new phase, either ferromagnetic or antiferromagnetic--- would satisfy the Nernst theorem.

Quite the same analysis  can be made on free particle systems. The reader should notice that the classical ideal gas allow to envision  a process of the type (a) and (b), moreover it satisfies (c) in contradiction with  the Nernst theorem. On its own,  quantum ideal gases\cite{landau-lifshitz-1968} gets vanishing thermal expansion coefficients and precludes experiment of the type (b) since entropy is necessarily a low bounded function. Yet, the quantum model still allows to envision type (a) experiments which do not happen to be in nature and which do preclude the meaning of the Nernst theorem. Of course in real systems ``ordering'' does always occur and the settlement of condensed phases seems unavoidable.

It is then likely that interactionless models ---either kinetic, magnetic or whichever--- do not accurately describe the properties of \emph{real} systems at sufficiently low temperatures since interactions can not be neglected.



\acknowledgements

The authors wish to thank to Mr. Jos\'e Antonio P\'erez G\'omez for long discussions and, specially, for enlightening them with the uniform convergence. The authors also thanks to Ms. Pilar N\'u\~nez for her kindly search of old bibliographic data.

This work was supported by Spanish \emph{Ministerio de Ciencia y Tecnolog\'\i a} project BFM2002-02237.

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\begin{document}

\title{Restricciones universales a la conversi{\'o}n del calor en trabajo que provienen del an{\'a}lisis del teorema de Nernst como un l{\'\i}mite uniforme}
\author{Jos\'{e} Mar\'{\i}a \surname{Mart\'{\i}n Olalla}}
\email[Correo electr{\'o}nico:~]{olalla@us.es}
\author{Alfredo \surname{Rey de Luna}}
\affiliation{Departamento de F\'{\i}sica de la Materia Condensada. Universidad de Sevilla. \\ Ap. Correos 1065 E-41080Sevilla. Spain}
\preprint{J.  Phys. A: Math. Gen. \textbf{36}(29) (2003) 7909--7921}

\begin{abstract}
Se analiza la relaci{\'o}n entre el teorema de Nernst y el enunciado de Kelvin-Planck del segundo principio de la termodin{\'a}mica. Se{\~n}alamos el hecho de que el cambio de entrop{\'\i}a tiende uniformemente a cero cuando la temperatura tiende a cero. El an{\'a}lisis de esta hip{\'o}tesis muestra que es equivalente al hecho de que la compensaci{\'o}n de una m{\'a}quina de Carnot escala con el calor absorbido del foco caliente de forma que el teorema de Nernst puede derivarse  del enunciado del segundo principio.
\end{abstract}
\pacs{05.70.-a,05.70.Ce}%, 11.30.-j, 65.40}
\keywords{Primer principio, Segundo principio, Tercer principio, Fundamentos de termodin{\'a}mica, Gases ideales, M{\'a}quinas de Carnot, Inaccesibilidad del cero absoluto, Calores espec{\'\i}ficos}

\maketitle


\section{Introducci{\'o}n}
\label{sec:intro}
La formulaci{\'o}n cl{\'a}sica del enunciado de Kelvin-Planck establece que:\cite[p\'ag.~89]{planck-1927}
\begin{quote}
es imposible construir una m{\'a}quina que trabaje de forma c{\'\i}clica y no produzca otros efectos m{\'a}s que la elevaci{\'o}n de un cuerpo y el enfriamiento de una fuente
\end{quote}
Existen otras formulaciones que son esencialmente equivalentes a {\'e}sta  --- enunciados de Kelvin\cite{thomson-1853}, Clausius\cite{clausius-1854} y Carath{\'e}odory\cite{caratheodory-1909}--- pero, para nuestro an{\'a}lisis, nos referiremos a la formulaci{\'o}n se{\~n}alada arriba. Esencialmente el enunciado requiere ``otra'' fuente para realizar el proceso se{\~n}alado.

El desarrollo del principio necesita del concepto de   ``fluido de trabajo'', la sustancia que realiza el ciclo. Sus propiedades se eliminan normalmente del an{\'a}lisis puesto que el estado inicial y final del fluido coinciden y ``s{\'o}lo ha servido como agente para llevar a cabo cambios en sus alrededores''\cite[p{\'a}g.~68]{planck-1927}. Sin embargo, el fluido \emph{debe} ser capaz de realizar el proceso que se desea.

Mostraremos en este trabajo que una propiedad general de la materia restringe  la habilidad de los fluidos de trabajo para realizar ciclos termodin{\'a}micos y, por tanto, restringe lo que el enunciado de Kelvin-Planck permite. Mostraremos que esta restricci{\'o}n proviene, realmente, de una interpretaci{\'o}n adecuada del enunciado rese{\~n}ado arriba.

La propiedad general de la que estamos hablando se conoce hoy en d{\'\i}a como la tercera ley de la termodin{\'a}mica. La necesidad y car{\'a}cter de tal ley ha sido objeto de discusi{\'o}n desde el principio del siglo pasado. El estudio de algunos problemas qu{\'\i}micos llev{\'o} a Nernst\cite{nernst-1906,nernst-1924} a descubrir su \emph{teorema del calor} el cual establece que\cite[p\'ag.~85]{nernst-1924}:
\begin{quote}
en las proximidades del cero absoluto \emph{todo} proceso transcurre sin alteraci{\'o}n de entrop{\'\i}a
\end{quote}
El teorema ---que cl{\'a}sicamente no proviene del enunciado de Kelvin-Planck\cite{einstein-1921,epstein-1937}--- est{\'a} avalado por una ingente cantidad de datos experimentales. Hemos elegido esta antigua ---aunque meritoria--- versi{\'o}n del teorema por las razones que se ver{\'a}n en  \sref{sec:space} pero no vemos otra raz{\'o}n para que se haya olvidado este enunciado m{\'a}s que el hecho de que se refiere a propiedades de ``procesos'' en vez de propiedades de ``sistemas'' tal y como hoy en d{\'\i}a se recoge en la literatura cient{\'\i}fica\cite{kestin-1968ii,landau-lifshitz-1968,landsberg-78,baierlein-99,zemansky-1968}.

Nernst deriv{\'o} su teorema de dos observaciones de car{\'a}cter general. La primera es el llamado \emph{principio de inaccesibilidad del cero absoluto} que se{\~n}ala que ning{\'u}n proceso puede disminuir la temperatura de un sistema hasta el cero absoluto. El segundo es el hecho de que los calores espec{\'\i}ficos tienden a cero conforme la temperatura tiende a cero.

Debe reconocerse que Planck\cite{planck-1927} se{\~n}al{\'o} que estas observaciones debieron haber llevado a una conclusi{\'o}n m{\'a}s restrictiva: ``cuando la temperatura de disminuye indefinidamente, la entrop{\'\i}a de un cuerpo qu{\'\i}micamente homog{\'e}neo de densidad finita se aproxima indefinidamente a un valor que es independiente de la presi{\'o}n, estado de agregaci{\'o}n y de las modificaciones qu{\'\i}micas.'' La formulaci{\'o}n de Planck evita que $\Delta S\to0$ mientras $S\to-\infty$ cuanto $T\to0$ y expresa que el valor absoluto de la entrop{\'\i}a est{\'a} acotado en el cero absoluto.

Sin embargo, nuestro trabajo se relaciona {\'u}nicamente con el an{\'a}lisis de  $\Delta S$ puesto que s{\'o}lo trata con el problema de la conversi{\'o}n del calor en trabajo. Este problema es invariante frente a una traslaci{\'o}n del valor de la entrop{\'\i}a y, por eso, la formulaci{\'o}n de Planck est{\'a} fuera de nuestro {\'a}mbito de trabajo. Lo mismo puede decirse de la anulaci{\'o}n de los calores espec{\'\i}ficos.

La presentaciones actuales del tercer principio normalmente lo relacionan con las propiedades microsc{\'o}picas de los sistemas objeto de estudio\cite{balian-91,huang-87,mafe-ajp-98,roseinnes-ajp-99,wu-pre-98}. Tambi{\'e}n se han hecho algunos esfuerzos para clarificar su significado macrosc{\'o}pico\cite{falk-pr-1959,yan-chen-jpa-88,landsberg-jpa-89,oppenheim-jpa-89,liboff-physicsessays-94,landsberg-ajp-97,belgiorno-pra-03a}.

