Additivity and Extensivity of the Clausius Entropy and Other Thermodynamic Quantities

1. Introduction

There are several ways of introducing a quantity called entropy, both statistical and non-statistical. This paper is only concerned with the non-statistical entropy as introduced by Clausius. (The original works by Clausius, [1] and [2], have been incorporated into many textbooks, like [3].) Instead of referring to this quantity as Clausius entropy, we will usually simply call it entropy.

Most textbooks state that the entropy S is both additive and extensive. 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 both additive and extensive. This would be unsatisfactory if additivity and extensivity can either be proved directly or be derived from weaker assumptions. Also, other textbooks claim that additivity and extensivity of the entropy S are a consequence of additivity and extensivity of the internal energy U. However, this statement does not seem to be universally true, as can be seen by contemplating additivity in the case of 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 is the 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.

The goal of this paper is to improve clarity on the subject of additivity and extensivity. Its outline is as follows: First, we show that the entropy and the internal energy can be taken to be additive for systems that exist next to each other, contemporaneously, and without relative speed. Then, after some remarks regarding the scaling of thermodynamic systems, we show that any thermodynamic state quantity is extensive if it is additive and if its volume density is finite. We argue that for many state quantities the assumption of extensivity can be replaced by the assumption of finite density. Having so far utilized only rational-valued scaling factors, we conclude with a justification of real-valued scaling factors.

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 [3] or other textbooks.

2. Additivity

The defining characteristic of a thermodynamic state quantity $\mathsf{\Phi }$ is that its line integral vanishes for a closed curve, $\oint \mathrm{d}\mathsf{\Phi }=0$.

Moreover, if $\mathsf{\Phi }$ is additive, then for n thermodynamic systems ${\alpha }_i$, $1\le i\le n$, the value $\mathsf{\Psi }\left(\alpha \right)$ of their composite system $\alpha$ is calculated as $\mathsf{\Psi }\left(\alpha \right)={\sum }_{i=1}^n\mathsf{\Psi }\left({\alpha }_i\right)$. Recombination (of several systems to create a composite system) and subdivision (of a system to create several subsystems) are inverse operations. Therefore, after a change of perspective, every system $\alpha$ can be seen as the composite of its n subsystems ${\alpha }_i$, which again leads to the equation $\mathsf{\Psi }\left(\alpha \right)={\sum }_{i=1}^n\mathsf{\Psi }\left({\alpha }_i\right)$.

Theorem 2.1.

Let Φ be a thermodynamic state quantity. Let there be n thermodynamic systems ${\alpha }_i$ such that $\oint \mathrm{d}{\mathsf{\Phi }}_i=0$ holds for each i, $1\le i\le n$. In this case, mathematics always allows additivity of Φ. However, physics might prohibit additivity of Φ in some or all instances.

Proof. 

Summation of all equations $0=\oint \mathrm{d}{\mathsf{\Phi }}_i$, $1\le i\le n$, yields $0={\sum }_{i=1}^n\oint \mathrm{d}{\mathsf{\Phi }}_i$. 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 $\gamma$, and the integral receives the common limits 0 and 1. We may then write $0={\sum }_{i=1}^n{\int }_0^1\left(\mathrm{d}{\mathsf{\Phi }}_i/\mathrm{d}\gamma \right)\mathrm{d}\gamma$. Both integration and differentiation are linear operations, which leads to $0={\int }_0^1\left(\mathrm{d}\left[{\sum }_{i=1}^n{\mathsf{\Phi }}_i\right]/\mathrm{d}\gamma \right)\mathrm{d}\gamma$. We can now return to the unparameterized notation: $0=\oint \mathrm{d}\left[{\sum }_{i=1}^n{\mathsf{\Phi }}_i\right]$. This means that $\mathrm{d}\left[{\sum }_{i=1}^n{\mathsf{\Phi }}_i\right]$ is an exact differential of a state quantity ${\sum }_{i=1}^n{\mathsf{\Phi }}_i$, which we can define as the value of $\mathsf{\Phi }$ for the composite of the n systems. So far, all arguments have been mathematical. There might still be physical arguments prohibiting the additivity of $\mathsf{\Phi }$. □

Remark 1 on theorem 2.1.

