Additivity of the Clausius Entropy

1. Introduction

There are several ways of introducing a quantity called entropy, both statistical and nonstatistical. Most of these entropies are additive in typical equilibrium scenarios, like the Boltzmann-Gibbs entropy [1] or the Lieb-Yngvason entropy [2]. However, some entropies are nonadditive, like the Tsallis entropy [3]. This paper is only concerned with the nonstatistical entropy as introduced by Clausius. (The original works by Clausius, [4] and [5], have been incorporated into many textbooks, like [6].) Instead of referring to this quantity as Clausius entropy, we will simply call it entropy.

Most textbooks state that the entropy S is additive. However, there are some possible points of criticism with regard to the presentation of many of those textbooks. For example, some of them either postulate or assume that the entropy is additive. This would be unsatisfactory if additivity could either be proved directly or be derived from weaker assumptions. Also, other textbooks claim that additivity of the entropy S is a consequence of additivity of the internal energy U. However, this statement does not seem to be universally true, as can be seen by considering two thermodynamic systems reversibly exchanging heat with their environment: We obtain the equations $\mathrm{d}{U}_1=T_1\mathrm{d}{S}_1$ and $\mathrm{d}{U}_2=T_2\mathrm{d}{S}_2$ for the two systems, where T denotes temperature. Let the internal energy be additive, $\mathrm{d}U=\mathrm{d}{U}_1+\mathrm{d}{U}_2$. For $T_1=T_2$, we can then obviously infer additivity of the entropy, $\mathrm{d}S=\mathrm{d}{S}_1+\mathrm{d}{S}_2$. For $T_1\ne T_2$, however, inferring additivity of the entropy does not seem possible.

This paper shows that there are conditions which lead to additivity of the entropy. Moreover, it is suggested that these conditions can be taken to be fulfilled for not too large composites of disjoint, adjacent thermodynamic systems existing contemporaneously, and without relative speed.

This paper is conceived as a contribution to the theory of classical thermodynamics, as derived from the laws of thermodynamics (and some additional assumptions). See [6] or other textbooks.

2. Additive Entropy

We begin with the following definition.

Definition 2.1 (Clausius cycle).  

If all heat exchange occurring during a cycle process Ξ performed by a thermodynamic system α is neutralized by the heat exchanged by a number Carnot cycles with their cold heat baths such that the composite of Ξ and all Carnot cycles may exchange heat only with the common hot heat bath $HB$ of the Carnot cycles, then this composite is called a Clausius cycle.

Remark on definition 2.1.  

The concept of a Clausius cycle implies the assumption that each heat exchange $\delta Q$ occurring at a temperature T during Ξ can be neutralized by the heat that a dedicated Carnot cycle exchanges with its cold heat bath at the temperature $T_c=T$; it is even assumed that this neutralization remains possible if $\delta Q$ is exchanged nonisothermally, $\mathrm{d}T\ne 0$. Obviously, a great number of Carnot cycles might be required for this purpose. Moreover, all these Carnot cycles must share a common hot heat bath $HB$. It is clear that a Clausius cycle can hardly be implemented in reality. Rather, it is a theoretical concept. Only the thermodynamic system α performing the cycle Ξ is supposed to exist in reality. Ξ may be any cycle; for example, it need not be reversible or quasistatic.

By considering a Clausius cycle, one can obtain (at least) two results (see [4] and [5], or [6]): (1) The Clausius cycle cannot accept heat exchange from $HB$, as this would constitute a violation of the Kelvin(-Planck) statement of the second law of thermodynamics. This then leads to the well-known Clausius inequality. (2) The Clausius cycle is reversible if and only if $\Xi$ is reversible. In this case, the heat $Q^{\left(HB\right)}$ that the reversible Clausius cycle exchanges with $HB$ must vanish, and this can be written as: $Q^{\left(HB\right)}=T^{\left(HB\right)}\oint \mathrm{d}S=0$, where $T^{\left(HB\right)}>0$ is the temperature of $HB$. This then leads to $0=\oint \mathrm{d}S$, which shows that the entropy S is a state quantity.

Theorem 2.2. 