El objetivo de este trabajo es una revisi{\'o}n de la descripci{\'o}n matem{\'a}tica del enunciado del teorema de Nernst se{\~n}alado anteriormente as{\'\i} como de sus consecuencias f{\'\i}sicas. Analizaremos observaciones macrosc{\'o}picas en el rango de la f{\'\i}sica de muy bajas temperaturas. Al hacer esto no estableceremos hip{\'o}tesis alguna sobre la constituci{\'o}n de los sistemas objeto de estudio. 

Es tambi{\'e}n un objetivo de este trabajo un an{\'a}lisis energ{\'e}tico de las consecuencias del teorema de Nernst. Brevemente, el motor m{\'a}s simple y eficiente contiene dos procesos en los que se intercambia entrop{\'\i}a. Este tipo de procesos est{\'a} restringido por el teorema de Nernst en las cercan{\'\i}as del cero absoluto. Probaremos que tal restricci{\'o}n lleva a una condici{\'o}n adicional que toda m{\'a}quina debe satisfacer.


\section{Limitaciones a la descripci{\'o}n del teorema de Nernst}
\label{sec:limit}

La formulaci{\'o}n cl{\'a}sica del teorema de Nernst establece que:\cite{kestin-1968ii} ``el cambio de entrop{\'\i}a asociado con cualquier proceso isotermo entre dos estados de un sistema en equilibrio interno tiende a cero cuando la temperatura tiende a cero,'' que se expresa matem{\'a}ticamente como::
\begin{equation}
\label{eq:weak}
    \forall\mecanica_1\mecanica_2\in\mathcal{D}\quad\lim_{T\to0^+}[S(T,\mecanica_1)-S(T,\mecanica_2)]=0%\Longleftrightarrow
\end{equation}
donde $\mecanica$ representa a cualquier variable mec{\'a}nica como el volumen, presi{\'o}n , campo magn{\'e}tico y  $\mathcal{D}\subseteq\mathbb{R}$ es su dominio de definici{\'o}n.\footnote{El propio Nernst describi{\'o} \eref{eq:weak} a trav{\'e}s del enunciado expuesto en \sref{sec:intro}.\cite[p\'ag.~85]{nernst-1924}} 

Landau y Lifshitz\cite{landau-lifshitz-1968} mostraron la importancia de mantener  $\mecanica_1,\mecanica_2$ fijas en \eref{eq:weak}. De otra forma, dicen, si, por ejemplo, $\mecanica_1$ tiende a infinito el teorema puede no ser v{\'a}lido.  En general puede decirse que la descripci{\'o}n del teorema funciona bien mientras que los valores de $\mecanica$ est{\'a}n fijos pero entra en contradicciones si se analizan l{\'\i}mites dobles como $T\to0$ y $\mecanica\to\infty$.

Un ejemplo acad{\'e}mico de este problema lo proporciona el siguiente modelo:
\begin{equation}
  \label{eq:modelo}
S(T,\mecanica)=S_0+\chi T\mecanica^g \Longrightarrow S(T,\mecanica_1)-S(T,\mecanica_2)=\chi T(\mecanica_1^g-\mecanica_2^g)\quad
\end{equation}
con $\mecanica\in\mathbb{R}^+$. En esta expresi{\'o}n  $\chi$ es una constante positiva que provee las dimensiones apropiadas. Obs{\'e}rvese que  \eref{eq:modelo} satisface \eref{eq:weak} pero el l{\'\i}mite doble $T\to0,\mecanica\to\infty$ depende del camino por el que se recorre. \Fref{fig:modelo} muestra un diagrama $T-S$ para este modelo.

El modelo no satisface el principio de inaccesibilidad del cero absoluto. Primero, es cierto que para unos valores fijos de $\mecanica_1$ y $\mecanica_2$ el cero absoluto s{\'\i} es inaccesible\cite{zemansky-1968}; esta proposici{\'o}n es equivalente a  \eref{eq:weak}. Sin embargo no hay necesidad de hacer esto si lo que se desea es acceder al cero absoluto; y no es menos cierto que un cualquier proceso isentr{\'o}pico  no tiene fin puesto que ---si $g>0$--- al aumentar indefinidamente el par{\'a}metro mec{\'a}nico se disminuye indefinidamente la temperatura del sistema en un {\'u}nico paso a trav{\'e}s de $T\mecanica^g=\mathrm{cte}$. La ``inaccesibilidad'' de la isoterma cero ser{\'\i}a una cuesti{\'o}n de limitaciones pr{\'a}cticas ---c{\'o}mo conseguir un $\mecanica$ infinito--- en vez de una restricci{\'o}n expuesta por una ley de la naturaleza ---estrictamente, el cero absoluto es accesible de forma asint{\'o}tica.---  

\begin{figure}[tb]
  \centering
  \includegraphics[bb=131 406 465 720,scale=0.7]{modelo.eps}
  \caption{Diagrama $T-S$ para el modelo \eref{eq:modelo}. Se muestran l{\'\i}neas de iso-$\mecanica$. La inaccesibilidad del cero absoluto y \eref{eq:weak} se satisfacen para valores dados de $\mecanica_1$ y $\mecanica_2$. Sin embargo, un proceso isentr{\'o}pico ---flecha horizontal--- disminuye la temperatura del sistema en forma arbitraria. Cualquier alteraci{\'o}n de entrop{\'\i}a es tambi{\'e}n posible en las proximidades de $T=0$ ---flecha vertical--- con tal que $\mecanica$ cambie de forma apropiada.}
  \label{fig:modelo}
\end{figure}
%The model does not satisfy the vanishing of the specific heat $c_\mecanica(T,\mecanica)=T\mecanica^g$ since this goal is not a sole matter of $T$ but also of $\mecanica$ and, again, the limit $\mecanica\to\infty$ get into troubles.

Finalmente, el modelo tampoco satisface la palabras de Nernst  ---v{\'e}ase \sref{sec:intro}.--- No importa cuan cerca estemos del cero absoluto, \emph{cualquier alteraci{\'o}n} de la entrop{\'\i}a es posible con tal que la variable mec{\'a}nica aumente suficientemente.

El gas ideal se comporta de forma similar a este modelo. Una descripci{\'o}n cl{\'a}sica de las part{\'\i}culas lleva a una entrop{\'\i}a que sin l{\'\i}mite inferior y que no satisface el teorema de Nernst. Por el contrario, una descripci{\'o}n cu{\'a}ntica del problema lleva\cite{landau-lifshitz-1968,huang-87,wu-pre-98} a \eref{eq:modelo} donde $\mecanica$ es el volumen y $g=2/3$ para fermiones. Un an{\'a}lisis profundo del modelo muestra que la elecci{\'o}n de una descripci{\'o}n cu{\'a}ntica o cl{\'a}sica est{\'a} regida por la condici{\'o}n\cite{landau-lifshitz-1968,balian-91}:
$$
\frac{V}{N}\left(\frac{mkT}{2\pi\hbar^2}\right)^{3/2}\gg1%\frac{(2\pi m)^{3/2}}{N_ah^3}\frac{V}{N}T^{3/2}\gg1
$$
conocida como ``l{\'\i}mite cl{\'a}sico.'' Aqu{\'\i} $N$ es el n{\'u}mero de part{\'\i}culas, $\hbar$ es la constante de Dirac, $m$ es la masa de las part{\'\i}culas y  $k$ la constante de Boltzmann. As{\'\i}, la competencia entre $T\to0$ y $V\to\infty$ es, de nuevo, crucial. Probablemente estas contradicciones s{\'o}lo reflejan el hecho de que las interacciones son necesarias en las cercan{\'\i}as del cero absoluto y modelos microsc{\'o}picos ideales pueden no representar fielmente los datos experimentales sobre sistemas macrosc{\'o}picos\cite{balian-91}.

