The essential postulates of classical thermodynamics are formulated, from which the second law is deduced as the principle of increase of entropy in irreversible adiabatic processes that take one equilibrium state to another. The entropy constructed here is defined only for equilibrium states and no attempt is made to define it otherwise. Statistical mechanics does not enter these considerations. One of the main concepts that makes everything work is the comparison principle (which, in essence, states that given any two states of the same chemical composition at least one is adiabatically accessible from the other) and we show that it can be derived from some assumptions about the pressure and thermal equilibrium. Temperature is derived from entropy, but at the start not even the concept of ‘hotness’ is assumed. Our formulation offers a certain clarity and rigor that goes beyond most textbook discussions of the second law.
The paper can be subdivided into two parts. In the first part, the authors deduce the existence of a universal entropy state function from a primitive notion of adiabatic accessibility. A key assumption in this derivation is the so-called Comparison Hypothesis, that postulates that for any two equilibrium states \(X\) and \(Y\) of a given thermodynamic system, either \(X\) is adiabatically accessible from \(Y\) or \(Y\) is adiabatically accessible from \(X\).
The second part is devoted to the derivation of the Comparison Hypothesis (which becomes thus a Comparison Principle) for an important class of thermodynamic systems. The main ingredients for this derivation are: - The First Law of Thermodynamics and, in particular the concept of internal energy. - The notion of thermal equilibrium and its transitivity (sometimes called the “Zero-th Law”). - An assumed convex structure in the state space of thermodynamic systems.
So, how does such a formal endeavor improve our understanding of Nature? On one side, entropy is clarified to represent nothing more than a numerical encoding of adiabatic accessibility. Its existence is independent of the concepts of energy, heat or temperature that appear in the usual formulations of Thermodynamics. On the other side, the classical statements of the Second Law of Kelvin-Planck and Clausius, which are explicitly formulated in terms of heat and temperature, are seen to be theorems on the adiabatic (in)accessibility of certain states, that follow from how the concepts of internal energy and thermal equilibrium interact with the primitive notion of adiabatic accessibility. Both of these are major advancements, because they clarify the physical content and universality of the Second Law of Thermodynamics.
The last Section of (Lieb and Yngvason 1999) provides a useful summary of the formalism and of the logical paths followed by the mathematical derivations. The rest of this post are some personal notes on the original reference.
Formal framework
Systems are described as collections of thermodynamic states, with two binary operations that have the operative interpretations of “composition” (of non-interacting systems) and “scaling”, respectively. In detail:
Systems. A system is just a set \(\Gamma\), whose elements correspond to physical (thermodynamic) states.
Composition. A binary operation that takes as input two thermodynamic states \(X_1\) and \(X_2\) of two systems \(\Gamma _1\) and \(\Gamma _2\), and returns a new state \(X_1 \oplus X_2 \in \Gamma _1 \oplus \Gamma _2\), where the latter set can be informally thought as the Cartesian product \(\Gamma _1 \times \Gamma _2\). Since the ordering in the enumeration of systems is inconsequential from the physical point of view, we assume that compositions are equipped with an equivalence relation \(\stackrel{\oplus}{\sim}\), such that \(X_1 \oplus X_2 \stackrel{\oplus}{\sim} X_2 \oplus X_1\) and that \((X_1 \oplus X_2)\oplus X_3 \stackrel{\oplus}{\sim} X_1 \oplus (X_2\oplus X_3)\).
Scaling. A binary operation that takes as input a positive number \(\lambda\) and a thermodynamic state \(X\in \Gamma\), and returns a new state $X $ as output. The operation is assumed to satisfy \(t(sX) = (ts)X\), as well as \(t(X_1\oplus X_2)=(tX_1) \oplus (tX_2)\).