The theorem remains true independent of the particular absolute value scale that each system ${\alpha }_i$ might use for the quantity ${\mathsf{\Phi }}_i$, $1\le i\le n$. Also, it remains true independent of whether there is an exchange of energy or matter between the systems.

Remark 2 on theorem 2.1.

The theorem suggests a shift in how additivity is perceived. Traditionally, classical thermodynamics has to either prove or assume that a state quantity is additive. With theorem 2.1, additivity can be taken for granted unless prohibited by physical arguments.

Theorem 2.1 only refers to the generic state quantity $\mathsf{\Phi }$. We now apply this theorem to some particular state quantities.

We begin with the entropy S. The following definition will help avoid lengthy repetitions.

Definition 2.2 (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 of the Carnot cycles, then this composite is called aClausius cycle.

Remark on definition 2.2.

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 non-isothermally, $\mathrm{d}T\ne 0$.) 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. 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.

Considering the Clausius cycle in definition 2.2, Clausius was able to infer the equation $\oint \mathrm{d}S=0$, provided that $\Xi$ is reversible ([1] and [2], or [3]). This shows that the entropy S is a state quantity.

Above theorem 2.1 then allows additivity of the entropy, but as yet from a mathematical perspective only. We now have to look for physical criteria which would prohibit additivity. For this purpose, we consider n Clausius cycles, belonging to n thermodynamic systems ${\alpha }_i$ performing reversible cycle processes ${\Xi }_i$, $1\le i\le n$. We know that $\oint \mathrm{d}{S}_i=0$ holds for all i, $1\le i\le n$. From a physical perspective, if the Clausius cycles have separate and disconnected hot heat baths, then the Clausius cycles themselves seem separate and disconnected, too. Under these circumstances, additivity of the entropy may be prohibited. On the other hand, if all n Clausius cycles share a common hot heat bath, then from a physical perspective the scenario seems not fundamentally different from the scenario of a single Clausius cycle. This in turn justifies the following conclusion.

Conclusion 2.3.

Let there be n Clausius cycles, belonging to n thermodynamic systems ${\alpha }_i$ performing reversible cycle processes ${\Xi }_i$, $1\le i\le n$. Also, let all Clausius cycles share a common hot heat bath. In this case, the entropy is additive.

The main requirement of conclusion 2.3 is that the Clausius cycles can be supposed to share a common hot heat bath. We now look for physical criteria which either allow or prohibit fulfillment of this requirement. Among the potentially prohibitive criteria could be listed: (a) great distance in space between the systems, (b) great distance in time between the systems, (c) great relative speed between the systems. This in turn justifies the following conclusion.

Conclusion 2.4.

The entropy is additive for systems that exist next to each other, contemporaneously, and without relative speed.

The entropy might also be (at least approximately) additive for negligible distance in space between the systems. Similarly, it might also be (at least approximately) additive for negligible relative speed between the systems. However, it is outside the scope of this paper to discuss what exactly qualifies as negligible distance in space or negligible relative speed. We do not discuss the case of systems existing at different points of time.

We now turn to the internal energy U. According to the first law of thermodynamics, U is a state quantity. Textbooks usually explicitly state that U is additive, often as an adjunct to the first law. In this case, additivity is postulated (or assumed). This section suggests another approach: Theorem 2.1 mathematically allows additivity of the internal energy. One then has to look for physical criteria which either allow or prohibit additivity. Again, there seem to be no prohibitive physical criteria for systems that exist next to each other, contemporaneously, and without relative speed: Under these circumstances the physical scenario of a composite system does not seem fundamentally different from that of a single system.

Conclusion 2.5.