Hemos mostrado que un modelo que satisfaga \eref{eq:weak} ---la descripci{\'o}n cl{\'a}sica del teorema de Nernst--- no conduce a la inaccesibilidad del cero absoluto. A{\'u}n peor no se ajusta a las palabras de Nernst. Nuevas hip{\'o}tesis son necesarias para una descripci{\'o}n matem{\'a}tica precisa y completa de las leyes emp{\'\i}ricas observadas a muy bajas temperaturas.


\section{El teorema de Nernst como un l{\'\i}mite uniforme}
\label{sec:space}

La descripci{\'o}n matem{\'a}tica del teorema de Nernst expuesta en \sref{sec:intro} mejora notablemente considerando la siguiente hip{\'o}tesis I: \emph{el cambio isotermo de entrop{\'\i}a tiende \emph{uniformemente} a cero cuando la temperatura tiende a cero}:
\begin{equation}
  \label{eq:uniform}
  \forall\epsilon>0\quad\exists\delta(\epsilon)>0:\: T<\delta\Rightarrow \left|S(T,\mecanica_1)-S(T,\mecanica_2)\right|<\epsilon
\end{equation}
La clave de la ``convergencia uniforme'' es\cite{jones-72,jones-82} que el \emph{mismo}  $\delta(\epsilon)$ sirve \emph{para cualquier} $\mecanica_1,\mecanica_2$ que pertenezcan a  $\mathcal{D}$.  Es inmediato que \eref{eq:uniform} satisface el enunciado puesto en \sref{sec:intro}. Mostraremos que es la mejor forma de expresar matem{\'a}ticamente la proposici{\'o}n de Nernst.


N{\'o}tese que, cl{\'a}sicamente, el teorema de Nernst restringe la dependencia funcional del cambio isotermo de entrop{\'\i}a $\Delta S=S(T,\mecanica_2)-S(T,\mecanica_1)$  en $T$ de forma que converge a cero en el cero absoluto. La condici{\'o}n de ``uniformidad'' que aqu{\'\i} introducimos, significa esencialmente que ninguna valor de $\mecanica$ puede comprometer esta convergencia. Es decir, no es posible una divergencia accidental debida a un valor particular de $\mecanica$ en las cercan{\'\i}as del cero absoluto. En este sentido, el teorema de Nernst tambi{\'e}n restringe la dependencia funcional de  $\Delta S(T,\mecanica)$ en $\mecanica$. 


\paragraph{El papel de  $x$:}
\label{con:x}
Si la hip{\'o}tesis se toma en consideraci{\'o}n la variable mec{\'a}nica no desempe{\~n}a ning{\'u}n papel en la descripci{\'o}n de problema. Esta es una consecuencia primaria de la convergencia uniforme puesto que dado $\epsilon$ entonces $\delta$ es s{\'o}lo una propiedad del sistema objeto de estudio sin considerar los valores de $\mecanica$. El lector debe notar que {\'e}ste es uno de los puntos claves de la formulaci{\'o}n del teorema como ley natural que no dependa de la configuraci{\'o}n de sistema objeto de estudio.

En la descripci{\'o}n cl{\'a}sica del teorema \eref{eq:weak} $\delta$ es una funci{\'o}n $\epsilon,\mecanica_1,\mecanica_2$ de forma que la variable mec{\'a}nica s{\'\i} desempe{\~n}a un papel. Este papel normalmente no se considera en el problema pero es de gran importancia a la hora de considerar, por ejemplo, l{\'\i}mites dobles.



\paragraph{Existencia de regiones inaccesibles en un diagrama $T-S$:}
\label{con:inaccessible}
\Eref{eq:uniform} asegura que $S(T,\mecanica_1)-S(T,\mecanica_2)$ est{\'a} acotada en las cercan{\'\i}as del cero absoluto de forma que tiene un supremo:
\begin{equation}
  \label{eq:sigma}
  \sigma(T)=\underset{\mecanica\in\mathcal{D}}{\sup}\big\{S(T,\mecanica_1)-S(T,\mecanica_2)\big\}
\end{equation}
La funci{\'o}n $\sigma$ existe y es mon{\'o}tona creciente por lo menos en las cercan{\'\i}as de $T=0$. La funci{\'o}n depende de las propiedades termof{\'\i}sicas del sistema objeto de estudio. Lo que sigue describe las propiedades m{\'a}s relevantes de esta funci{\'o}n en estas cercan{\'\i}as.

Consideremos un sistema que satisface la hip{\'o}tesis I cuyo estado de equilibrio est{\'e} definido por una temperatura y un valor del par{\'a}metro mec{\'a}nico. La entrop{\'\i}a del sistema vale  $S(T,\mecanica)$. Aum{\'e}ntese la entrop{\'\i}a del sistema de forma isoterma; la existencia de $\sigma$ asegura que la entrop{\'\i}a final no puede exceder de $S(T,\mecanica)+\sigma$. El mismo argumento sirve para una disminuci{\'o}n de entrop{\'\i}a. Por lo tanto, $S(T,\mecanica)$ es una funci{\'o}n de $\mecanica$ con cota superior en inferior para una temperatura dada, y existen las siguientes funciones:
\begin{equation}
  \label{eq:sigma2}
  \umax{S}(T)=\underset{\mecanica\in\mathcal{D}}{\sup}\{S(T,\mecanica)\};\quad\umin{S}(T)=\underset{\mecanica\in\mathcal{D}}{\inf}\{S(T,\mecanica)\}
\end{equation}

Como la condici{\'o}n de estabilidad $(\partial S/\partial T)_\mecanica>0$ es v{\'a}lida en las cercan{\'\i}as de  $T=0$ ---excepto quiz{\'a} en  $T=0$--- las funciones anteriores son crecientes en $T$ de forma que estados del tipo $\{T,S>\umax{S}(T)\}$ y $\{T,S<\umin{S}(T)\}$ no pueden existir.  As{\'\i}, los estados de equilibrio no llenan el plano $T-S$ y dos fronteras aparecen si se cumple el teorema de Nernst como una condici{\'o}n uniforme.

Con m{\'a}s detalle podr{\'\i}a decirse que uno de los grandes objetivos del tercer principio de la termodin{\'a}mica es asegurar que la entrop{\'\i}a tiene un {\'u}nico valor en la isoterma cero\cite{kestin-1968ii}.  En la formulaci{\'o}n cl{\'a}sica del teorema, se excluyen los puntos del tipo $\{T=0,S\neq S_0\}$\cite[Figura~23.5]{kestin-1968ii} en un diagrama $T-S$  (estos puntos se representan por el s{\'\i}mbolo  $\times$ en \fref{fig:f}). Desde un punto de vista f{\'\i}sico ``cuando un punto es excluido debemos demandar lo mismo en una peque{\~n}a regi{\'o}n alrededor del punto''\cite{caratheodory-1909}\cite[p\'ag.~236]{kestin-1976}.\footnote{Aqu{\'\i}, Carath\'edory est{\'a} hablando sobre la inaccesibilidad adiab{\'a}tica y prepara su conocido Axioma~II. Debe destacarse que el axioma dice, literalmente, que``(\ldots) existen estados a cuyas cercan{\'\i}as no puede accederse \emph{arbitrariamente} por un proceso adiab{\'a}tico'' en vez del concepto ``estados que son \emph{inaccesibles} por un proceso adiab{\'a}tico.'' El concepto ``arbitrariamente pr{\'o}ximo'' desempe{\~n}an un papel importante en la discusi{\'o}n posterior.} Por el contrario si una cercan{\'\i}a de  $\{T=0,S\neq S_0\}$ pudiera alcanzarse la exclusi{\'o}n de estos puntos ---aislados--- ser{\'\i}a enteramente ficticia.