Some care is needed in the operative interpretation of the scaling operation for systems that are not scale invariant, such as a droplet of liquid with a non-negligible surface energy. This particular example is discussed in detail by the authors in Sec. 3.1, in connection with simple systems. In short, for such systems we should interpret the state space as a submanifold of an extended state space corresponding to a scale invariant system. It is for this latter system that the axioms of (Lieb and Yngvason 1999) are assumed to hold.
Axioms of adiabatic accessibility
A central notion in (Lieb and Yngvason 1999) is that of adiabatic accessibility. From the operative point of view this corresponds to the possibility of transforming a thermodynamic state into another without supplying heat to the system. Mathematically, adiabatic accessibility is a relation \(\prec\) on the set of all thermodynamic states that satisfies (among other things) the axioms of a pre-order: reflexivity and transitivity.
More in detail, \(\prec\) is assumed to satisfy the following axioms:
- (A1) Reflexivity. \(X \prec X\) for any \(X\).
- (A2) Transitivity. \(X\prec Y\) and \(Y \prec Z\) implies \(X \prec Z\) for all \(X\), \(Y\) and \(Z\).
- (A3) Consistency. \(X\prec Y\) and \(X^\prime \prec Y ^\prime\) implies \(X\oplus X^\prime \prec Y \oplus Y^\prime\) for all \(X\), \(X^\prime\), \(Y\) and \(Y^\prime\)
- (A4) Scaling invariance. \(X\prec Y\) implies \(\lambda X \prec \lambda Y\) for all \(X\), \(Y\) and \(\lambda >0\).
- (A5) Splitting and recombination. \(X \prec (\lambda X,(1-\lambda)X) \prec X\) for all \(X\) and \(0<\lambda <1\).
- (A6) Stability: if \[ (X,\varepsilon _n Z) \prec (Y,\varepsilon_n W) \] holds for some sequence \(\varepsilon _n \to 0\), then \(X \prec Z\).
Existence and uniqueness of entropy
The first part of (Lieb and Yngvason 1999) is concerned with the existence and uniqueness of the entropy function. The authors prove that, under axioms A1-A6 characterizing adiabatic accessibility, the two following conditions are equivalent:
- (i) Given states \(X_i\) and \(Y_i\) of a system \(\Gamma\) and positive numbers \(\lambda _i\) for \(i = 1,\,2,\,\dots,\,n\), the two states \(X = \lambda_1 X_1 \oplus \lambda_2 X_2 \oplus \cdots \oplus \lambda _n X_n\) and \(Y = \lambda_1 Y_1 \oplus \lambda_2 Y_2 \oplus \cdots \oplus \lambda _n Y_n\) are always comparable, in the sense that either \(X\prec Y\) or \(Y \prec X\). This is expressed by saying that “the comparison hypothesis holds on multiple scaled copies of \(\Gamma\)”.
- (ii) There exists a function \(S_\Gamma \colon \Gamma \to \mathbb R\) with the following property: if \(X = \lambda_1 X_1 \oplus \lambda_2 X_2 \oplus \cdots \oplus \lambda _n X_n\) and \(Y = \gamma_1 Y_1 \oplus \gamma_2 Y_2 \oplus \cdots \oplus \gamma _m Y_m\), where \(\sum _{i=1}^n \lambda _i = \sum _{j=1}^m \gamma _j\), we have \(X\prec Y\) if and only if1:
\[ \sum _{i=1}^n \lambda _iS(X_i)\leq \sum _{i=1}^m \gamma _iS(Y_i). \] Furthermore, the function \(S\) is given by \(S = a S^* + b\) for some constants \(a,b\) with \(a>0\), where2:
\[ S^*(X) = \sup \{\lambda \colon (1-\lambda)X_0\oplus \lambda X_1 \prec X\} \] where \(X_{0,1}\in \Gamma\) are two reference states such that \(X_0 \prec X_1\) but the reverse does not hold (the definition enforces \(S^*(X_0)=0\) and \(S^*(X_1) = 1\)).