Let there be n Clausius cycles, belonging to n thermodynamic systems ${\alpha }_i$, $1\le i\le n$. Also, let all Clausius cycles share a common hot heat bath $HB$. In this case, a Clausius inequality for the composite of the n systems exists. Moreover, the entropy of the composite can be defined as ${\sum }_{i=1}^n{S}_i$, where $S_i$ is the entropy of ${\alpha }_i$, $1\le i\le n$. This is true regardless of whether there is exchange of energy or matter between the systems. Also, it is true regardless of the particular absolute entropy scales used by the n systems.

Proof. 

The composite of the n Clausius cycles cannot accept heat exchange from $HB$, as this would constitute a violation of the Kelvin(-Planck) statement of the second law of thermodynamics. This leads to a Clausius inequality, in a way completely analogous to the case of a single system. Moreover, if all Clausius cycles are reversible, then the heat $Q^{\left(HB\right)}$ that their composite exchanges with $HB$ can obviously be written as: $Q^{\left(HB\right)}=T^{\left(HB\right)}{\sum }_{i=1}^n\oint \mathrm{d}{S}_i=0$. This leads to $0={\sum }_{i=1}^n\oint \mathrm{d}{S}_i$. Naturally, the state space $\Gamma$ of the composite system is the composite of the state spaces ${\Gamma }_i$ of the constituent systems, $\Gamma ={\Gamma }_1\times \dots \times {\Gamma }_n$. Each of the n line integrals refers to a closed curve in its respective state space, and the composite of these n curves describes a curve in the composite state space $\Gamma$. Under these circumstances, we can swap the order of summation and integration, $0=\oint \mathrm{d}\left[{\sum }_{i=1}^n{S}_i\right]$. This means that $\mathrm{d}\left[{\sum }_{i=1}^n{S}_i\right]$ is the exact differential of a state quantity ${\sum }_{i=1}^n{S}_i$, which we can define as the entropy for the composite of the n systems. This reasoning holds independent of any exchange of energy or matter between the n systems. Also, it holds independent of the particular absolute entropy scale used by each of the n systems. □

Remark on theorem 2.2. 

We offer an alternative derivation for $0={\sum }_{i=1}^n\oint \mathrm{d}{S}_i=\oint \mathrm{d}\left[{\sum }_{i=1}^n{S}_i\right]$ that does not immediately require considering state spaces: We start with $0={\sum }_{i=1}^n\oint \mathrm{d}{S}_i$. The value of each line integral is independent of its parameterization. Moreover, it is always possible to find a common parameterization for all n line integrals. For example, if ${\gamma }_i$ is the parameter for the integral i, and the limits of the integral are ${\gamma }_i^{\left(0\right)}$ and ${\gamma }_i^{\left(1\right)}$, then the substitution ${\gamma }_i=\left({\gamma }_i^{\left(1\right)}-{\gamma }_i^{\left(0\right)}\right)·\gamma +{\gamma }_i^{\left(0\right)}$ allows replacing the parameter ${\gamma }_i$ with a common parameter γ, and the integral receives the common limits 0 and 1. We may then write $0={\sum }_{i=1}^n{\int }_0^1\left(\mathrm{d}{S}_i/\mathrm{d}\gamma \right)\mathrm{d}\gamma$. Both integration and differentiation are linear operations, which leads to $0={\int }_0^1\left(\mathrm{d}\left[{\sum }_{i=1}^n{S}_i\right]/\mathrm{d}\gamma \right)\mathrm{d}\gamma$. We can now return to the unparameterized notation, $0=\oint \mathrm{d}\left[{\sum }_{i=1}^n{S}_i\right]$.

The main requirement of theorem 2.2 is that the Clausius cycles can be supposed to share a common hot heat bath $HB$. We now look for physical criteria which might prohibit fulfillment of this requirement. For example, if $HB$ has finite dimensions, then theorem 2.2 might no longer hold for very large composite systems. Further prohibitive criteria might be: (a) great distance in space between the systems, (b) great distance in time between the systems, (c) great relative speed between the systems. In turn, we suggest that theorem 2.2 is valid if absence of these prohibitive criteria is ensured. This leads to the following conclusion.

Conclusion 2.3. 

The entropy is additive for not too large composites of disjoint, adjacent systems existing contemporaneously, and without relative speed.

3. Results and Conclusions

It has been shown that entropic additivity can be taken for granted under certain conditions. A more refined examination of entropic additivity could possibly be supplied by other thermodynamic theories, like relativistic thermodynamics and nonlocal thermodynamics.