As{\'\i}, el diagrama $T-S$ (v{\'e}ase \fref{fig:f}) consiste en una regi{\'o}n ~I de valores permitidos  de $\{T,S\}$, y la regi{\'o}n~II, prohibida. La existencia de esta regi{\'o}n prohibida es una consecuencia de la hip{\'o}tesis I. Las fronteras ---que pueden o no ser accesibles--- no coinciden con el eje $T=0$. En \fref{fig:f}  y en la discusi{\'o}n precedente hemos hecho uso de la hip{\'o}tesis de Planck por claridad. El argumento tambi{\'e}n vale si la entrop{\'\i}a $S_0$ tiende a $-\infty$ mientras que, simult{\'a}neamente, $\Delta S$ tiende a cero.

\begin{figure}[tbp]
  \centering
\includegraphics[bb=143 381 492 726,scale=0.6]{nernst.eps}
\caption{El diagrama $T-S$ y el teorema de Nernst. El s{\'\i}mbolo  $\times$ representa los puntos excluido cl{\'a}sicamente por el teorema\protect\cite[Figura~23.5]{kestin-1968ii}. Un an{\'a}lisis m{\'a}s pausado del teorema revela que existe una regi{\'o}n~I cuyos puntos representan estados de equilibrio, y una regi{\'o}n~II que no lo hace. El ciclo 1-2-3-4-1 es una m{\'a}quina constituida por dos isotermas 1-2 y 3-4 y dos procesos 2-3 y 4-1 que difieren en un transporte paralelo de entrop{\'\i}a $S'=S+\Lambda$. El {\'a}rea 1-2-3-4-1 es igual a $\Lambda[T-\sigma^{-1}(\Lambda)]$ como en una m{\'a}quina de Carnot. La figura sirve incluso si $S_0\to-\infty$ mientras que $\Delta S\to0$}
\label{fig:f}
\end{figure}



El siguiente punto es conocido, sin embargo la hip{\'o}tesis~I enriquece y clarifica su significado.

\paragraph{Procesos que terminan (enunciado de inaccesibilidad):}
\label{con:unattainability}

Consideremos un proceso isentr{\'o}pico $S=\Sigma_0$ que empieza a una temperatura tal que  $\umin{S}<\Sigma_0<\umax{S}$. El proceso continua hasta una temperatura  $T_1$ definida por $\umax{S}(T_1)=\Sigma_0$. Esta temperatura no es cero.

Aqu{\'\i}, podemos disminuir la entrop{\'\i}a del sistema isot{\'e}rmicamente hasta que la condici{\'o}n  $\Sigma_1=\Sigma_0-\sigma(T_1)=\umin{S}(T_1)$ se alcance. En este punto, un nuevo proceso isentr{\'o}pico enfr{\'\i}a el sistema hasta $T_2$ definida por $\umax{S}(T_2)=\Sigma_1$.

El proceso en escalera sin fin que lleva al cero absoluto ha quedado definido.


\paragraph{Anulaci{\'o}n de los coeficientes de expansi{\'o}n t{\'e}rmica}
\label{con:expansion}
 
Los coeficientes de expansi{\'o}n t{\'e}rmica se relacionan con la derivada $(\partial S/\partial\mecanica)_T$ a trav{\'e}s de relaciones de Maxwell\cite{kestin-1968ii}. De  \eref{eq:uniform} se deriva  la anulaci{\'o}n de estas derivadas puesto que:
$$
\lim_{T\to0}\left(\frac{\partial S}{\partial\mecanica}\right)_T=\lim_{T\to0}\lim_{\mecanica'\to\mecanica}\frac{S(T,\mecanica')-S(T,\mecanica)}{\mecanica'-\mecanica}\quad\quad\forall\mecanica\in\mathcal{D}
$$
Si el l{\'\i}mite doble existe, puede calcularse en el orden que se desee. Tomando primero el l{\'\i}mite en $T$ e invocando \eref{eq:uniform} se tiene, necesariamente que\cite{kestin-1968ii}:
\begin{equation}
  \label{eq:alfa}
  \lim_{T\to0}\left(\frac{\partial S}{\partial\mecanica}\right)_T=0\quad\quad\forall\mecanica\in\mathcal{D}
\end{equation}


Debe destacarse que las propiedades c y d son hechos f{\'\i}sicos ampliamente confirmados por la experiencia\cite{nernst-1924,kestin-1968ii}, al contrario que las proposiciones a y b que son de car{\'a}cter matem{\'a}tico.  Estos hechos f{\'\i}sicos son los que dan soporte a la hip{\'o}tesis. Sin embargo, la m{\'a}s importante de las consecuencias de esta hip{\'o}tesis tiene que ver con el problema de la conversi{\'o}n del calor en trabajo y ser{\'a} explicada con detalle en la siguiente secci{\'o}n.

\section{El teorema de Nernst y la producci{\'o}n de trabajo}
\label{sec:work}

En la secci{\'o}n precedente hemos mostrado que el teorema de Nernst fuerza la existencia de regiones prohibidas en el plano $T-S$. Ahora derivaremos consecuencias teniendo en cuenta que la condici{\'o}n de uniformidad introducida en la secci{\'o}n anterior hace que $\mecanica$ no desempe{\~n}e papel en el problema. 

Consideramos la siguiente cuesti{\'o}n: queremos construir un motor que produzca un trabajo mec{\'a}nico $W$ usando un fluido de trabajo dado y que extraiga una cantidad de calor dada de un foco t{\'e}rmico de temperatura tambi{\'e}n dada; \emph{\textquestiondown cu{\'a}l es la m{\'\i}nima cantidad de calor que ha de depositarse en el foco fr{\'\i}o?}\footnote{La cuesti{\'o}n analiza el comportamiento de $W$ o ---$Q'$--- puesto que el foco caliente y el calor absorbido est{\'a}n dados. Es tambi{\'e}n normal preguntarse por el comportamiento de $W$ cuando el foco fr{\'\i}o y caliente est{\'a}n dados. Este es un problema relacionado con el concepto de irreversibilidad\cite{landau-lifshitz-1968}  y esta fuera del objeto de esta discusi{\'o}n.}

La lectura cl{\'a}sica del enunciado de Kelvin-Planck s{\'o}lo precisa que el calor cedido al foco fr{\'\i}o ---en adelante ``compensaci{\'o}n''--- debe ser no nulo $Q'\neq 0$. Parece que en lo que se refiere a este enunciado un valor de $Q'$ despreciable es v{\'a}lido. As{\'\i}, $\umin{Q'}=0^+$ ---i.e. arbitrariamente pr{\'o}ximo a cero pero mayor que cero.--- Esta respuesta es independiente de $Q,T$ y del fluido de trabajo y proviene del hecho de que se tiene la impresi{\'o}n de que un ciclo de Carnot  puede situarse libremente dentro de un diagrama $T-S$ puesto que no hay m{\'a}s restricci{\'o}n m{\'a}s que $Q'\neq 0$.

Una m{\'a}quina que tuviera $\umin{Q'}=0^+$ dar{\'\i}a un rendimiento $\eta=W/Q$ tan pr{\'o}ximo a la unidad como se quiera. Nunca se ha construido una m{\'a}quina tan eficiente, ahora mostraremos que esto se debe a limitaciones fundamentales de la materia, con independencia de las limitaciones de {\'\i}ndole pr{\'a}ctico que pudieran surgir.

Como regla general, $\sigma$ \eref{eq:sigma} es no nula para temperaturas no nulas y de  \eref{eq:uniform} y \eref{eq:sigma} se obtiene:
\begin{equation}
  \label{eq:limsigma}
  \forall\epsilon>0\quad\exists\delta(\epsilon)>0:\quad T<\delta\Longrightarrow \sigma(T)<\epsilon
\end{equation}
Es decir, $\lim_{T\to0^+}\sigma(T)=0$.


En \eref{eq:limsigma} no hay ninguna necesidad de colocar los s{\'\i}mbolos de valores absolutos puesto que $T$ y $\sigma$ son cantidades positivas. De \eref{eq:limsigma} y del argumento precedente se deduce inmediatamente la existencia de la funci{\'o}n inversa $\sigma^{-1}(\Lambda)$. De hecho, la funci{\'o}n inversa no es m{\'a}s que una representaci{\'o}n adecuada del par{\'a}metro $\delta(\epsilon)$. La funci{\'o}n inversa se{\~n}ala la temperatura a la que la anchura en entrop{\'\i}a de los estados accesibles es, justamente, $\Lambda$. Esta temperatura depende de las propiedades termof{\'\i}sicas del sistema objeto de estudio.