Convexity
- (A7) Convex combination. The space \(\Gamma\) is a locally convex subset of a vector space, meaning that \(tX + (1-t)Y \in \Gamma\) for all \(X,Y\in \Gamma\) and \(0<t<1\). Furthermore, we have that \(t X \oplus (1-t)Y\prec t X + (1-t)Y\).
It is easily seen that if an entropy function (in the sense of the previous section) exists, then the second part of the previous axiom (involving \(\prec\)) is equivalent to the requirement that entropy is a concave function, i.e. \(S(t X + (1-t)Y)\geq tS(X)+(1-t)S(Y)=S(tX\oplus(1-t)Y)\).
The convex combination of states is most easily interpreted for simple systems (see below), and corresponds to a state in which energy and work coordinates take the intermediate values \(t\xi(X)+(1-t)\xi(Y)\). The possibility of forming convex combinations of states and their adiabatic accessibility is quite obvious for systems such as gas or liquids. For example, suppose we are given \(n_i\) moles of gas with energies \(n_iU_i\) and volumes \(n_iV_i\), where \(n_1 + n_2 = 1\), and the two gases are separated by a rigid, adiabatic wall. The convex combination is the state of 1 mole of gas that is produced by simply removing the barrier. For solids or more complicated systems, the axiom looks much more questionable, but can be at least experimentally tested for any given system, by exploiting its equivalence with the concavity of entropy.
Simple systems
The discussion moves on to concrete systems whose state space is assumed to be an open convex subset of \(\mathbb R^{n+1}\) for some \(n>0\), in agreement with A7. The points of these systems are denoted by pairs \((U,V)\), where \(U\in \mathbb R\) and \(V \in \mathbb R ^n\) have the physical interpretation of energy and extensive work coordinates, respectively. The possibility of a parametrization with internal energy as one coordinate is the content of the First Law of Thermodynamics.
Simple systems are characterized by the following axioms (the separation of S2 into S2a and S2b is mine):
- (S1) Irreversibility. For each \(X\) there is a point \(Y\) such that \(X\prec Y\). (This axiom is actually implied by the stronger axiom T4, see below.)
- (S2a) Tangent Plane. For each \(X\), the forward sector \(A_X=\{Y\in \Gamma\colon X\prec Y\}\) has a unique tangent plane parametrized by an equation of the form:
\[ U-U(X)+\sum _{i=1}^n p_i(X)\cdot(V_i-V_i(X))=0 \] - (S2b) Lipschitz Continuity. The functions \(p_i\) in the previous axiom are locally Lipschitz continuous functions. - (S3) Connectedness of the boundary. The boundary of any forward sector is connected.
Using only axioms (S1) and (S2a), the Kelvin-Planck statement of the Second Law of Thermodynamics can be proved, namely:
No process is possible, the sole result of which is a change in the energy of a simple system (without changing the work coordinates) and the raising of a weight.
Including also (S2b) and (S3), the main theorem of this section can be proved, stating that any two states of a simple system are comparable.
Thermal Equilibrium
(T1) Thermal Contact. For any two simple systems. For any two simple systems with state spaces \(\Gamma _1\) and \(\Gamma _2\), there is another simple system, the thermal join of \(\Gamma _1\) and \(\Gamma _2\), with state space: \[ \Gamma_1 \stackrel{T}{\oplus}\Gamma_2 \equiv \{(U_1+U_2,V_1,V_2)\,\colon (U_i,V_i)\in\Gamma_i\}, \] whose states satisfy: \[ \Gamma_1\oplus\Gamma_2\ni(U_1,V_1)\oplus(U_2,V_2) \prec (U_1+U_2,V_1,V_2)\in \Gamma_1 \stackrel{T}{\oplus}\Gamma_2. \]
(T2) Thermal Splitting. For any state of the thermal join \((U,V_1,V_2)\in \Gamma_1\stackrel{T}{\oplus}\Gamma_2\) there exist states \((U_i,V_i)\in \Gamma _i\) such that \(U_1 + U_2 =U\) and: \[ (U,V_1,V_2) \prec (U_1,V_1) \oplus (U_2,V_2). \] In particular, if \((U,V)\in \Gamma\) is a state of a simple system and \(\lambda \in (0,1)\), we have: \[ (U,\lambda V,(1-\lambda)V)\prec\lambda(U,V) \oplus (1-\lambda)(U,V) \] The right-hand side here is an element of \(\lambda \Gamma \oplus (1-\lambda) \Gamma\), while the left hand side is an element of the thermal join of scaled copies \(\lambda \Gamma \stackrel{T}{\oplus} (1-\lambda) \Gamma\).