The internal energy is additive for systems that exist next to each other, contemporaneously, and without relative speed.

We will not examine the additivity of all other possible state quantities. Besides, some of them have an origin outside of thermodynamics, and their additivity need not necessarily be examined in the framework of classical thermodynamics. For example, we take it for granted that the volume V is additive, at least for the scenarios discussed in this paper.

3. Scaled System

The concept of extensive state quantities is related to the concept of scaled thermodynamic systems. Therefore, we now insert a few remarks on the subject of scaling a system by a scaling factor $\lambda >0$. These remarks do not intend to be exhaustive.

Scaling is an operation by which a scaled system is obtained from an original system. For the scaled system, the values of some thermodynamic state quantities are all multiplied by the same scaling factor $\lambda$ compared to the respective values of the original system. These quantities are called extensive. Non-extensive state quantities are usually called intensive.

The scaling is successful only if the thus obtained scaled system is still a valid representation of the physical reality.

Not all systems are scalable. For example, a completely inhomogeneous system is usually not scalable. Also, if there is an equation of state and multiplying all extensive quantities by a factor $\lambda \ne 1$ leads to a violation of this equation, then the system is usually not scalable.

The scalability of a system typically requires that the scaling factor is neither too close to zero nor too big: For example, if volume is an extensive quantity, then for $\lambda \to 0$ surface effects might no longer be negligible; more importantly, for $\lambda \to 0$ the scaled system is likely to become a microscopic system without the thermodynamic behavior of the original system. Likewise, for $\lambda \to \infty$, the physical behavior of the scaled system is likely to become very different from that of the original system; for example, if mass is an extensive quantity, then the scaled mass of what was formerly an ideal gas may become so large that the system will start to behave like a planet.

Not all inhomogeneous systems are unscalable. For example, a gas in the presence of gravity acting along the z-axis will likely become inhomogeneous and therefore unscalable along the z-axis. However, the system may still be scalable along the x-axis or y-axis.

It is obvious that the scaling of a system is in itself not a physical operation. In fact, from the point of view of physics, the only important aspect appears to be that the obtained scaled system is physically valid. The way that the scaled system is obtained, however, seems unimportant from the point of view of physics.

The shape of the scaled system might bear no resemblance to that of the original system. More generally, there is a vast and varied number of both scalable systems and scaled systems. This makes it difficult to establish scaling procedures that are exact and detailed on the one hand, as well as generally valid on the other hand. For this reason, the following scaling procedures are only generic.

Assumption 3.1.

During the scaling of a thermodynamic system, the following procedures with regard to the system itself or any of its subsystems are allowed: (1) Subdivision into subsystems. (2) Removal. (3) Copying. (4) Rearrangement in space. (5) Recombination to a composite system.

Remark on assumption 3.1.

Other procedures might also be allowed, like giving homogeneous systems a different shape while keeping the values of all state quantities unchanged. However, we will only use the procedures mentioned in the assumption.

4. Extensivity

The subject of this section is to show that any state quantity adhering to the following definition is extensive, given a scalable system.

Definition 4.1.

Let Ψ denote a thermodynamic state quantity that fulfills the following requirements: (a) Ψ is additive. (b) The volume density of Ψ is always finite: $0\le \left|\mathsf{\Psi }\right|/V<\infty$, where $0<V<\infty$. In particular, the density remains finite for $V\to 0$.

Remark on definition 4.1.

Requirement (b) seems necessary: Without it, the value $\mathsf{\Psi }\ne 0$ of a system could be condensed at one point inside its volume $V>0$, instead of being in some way distributed throughout its volume. Also, without it, $\left|\mathsf{\Psi }\right|=\infty$ would be possible. Both could lead to problems in the proof of theorem 4.2 below.

We now show that the quantity $\mathsf{\Psi }$ is extensive. However, as said in Section 3, there are many different types of scalable systems, and we cannot examine all of them. For this reason, the following theorem is restricted to a special type of scalable system. In most cases, it is more or less obvious what procedures need to be followed in order to obtain the scaled system. Often, the procedures described in assumption 3.1 will be sufficient, and they will be used in the proof of the following theorem. However, these procedures are relatively generic, which will render the proof somewhat heuristic.