Consideremos entonces la pregunta se{\~n}alada al principio de esta secci{\'o}n pero consideremos ahora se cumplen el enunciado de Kelvin-Planck y el teorema de Nernst tal y como se expuso en \sref{sec:intro}. Por tanto, las restricciones expuestas en \sref{sec:space}, y mostradas en  \fref{fig:f}, son v{\'a}lidas. El fluido de trabajo realiza un ciclo en el que extrae una cantidad de entrop{\'\i}a  $\Lambda=Q/T$ de la fuente caliente. Para ello es necesario que $\Lambda<\sigma(T)$.

Esta entrop{\'\i}a debe devolverse en el foco fr{\'\i}o que recibir{\'\i}a una cantidad de calor $Q'$. Para alcanzar el m{\'a}ximo rendimiento, la temperatura de este foco debe ser la menor de las posibles capaces de intercambiar esta cantidad de entrop{\'\i}a con este fluido de trabajo. Siguiendo los argumentos expuestos en los p{\'a}rrafos anteriores se deduce que esta temperatura m{\'\i}nima viene dada por $\sigma^{-1}(\Lambda)$ que es una propiedad del fluido objeto de estudio. Por tanto, 
%\nopagebreak[5]
\begin{eqnarray}
  \label{eq:eta}
  \forall Q\neq0,T\neq0:\, \frac{Q}{T}=\Lambda<\sigma(T)\nonumber\\
\Longrightarrow\exists \sigma^{-1}(\Lambda):\, Q'\geq\Lambda\times\sigma^{-1}(\Lambda)=\umin{Q'}
\end{eqnarray}
Esta m{\'a}quina se representa en  \fref{fig:f} por el ciclo $1-2-3-4-1$ constituido por dos isotermas y dos procesos en los que se produce un transporte paralelo de entrop{\'\i}a; los intercambios de energ{\'\i}a y entrop{\'\i}a en $2-3$ se cancelan con los de $4-1$ de forma que las fuentes involucradas en estos procesos no desempe{\~n}an ning{\'u}n papel en el problema.\footnote{Ser{\'\i}a posible disminuir $Q'$ si se considera el ciclo  $1-2-3-5-4-1$ ---v{\'e}ase \fref{fig:f}--- porque la entrop{\'\i}a ser{\'\i}a intercambiada con focos de menor temperaturas; sin embargo la condici{\'o}n de dos focos se rompe. Incluso en este caso, la compensaci{\'o}n se expresa por  $\sigma''\Lambda$ donde $\sigma''$ es una temperatura desconocida perteneciente al intervalo entre $T_5$ y $T_4=\sigma^{-1}(Q/T)$. La discusi{\'o}n que sigue es v{\'a}lida, igualmente, para esta compensaci{\'o}n} El trabajo producido es igual a   $\umax{W}=\Lambda\times\left[T-\sigma^{-1}(\Lambda)\right]$.

El valor de  $\umin{Q'}$ es una funci{\'o}n de los par{\'a}metros del problema: $Q, T$ y el fluido de trabajo que entra a trav{\'e}s de $\sigma^{-1}$. Adem{\'a}s la compensaci{\'o}n m{\'\i}nima es s{\'o}lo funci{\'o}n del intercambio de entrop{\'\i}a $\Lambda$. 

Aunque el valor de $\umin{Q'}$ depende, para cada situaci{\'o}n, de las propiedades del fluido de trabajo, es un hecho muy destacable que, como regla general, $\umin{Q'}$ no puede ser una cantidad arbitrariamente pr{\'o}xima a cero para un $\Lambda$ dado puesto que, seg{\'u}n el teorema de Nernst, $\sigma^{-1}$ tampoco lo es (v{\'e}ase \fref{fig:f}).

\begin{figure}[tbp]
  \centering 
 \includegraphics[bb=149 444 457 721,scale=0.6]{qfrio.eps}
  \caption{Diagrama $\Lambda-Q'$ para una sustancia de trabajo. Un punto de la regi{\'o}n I, pro ejemplo $a$, representa un conjunto de m{\'a}quinas que depositan todas ellas la misma cantidad de calor $Q'$ en el foco fr{\'\i}o, e intercambian la misma cantidad de entrop{\'\i}a entre el foco caliente y el fr{\'\i}o. La l{\'\i}nea gruesa ---que depende del fluido de trabajo--- representa \eref{eq:eta} y tiende a  $\{0,0\}$ con pendiente cero. Para un fluido de trabajo, es imposible construir una m{\'a}quina que se sit{\'u}e en la regi{\'o}n~II. El s{\'\i}mbolo $\times$ representa la restricci{\'o}n expuesta por el enunciado de Kelvin-Planck en su lectura cl{\'a}sica.}
  \label{fig:etamax}
\end{figure}
El car{\'a}cter universal de  \eref{eq:eta} permite dibujar un diagrama  $\Lambda-Q'$ (v{\'e}ase  \fref{fig:etamax}). Obs{\'e}rvese que en el contexto de la lectura cl{\'a}sica del enunciado de Kelvin-Planck este diagrama no tendr{\'\i}a m{\'a}s restricci{\'o}n que la exclusi{\'o}n de los puntos  $\{\Lambda\neq0,Q'=0\}$, que son destacados en la figura por el s{\'\i}mbolo $\times$. Pero si ahora tenemos en cuenta el teorema de Nernst es evidente que puntos del tipo $\{\Lambda,Q'<\umin{Q'}(\Lambda)\}$ tampoco son posibles. Estos puntos definen una regi{\'o}n  cuya frontera viene dada por  \eref{eq:eta}; la analog{\'\i}a entre las regiones~I y II y la frontera dibujadas en   \fref{fig:f} con aquellas de \fref{fig:etamax} es inmediata. Del teorema de Nernst se deduce f{\'a}cilmente que $\umin{Q'}$ tiende a cero con pendiente cero cuando $\Lambda$ tiende a cero. Resumiendo el significado de  \fref{fig:etamax} debe destacarse que el teorema de Nernst excluye una regi{\'o}n alrededor de los puntos realmente excluidos por la lectura cl{\'a}sica del enunciado de Kelvin-Planck. Esto est{\'a} de acuerdo con las palabras de Carath\'eodory expuestas en \sref{sec:space}.

El l{\'\i}mite $\Lambda\to0^+$ es la {\'u}nica forma de conseguir  $\umin{Q'}\to0^+$. En este caso, y tomando $T$ como una constante acotada, es claro que $Q$ tambi{\'e}n tiende a cero y, como consecuencia, tambi{\'e}n el trabajo. Por tanto:
  \begin{equation}
    \label{eq:qmin}
    Q'\to0^+\Longrightarrow W\to0^+
  \end{equation}
Esta proposici{\'o}n contiene la esencia del teorema de Nernst. Su significado m{\'a}s dram{\'a}tico se observa mejor si se nota que  $Q'\to0^+$ y $\eta\to1^-$ son equivalentes. Se{\~n}ala por tanto que  \emph{conforme se consigue la m{\'a}quina m{\'a}s eficiente, la producci{\'o}n de trabajo va disminuyendo hasta cero}. 

Debe destacarse una vez m{\'a}s que en la discusi{\'o}n precedente $\mecanica$ no desempe{\~n}a ning{\'u}n papel y en este sentido las restricciones de \fref{fig:etamax} y \eref{eq:qmin} son \emph{universales}. Este hecho proviene de la uniformidad. Si, al contrario, el teorema de Nernst se lee como un simple l{\'\i}mite ---sin el requerimiento de uniformidad--- entonces \eref{eq:qmin} y \fref{fig:etamax} son s{\'o}lo v{\'a}lidas para transiciones entre dos valores dados de $\mecanica$  ---v{\'e}ase \cite[Figura~23.9]{kestin-1968ii}\cite{liboff-physicsessays-94}.--- Por lo tanto, \eref{eq:weak} no conduce a ninguna restricci{\'o}n real ya que la frontera de \fref{fig:etamax} puede situarse tan pr{\'o}xima a la restricci{\'o}n expuesta por la lectura cl{\'a}sica del enunciado de Kelvin-Planck con tal de que se consideren valores apropiados de $\mecanica_1$ y $\mecanica_2$. As{\'\i}, la mejora del rendimiento de cualquier m{\'a}quina ser{\'\i}a una cuesti{\'o}n de limitaciones pr{\'a}cticas si \eref{eq:weak} fuera valida.\footnote{Si la sustancia de trabajo se comporta como  \eref{eq:modelo}, la m{\'a}quina conseguir{\'\i}a mejores eficiencias si se alcanzan valores mayores de  $\mecanica$.}

\section{El teorema de Nernst y el enunciado del segundo principio}
\label{sec:second}
 
\Sref{sec:work} muestra claramente que el teorema de Nernst restringe la conversi{\'o}n del calor en trabajo en una forma que es independiente de la configuraci{\'o}n mec{\'a}nica del sistema. Es deseable un enlace entre el enunciado del segundo principio y el teorema.

Uno de los resultados m{\'a}s importantes  de  \sref{sec:work} es el importante papel que desempe{\~n}a  $Q'$, la compensaci{\'o}n, en el problema de la conversi{\'o}n del calor en trabajo. Su importancia proviene del hecho de que su valor m{\'\i}nimo viene dado por una expresi{\'o}n universal  \eref{eq:eta}.  

Desde un punto de vista estrictamente hist{\'o}rico el papel del foco fr{\'\i}o has sido la clave de los enunciados del segundo principio. As{\'\i}, el primer enunciado ---debido a Kelvin\cite{thomson-1853},--- expresa que es imposible construir una m{\'a}quina que, a la vez, produzca trabajo y enfr{\'\i}e la m{\'a}s fr{\'\i}a de las fuentes disponibles, sin importar qu{\'e} ocurra con otras fuentes ---m{\'a}s calientes--- (v{\'e}ase \fref{fig:staircase}).

Planck simplific{\'o} el enunciado al notar que es imposible construir una m{\'a}quina que produzca trabajo enfriando s{\'o}lo una fuente\cite{planck-1927}. En este sentido, una fuente m{\'a}s fr{\'\i}a debe calentarse en una cierta cantidad ---compensaci{\'o}n--- (v{\'e}ase \fref{fig:staircase}).  Sin embargo, como no incluy{\'o} ninguna condici{\'o}n sobre el ``tama{\~n}o'' de la compensaci{\'o}n, se espera que pueda ser tan peque{\~n}a como sea posible: esta es la lectura ``cl{\'a}sica'' del enunciado de Kelvin-Planck que se ha mantenido desde su formulaci{\'o}n por Planck.

Pero adem{\'a}s, si el teorema de Nernst entra en consideraci{\'o}n a trav{\'e}s de la hip{\'o}tesis~I, se obtiene la {\'u}ltima restricci{\'o}n del problema: una compensaci{\'o}n m{\'\i}nima queda determinada por las propiedades de la sustancia de trabajo y el intercambio de entrop{\'\i}a (v{\'e}ase \eref{eq:eta} y \fref{fig:staircase}): \emph{si se quiere transformar una cantidad finita de calor en trabajo hay que pagar, efectivamente, un impuesto ---una compensaci{\'o}n--- es decir el impuesto no puede hacerse indefinidamente peque{\~n}o por voluntad del experimentador}.\footnote{En la naturaleza no existen ``para{\'\i}sos fiscales''.} El texto con {\'e}nfasis es un enunciado informal de  \eref{eq:qmin}.

\begin{figure}[tb]
  \centering  
  \includegraphics[bb=137 446 592 726,scale=0.5]{staircase.eps}
  \caption{La secuencia de enunciados del segundo principio. El lector debe reconocer las restricciones sobre la compensaci{\'o}n que provienen de la violaci{\'o}n de los diferentes enunciados. De izquierda a derecha, una violaci{\'o}n del enunciado de Kelvin, una violaci{\'o}n de la lectura cl{\'a}sica del enunciado de Kelvin-Planck, y una violaci{\'o}n de una lectura m{\'a}s razonada de este enunciado y que incluye el teorema de Nernst. En este {\'u}ltimo caso $\umin{Q'}$ viene dado por \eref{eq:eta}.}  \label{fig:staircase}
\end{figure}

El lector debe notar que, de hecho,  \eref{eq:qmin}  es algo m{\'a}s que el simple enunciado del teorema de Nernst puesto que incide sobre el problema del enunciado de Kelvin-Planck. Es el dintel que corona el leitmotiv de los principios de la termodin{\'a}mica al se{\~n}alar la {\'u}ltima restricci{\'o}n sobre la compensaci{\'o}n. Si esta relaci{\'o}n se describiera en forma negativa nos encontrar{\'\i}amos con algo muy parecido a un enunciado del segundo principio: \emph{prohibe la producci{\'o}n de un trabajo \emph{finito} sin una compensaci{\'o}n \emph{finita}.}



\section{Del enunciado de Kelvin-Planck al teorema de Nernst}
\label{sec:inverse}


Las secciones anteriores han analizado el teorema de Nernst a trav{\'e}s de la hip{\'o}tesis~I poniendo de manifiesto su relaci{\'o}n con el problema de la conversi{\'o}n del calor en trabajo. Ahora deseamos realizar el viaje contrario: comenzando por el enunciado de Kelvin-Planck, \textquestiondown bajo qu{\'e} circunstancias se obtiene la hip{\'o}tesis~I?

En nuestra opini{\'o}n puede decirse que la lectura cl{\'a}sica del enunciado de Kelvin-Planck es ``cruda'' en el sentido de que la causa ---el calor absorbido por de la fuente caliente--- y el efecto inevitable ---la compensaci{\'o}n m{\'\i}nima--- se han considerado desacopladas desde los comienzos de la termodin{\'a}mica. Una lectura m{\'a}s razonada del enunciado del segundo principio debi{\'o} haber concluido que debe existir un cierto acoplamiento entre $\umax{W}$ o $\umin{Q}$ y $Q$ puesto que s{\'o}lo una cantidad inifinitesimalmente peque{\~n}a de $Q$ puede dar lugar a una compensaci{\'o}n infinitesimalmente peque{\~n}a. Ahora se{\~n}alaremos que esta hip{\'o}tesis ---hip{\'o}tesis~II--- es suficiente para el teorema de Nernst:
\begin{quote}
  \emph{la compensaci{\'o}n de una m{\'a}quina de Carnot tiende a cero \emph{s{\'o}lo si} el calor absorbido del foco caliente tiende a cero}
\end{quote}

La importancia de la hip{\'o}tesis vendr{\'a} dada por las conclusiones que se deriven de ella pero, antes de nada el lector no debe tomarla como una hip{\'o}tesis adicional al enunciado de Kelvin-Planck puesto que est{\'a} incluida, impl{\'\i}citamente, en las palabras del enunciado. La clave de la cuesti{\'o}n es la palabra ``efectos'' que aparece en el enunciado. Se puede conseguir expl{\'\i}citamente la hip{\'o}tesis~II si se sustituye esta palabra por ``efectos finitos'' de forma que todo calor absorbido ``finito'' conduzca, necesariamente, a una compensaci{\'o}n ``finita''. Sin embargo ha de entenderse que todo ``efecto'' es, por definici{\'o}n, ``finito'' y por tanto est{\'a} modificaci{\'o}n es un pleonasmo incompatible con la descripci{\'o}n en palabras de una ley de la naturaleza. 

El hecho de que la hip{\'o}tesis~II conduzca el teorema de Nernst es, sorprendentemente, elemental. T{\'e}ngase en cuenta que la compensaci{\'o}n de una m{\'a}quina equivale a $Q'=T'\times\Lambda$ donde $T'$ es la temperatura del foco fr{\'\i}o y  $\Lambda$ la entrop{\'\i}a intercambiada. Esta expresi{\'o}n puede hacerse cero (1) si $T'$ tiende a cero, con independencia de lo que haga $\Lambda$, o (2) si  $\Lambda$ tiende a cero con independencia de lo que haga $T'$. La hip{\'o}tesis excluye la opci{\'o}n (1), es decir excluye cualquier cambio de entrop{\'\i}a en las cercan{\'\i}as del cero absoluto. Por tanto, el teorema de Nernst, tal y como se enunci{\'o} en  \sref{sec:intro} y se analiz{\'o} en \sref{sec:space} queda demostrado.