Theorem 4.2.

Let Ψ fulfill the requirements of definition 4.1. Let α be a thermodynamic system with a closed and bounded volume $V>0$. Also, let the intensive quantities of α be continuous within V. Finally, let α be scalable. Then Ψ is an extensive quantity for rational-valued scaling factors λ, $0<\lambda \in \mathbb{Q}$.

Proof. 

For reasons explained later, it might be necessary to subdivide $\alpha$ into $\beta$ and $\alpha \setminus \beta$ such that the subsystem $\beta$ contains the surface of $\alpha$, and $\left|\mathsf{\Psi }\left(\beta \right)\right|\approx 0$ holds. This can always be achieved: Since the surface of $\alpha$ has no volume, the volume of $\beta$ may get arbitrarily small; in this case, requirement (b) of definition 4.1 guarantees that $\left|\mathsf{\Psi }\left(\beta \right)\right|$ will get arbitrarily small, too. The additivity of $\mathsf{\Psi }$ then leads to $\mathsf{\Psi }(\alpha \setminus \beta )\approx \mathsf{\Psi }\left(\alpha \right)$. The subsystem $\beta$ is removed and will not be part of the scaled system. On the other hand, if it is not necessary to subdivide $\alpha$ into $\beta$ and $\alpha \setminus \beta$, then $\beta$ simply denotes an empty system. In both cases, we then subdivide $\alpha \setminus \beta$ into m (sufficiently) homogeneous subsystems. This is possible if m is chosen large enough, because of the continuity of the intensive quantities. Apart from homogeneity, there might be other specifications with respect to the subsystems, like a specific shape suitable to the desired scaled system; we cannot know them for this generic proof. The scaling factor can be written as $\lambda =a/b$ with $0<a\in \mathbb{N}$ and $0<b\in \mathbb{N}$. Each of the m subsystems is then further subdivided into b (sufficiently) equal subsubsystems. (A subsubsystem is a second-level subsystem: a subsystem of a subsystem.) This is possible due to the homogeneity and suitable shape of the subsystems. However, it might not be possible for subsystems that contain parts of the surface of $\alpha$, due to a potentially unsuitable shape of this surface; in this case, the subsystem $\beta$ containing the surface has been removed beforehand. For each subsystem i, $1\le i\le m$, the values ${\mathsf{\Psi }}_{ij}$ of its subsubsystems j, $1\le j\le b$, must be equal, because the subsubsystems themselves are equal. Due to the additivity of $\mathsf{\Psi }$, this leads to ${\mathsf{\Psi }}_{ij}=\left(1/b\right)·{\mathsf{\Psi }}_i$, where ${\mathsf{\Psi }}_i$ is the value of the subsystem i. We now distinguish between $a>b$ and $a\le b$: For $a>b$, we add $a-b$ copies of one of the equal subsubsystems for each of the m subsystems. For $a\le b$, we remove $b-a$ subsubsystems for each of the m subsystems. In both cases, all $m·a$ subsubsystems may then be rearranged in space and recombined in order to obtain the scaled system. Let $\lambda \alpha$ denote the scaled system. The additivity of $\mathsf{\Psi }$ allows calculating $\mathsf{\Psi }\left(\lambda \alpha \right)$ as the sum of all ${\mathsf{\Psi }}_{ij}$ for all $m·a$ subsubsystems: $\mathsf{\Psi }\left(\lambda \alpha \right)={\sum }_{i=1}^m{\sum }_{j=1}^a{\mathsf{\Psi }}_{ij}={\sum }_{i=1}^m{\sum }_{j=1}^a\left(1/b\right)·{\mathsf{\Psi }}_i=\left(a/b\right)·{\sum }_{i=1}^m{\mathsf{\Psi }}_i=\lambda ·\mathsf{\Psi }(\alpha \setminus \beta )\approx \lambda ·\mathsf{\Psi }\left(\alpha \right)$. □

Remark 1 on theorem 4.2.