Se concluye entonces que las hip{\'o}tesis I y II son proposiciones equivalentes de forma que el teorema de Nernst proviene, en {\'u}ltima instancia, de una lectura razonada del enunciado de Kelvin-Planck a trav{\'e}s de la hip{\'o}tesis II.\footnote{As{\'\i}, el segundo principio garantiza que la entrop{\'\i}a tiende a un {\'u}nico valor ---que no depende de $\mecanica$---- cuando la temperatura tiende a cero. El valor de la entrop{\'\i}a en el cero absoluto, o espec{\'\i}ficamente si es finito o infinito ---formulaci{\'o}n de Planck--- es ajeno al an{\'a}lisis realizado en este art{\'\i}culo puesto que el segundo principio s{\'o}lo es sensible a  variaciones de entrop{\'\i}a y el problema de su  valor absoluto queda indeterminado.} Esta proposici{\'o}n es v{\'a}lida, al menos en tanto que nos refiramos a sistemas capaces de convertir calor en trabajo.





\section{Discusi{\'o}n y conclusi{\'o}n}
\label{sec:discu}



En 1909 Carath\'eodory\cite{caratheodory-1909}, siguiendo una sugerencia de Born, consigui{\'o} traducir los enunciados cl{\'a}sicos del segundo principio ---que hablan del problema de la conversi{\'o}n del calor en trabajo--- en un enunciado que habla de las propiedades de un sistema aislado. Este trabajo presenta el camino contrario para el teorema de Nernst. Se ha supuesto que el teorema expresa propiedades de los sistemas en las cercan{\'\i}as de $T\to0$; el estudio que aqu{\'\i} presentamos (v{\'e}ase \sref{sec:work}, especialmente \eref{eq:eta} y \eref{eq:qmin}) relaciona el teorema con el problema de la producci{\'o}n de trabajo. Estar{\'\i}a expresando una propiedad universal que debe satisfacer toda motor t{\'e}rmico.

El lector puede preguntarse qu{\'e} hip{\'o}tesis hacen al teorema de Nernst independiente del enunciado de Kelvin-Planck y cu{\'a}les no. El objetivo del segundo principio es restringir la producci{\'o}n continua de trabajo se{\~n}alando una asimetr{\'\i}a fundamental: el trabajo puede disiparse en forma de calor pero lo contrario no. Este objetivo se manifiesta en un enunciado que expresa, en palabras, una restricci{\'o}n. Lo que hemos mostrado en este trabajo es que el grado de restricci{\'o}n importa y tiene influencia en las propiedades de los sistemas. La lectura cl{\'a}sica del enunciado de Kelvin-Planck asume s{\'o}lo que la compensaci{\'o}n es non nula. Con esta lectura, las propiedades de los sistemas en las cercan{\'\i}as de $T=0$ tiene que venir expresadas por una ley adicional e independiente. Sin embargo, una lectura m{\'a}s razonada, a trav{\'e}s de la hip{\'o}tesis~II presentada en \sref{sec:inverse} conduce a alguna de estas propiedades: la inaccesibilidad del cero absoluto y la anulaci{\'o}n de los coeficientes de expansi{\'o}n t{\'e}rmicos. Sin embargo, se{\~n}alamos de nuevo que la formulaci{\'o}n es insensible a la cuesti{\'o}n de si los calores espec{\'\i}ficos se anulan o no en el cero absoluto.

La nueva lectura del enunciado evita el embarazoso hecho de que $W$ debe diferir de $Q$ ---la cuesti{\'o}n es \textquestiondown en qu{\'e} cantidad deben diferir?--- se{\~n}alando que deben hacerlo en una cantidad mensurable que depende de la sustancia de trabajo la cual tambi{\'e}n desempe{\~n}a un papel en el problema.  Las palabras de  Carath\'eodory citadas en \sref{sec:space} vuelven a dar en el clavo: si la condici{\'o}n $W=Q$ ha de excluirse, $W=Q^-$ tambi{\'e}n debi{\'o} haberse excluido.\footnote{Presentamos una analog{\'\i}a entre las palabras de  Carath\'eodory y los resultados de este trabajo. No es intenci{\'o}n de los autores se{\~n}alar que lo uno derive de lo otro o al rev{\'e}s. Es una coincidencia que ambos problemas traten sobre la misma ley natural.} 

Una excepci{\'o}n a las hip{\'o}tesis aqu{\'\i} presentadas resultar{\'\i}a en un fallo de las consecuencias b y c expuestas en la secci{\'o}n \sref{sec:space} es decir: (i) una experiencia que permitiera aumentar $1/T$ indefinidamente, (ii) una experiencia que permitiera disminuir la entrop{\'\i}a de forma indefinida, o (iii) una sustancia cuyos coeficientes de expansi{\'o}n t{\'e}rmica no se anulasen en el cero absoluto.\footnote{El lector debe observar que la inaccesibilidad del cero absoluto no garantiza el teorema de Nernst. El modelo $S(T,\mecanica)=\chi\log T\times (\mecanica+a)/(\mecanica+b)$ con $\chi,a\neq b>0,\mecanica\in\mathbb{R}^+$ muestra un ejemplo matem{\'a}tico de sistema que excluye experimentos del tipo (i) y (ii) pero no satisface el teorema de Nernst.} 

Esta fuera del {\'a}mbito de este trabajo describir las relevancia microsc{\'o}pica de nuestras hip{\'o}tesis, es decir determinar qu{\'e} tipo de hamiltonianos ---interacciones--- conducen a una entrop{\'\i}a que se desvanece uniformemente en $T=0$ as{\'\i} como las propiedades de simetr{\'\i}as de estos hamiltonianos. Modelos reconocidos ---especialmente gases ideales--- no satisfacen estas hip{\'o}tesis. Sin embargo, el lector no debe concluir que esto sea una ``excepci{\'o}n'' a las hip{\'o}tesis puesto que {\'e}stas s{\'o}lo se fundamentan en datos experimentales macrosc{\'o}picos y no en el an{\'a}lisis de modelos.

La relaci{\'o}n entre modelos no interaccionantes y el teorema de Nernst ha sido sugerida recientemente\cite{balian-91,roseinnes-ajp-99} y parece inevitable la presencia de interacciones en las cercan{\'\i}as del cero absoluto. Esta sugerencia proviene del an{\'a}lisis del modelos de espines independientes que no satisface el teorema de Nernst puesto que su estado fundamental es degenerado. Sin embargo, este tipo de sistemas no se presenta en la naturaleza puesto que los s{\'o}lidos reales exhiben ordenaci{\'o}n y correlaci{\'o}n de espines a temperaturas suficientemente bajas. La ordenaci{\'o}n resultante ---usualmente una nueva fase ferromagn{\'e}tica o antiferromagn{\'e}tica--- s{\'\i} satisface el teorema.

Un an{\'a}lisis similar vale para sistemas de part{\'\i}culas. El lector debe notar que el gas ideal cl{\'a}sico permite imaginar procesos del tipo (i) y (ii), adem{\'a}s no satisface  (iii) en contradicci{\'o}n con el teorema de Nernst. Por su parte, el gas ideal cu{\'a}ntico\cite{landau-lifshitz-1968} dan coeficientes de expansi{\'o}n t{\'e}rmica nulos en el cero absoluto, y excluyen experiencias del tipo (ii) puesto que la entrop{\'\i}a est{\'a} acotada. Sin embargo a{\'u}n permiten experiencias del tipo (i) que no ocurren en la naturaleza y que violan en significado del teorema de Nernst. Por supuesto en sistemas reales la ``ordenaci{\'o}n'' est{\'a} siempre presente y la formaci{\'o}n de fases condensadas parece inevitable.

Es muy posible por tanto que modelos no interaccionantes ---sean cin{\'e}ticos, magn{\'e}ticos o de cualquier otra {\'\i}ndole--- no describen de forma precisa las propiedades de los sistemas \emph{reales} a temperaturas suficientemente bajas puesto que las interacciones no pueden ser despreciadas en este caso.

\renewcommand{\acknowledgmentsname}{Agradecimientos}
%\begin{acknowledgements}
\acknowledgements
 Los autores desean mostrar su gratitud para con D.~Jos\'e Antonio P\'erez G\'omez por las largas discusiones mantenidas y, especialmente, por iluminarlas con el concepto de convergencia uniforme. Tambi{\'e}n agradecen a D\~na.~Pilar N\'u\~nez su amable b{\'u}squeda de registros bibliogr{\'a}ficos antiguos. Este trabajo has sido financiado por el \emph{Ministerio de Ciencia y Tecnolog\'\i a} de Espa{\~n}a bajo el proyecto BFM2002-02237.
  


\bibliographystyle{apsrev}
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\end{document}


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  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Planck}(1927)}]{planck-1927}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Planck}},
  \emph{\bibinfo{title}{Treatise on Thermodynamics}}
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\bibitem[{\citenamefont{Thomson}(1853)}]{thomson-1853}
\bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Thomson}},
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\bibitem[{\citenamefont{Clausius}(1854)}]{clausius-1854}
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\bibitem[{\citenamefont{Carath{\'e}odory}(1909)}]{caratheodory-1909}
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\bibitem[{\citenamefont{Nernst}(1906)}]{nernst-1906}
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\bibitem[{\citenamefont{Nernst}(1924)}]{nernst-1924}
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  \emph{\bibinfo{title}{The New Heat Theorem: Its Foundations in Theory and
  Experimental}} (\bibinfo{publisher}{Dover}, \bibinfo{year}{1924}).

\bibitem[{\citenamefont{Einstein}(1921)}]{einstein-1921}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Einstein}}, in
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  (\bibinfo{publisher}{Gauthiers-Villars}, \bibinfo{year}{1921}), Rapports et
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\bibitem[{\citenamefont{Epstein}(1937)}]{epstein-1937}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Epstein}},
  \emph{\bibinfo{title}{A Textbook of Thermodynamics}}
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\bibitem[{\citenamefont{Kestin}(1968)}]{kestin-1968ii}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Kestin}},
  \emph{\bibinfo{title}{A course in Thermodynamics}}, vol.~\bibinfo{volume}{II}
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\bibitem[{\citenamefont{Landau and Lifshitz}(1968)}]{landau-lifshitz-1968}
\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Landau}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Lifshitz}},
  \emph{\bibinfo{title}{Statistical Physics}} (\bibinfo{publisher}{Pergamon
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\bibitem[{\citenamefont{Landsberg}(1978)}]{landsberg-78}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Landsberg}},
  \emph{\bibinfo{title}{Thermodynamics and Statistical Mechanics}}
  (\bibinfo{publisher}{Oxford Univ. Press}, \bibinfo{year}{1978}), ISBN
  \bibinfo{isbn}{0-19-851142-6}.

\bibitem[{\citenamefont{Baierlein}(1999)}]{baierlein-99}
\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Baierlein}},
  \emph{\bibinfo{title}{Thermal {P}hysics}} (\bibinfo{publisher}{Cambridge
  University Press}, \bibinfo{year}{1999}), ISBN \bibinfo{isbn}{0-521-65838-1}.

\bibitem[{\citenamefont{Zemansky}(1968)}]{zemansky-1968}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Zemansky}},
  \emph{\bibinfo{title}{Heat and Thermodynamics}}
  (\bibinfo{publisher}{McGraw-Hill}, \bibinfo{year}{1968}).

\bibitem[{\citenamefont{Balian}(1991)}]{balian-91}
\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Balian}},
  \emph{\bibinfo{title}{From microphysics to macrophysics}}
  (\bibinfo{publisher}{Springer Verlag}, \bibinfo{address}{Berlin Heidelberg},
  \bibinfo{year}{1991}), ISBN \bibinfo{isbn}{0-387-53266-8}.

\bibitem[{\citenamefont{Huang}(1987)}]{huang-87}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Huang}},
  \emph{\bibinfo{title}{Statistical {M}echanics}} (\bibinfo{publisher}{John
  Willey and Sons}, \bibinfo{year}{1987}), ISBN \bibinfo{isbn}{0-471-85913-3}.

\bibitem[{\citenamefont{Maf{\'e} and {de la Rubia}}(1998)}]{mafe-ajp-98}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Maf{\'e}}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{{de la Rubia}}},
  \bibinfo{journal}{American Journal of Physics} \textbf{\bibinfo{volume}{66}},
  \bibinfo{pages}{277} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Rose-Innes}(1999)}]{roseinnes-ajp-99}
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Rose-Innes}},
  \bibinfo{journal}{American Journal of Physics} \textbf{\bibinfo{volume}{67}},
  \bibinfo{pages}{27} (\bibinfo{year}{1999}).

\bibitem[{\citenamefont{Wu and Widom}(1998)}]{wu-pre-98}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Wu}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Widom}},
  \bibinfo{journal}{Physical Review E} \textbf{\bibinfo{volume}{57}},
  \bibinfo{pages}{5178} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Falk}(1959)}]{falk-pr-1959}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Falk}},
  \bibinfo{journal}{Physical Review} \textbf{\bibinfo{volume}{115}},
  \bibinfo{pages}{249} (\bibinfo{year}{1959}).

\bibitem[{\citenamefont{Yan and Chen}(1988)}]{yan-chen-jpa-88}
\bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Yan}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Chen}},
  \bibinfo{journal}{Journal of Physics A: Math. Gen.}
  \textbf{\bibinfo{volume}{21}}, \bibinfo{pages}{707} (\bibinfo{year}{1988}).

\bibitem[{\citenamefont{Landsberg}(1989)}]{landsberg-jpa-89}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Landsberg}},
  \bibinfo{journal}{Journal of Physics A: Math. Gen.}
  \textbf{\bibinfo{volume}{22}}, \bibinfo{pages}{139} (\bibinfo{year}{1989}).

\bibitem[{\citenamefont{Oppenheim}(1989)}]{oppenheim-jpa-89}
\bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Oppenheim}},
  \bibinfo{journal}{Journal of Physics A: Math. Gen.}
  \textbf{\bibinfo{volume}{22}}, \bibinfo{pages}{143} (\bibinfo{year}{1989}).

\bibitem[{\citenamefont{Liboff}(1994)}]{liboff-physicsessays-94}
\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Liboff}},
  \bibinfo{journal}{Physics Essays} \textbf{\bibinfo{volume}{7}},
  \bibinfo{pages}{95} (\bibinfo{year}{1994}).

\bibitem[{\citenamefont{Landsberg}(1997)}]{landsberg-ajp-97}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Landsberg}},
  \bibinfo{journal}{American Journal of Physics} \textbf{\bibinfo{volume}{65}},
  \bibinfo{pages}{269} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Belgiorno}(2003)}]{belgiorno-pra-03a}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Belgiorno}},
  \bibinfo{journal}{Journal of Physics A: Math. Gen.}  (\bibinfo{year}{2003}),
  \eprint{physics/0210037}.

\bibitem[{\citenamefont{Jones}(1972)}]{jones-72}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Jones}},
  \emph{\bibinfo{title}{Funciones Generalizadas}} (\bibinfo{publisher}{Urmo},
  \bibinfo{address}{Bilbao}, \bibinfo{year}{1972}).

\bibitem[{\citenamefont{Jones}(1982)}]{jones-82}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Jones}},
  \emph{\bibinfo{title}{The theory of generalised functions}}
  (\bibinfo{publisher}{Cambridge University Press}, \bibinfo{year}{1982}), ISBN
  \bibinfo{isbn}{0521237238}.

\bibitem[{\citenamefont{Kestin}(1976)}]{kestin-1976}
\bibinfo{editor}{\bibfnamefont{J.}~\bibnamefont{Kestin}}, ed.,
  \emph{\bibinfo{title}{The {S}econd {L}aw of {T}hermodynamics}},
  vol.~\bibinfo{volume}{5} of \emph{\bibinfo{series}{Benchmark Papers on
  Energy}} (\bibinfo{publisher}{Dowden, Hutchinson \& Ross, Inc.},
  \bibinfo{year}{1976}).

\end{thebibliography}