The procedures in the proof may not allow perfectly obtaining any desired scaled system. However, it should be possible to approximate the desired scaled system with arbitrary exactness if the number m of subsystems is chosen large enough. In this case, the subsystems may no longer be macroscopic. Instead, they may become purely mathematical, non-macroscopic extrapolations from greater, encompassing macroscopic subsystems. In general, non-macroscopic subsystems are no problem, as long as the resulting scaled system will be physically valid. However, the extrapolation must leave the values of the intensive quantities unchanged.

Remark 2 on theorem 4.2.

The proof requires continuous intensive quantities. Some systems show discontinuous intensive quantities, like a system with several phases. If scaling the complete system permits scaling each phase by the same scaling factor, then the theorem holds for each phase.

Remark 3 on theorem 4.2.

The theorem seems to confirm a statement made in [4] that additivity implies extensivity for rational-valued scaling factors, provided one accepts some assumptions similar to those made in assumption 3.1. In this context, [4] also cites [5]. However, the thermodynamic formalisms of both [4] and [5] are different from the one of classical thermodynamics, which means that their results are not easily transferable to this paper.

So far, only the generic quantity $\mathsf{\Psi }$ introduced in definition 4.1 has been utilized. We now turn to thermodynamic quantities usually considered to be extensive and examine whether they fulfill the requirements of definition 4.1. We will only examine the entropy S and the internal energy U.

According to conclusions 2.4 and 2.5, both S and U are additive. Many textbooks take for granted that the densities $S/V$ and $U/V$ are always finite. For sake of clarity, we introduce an explicit assumption.

Assumption 4.3.

The densities $\left|S\right|/V\ge 0$ and $\left|U\right|/V\ge 0$ are always finite, even for $V\to 0$.

Theorem 4.2 then claims that S and U are extensive.

In general, the extensivity of many thermodynamic quantities is assumed or postulated in textbooks. This section suggests another approach: A quantity is extensive if it is additive and if its density is finite. For those quantities that are not obviously additive, Section 2 claims that they are additive unless there are prohibitive physical counterarguments. This means that in most cases the assumption of extensivity can be replaced by the arguably weaker assumption of finite density.

5. Real-Valued Scaling Factor

So far, only rational-valued scaling factors have been considered. We now examine real-valued scaling factors $\lambda \in \mathbb{R}$.

From the point of view of experiment, we seem to be unable to ascertain whether the scaling factor of a scaled system is rational-valued or real-valued: The reading of a digital measuring equipment is usually a rational number, and even the reading of an analog measuring equipment is usually interpreted as a rational number. More generally, rational numbers and real numbers are so closely packed that no measuring equipment can likely distinguish between them.

This justifies approaching the subject of real-valued scaling factors from a purely mathematical point of view: The task is to construct real-valued scaling factors from rational-valued scaling factors. The more general task is to construct real numbers from rational numbers, and this construction has already been done in mathematics (e.g., [6] or [7]). We can therefore take it for granted that real-valued scaling factors exist.

6. Results and Conclusions

It has been shown that the entropy and the internal energy can be taken to be additive for systems existing next to each other, contemporaneously, and without relative speed. The question of whether they are still additive if these requirements are no longer fully met could possibly be answered by other thermodynamic theories, like relativistic thermodynamics.

A proof has been given that any thermodynamic state quantity is extensive if it is additive and if it has finite density. It has been argued that the heuristic nature of the proof is due to the vast number of possible scaling operations which impede increasing the rigor of the proof without impairing its generality. Nonetheless, a more rigorous proof could be looked for.

Still, for many thermodynamic quantities, the presented proof might allow replacing the traditional strong assumption of extensivity with the arguably weaker assumption of finite density.