(T3) Zero-th law. The relation \(\stackrel{T}{\sim}\) defined in the following is an equivalence relation: \(\Gamma _1 \ni(U_1,V_1)\stackrel{T}{\sim}(U_2,V_2)\in \Gamma _2\) if \((U_1 + U_2, V_1, V_2) \prec (U_1,V_1) \oplus (U_2,V_2)\).
(T4) Transversality. If \(\Gamma\) is a simple system and \(X\in \Gamma\), there exist \(X_0,X_1\in \Gamma\) such that \(X_0 \prec \prec X \prec \prec X_1\) and \(X_0 \stackrel{T}{\sim} X_1\).
(T5) Universal temperature range. If \(\Gamma _1\) and \(\Gamma _2\) are simple systems, and \(V_1\) is in the projection of \(\Gamma _1\) onto its work coordinates, for every \(X_2 \in \Gamma _2\) there is a \(U_1\) such that \((U_1,V_1) \stackrel{T}{\sim} X_2\)
The main result from these additional axioms is comparability in the product spaces of simple systems. Also, thermal equilibrium is found to be characterized by maximum entropy, in the sense that if \((U_i,V_i)\in \Gamma _i\) are states of simple systems with (consistent) entropies \(S_i\) respectively, then:
\[ (U_1,V_1)\stackrel{T}{\sim} (U_2,V_2) \iff S_1+S_2=\max_{U_1^\prime +U_2^\prime=U_1+U_2}\{S(U_1^\prime,V_1)+S(U_2 ^\prime ,V_2)\}. \]
If (T3) is assumed, and if we postulate that for each state \(X\) there is at least one state in thermal equilibrium with it - which is guaranteed by (T5) - then the second part of (T2) is equivalent to scaling invariance \(\stackrel{T}{\sim}\). In fact, transitivity and symmetry of \(\stackrel{T}{\sim}\) implies \(X\stackrel{T}{\sim}X\) for any \(X\). Then the second part of (T2) is equivalent to \(t X \stackrel{T}{\sim}(1-t)X\) for any \(t \in (0,1)\) and any \(X\), which is in turn equivalent to \(\lambda Y \sim Y\) for any \(Y\) and \(\lambda >0\).
Temperature
Temperature is a derived concept in the formalism of (Lieb and Yngvason 1999), by means of the usual bridge law \(T = \left(\frac{\partial U}{\partial S}\right)_V\). The differentiability of entropy is shown to be a consequence of the axioms stated so far.
Using the derived concept of temperature, the authors prove the Clausius inequality for a Carnot cycle:
\[ \frac{\delta Q_1}{T_1}+\frac{\delta Q_2}{T_2} \leq0, \] where \(T_1\) and \(T_2\) are the final temperatures of the reservoirs, that do not need to be assumed of infinite mass.
Chemical mixtures and reactions
The last result of the paper concerns thermodynamic processes that connect states of different systems. The notion of connection of state spaces is introduced: two state spaces \(\Gamma\) and \(\Gamma ^\prime\) are said to be connected if there are state spaces \(\Gamma _0,\,\Gamma_1,\,\dots,\Gamma_N\) and states \(X_i,Y_i \in \Gamma _i\), and \(X\in \Gamma\) and \(Y\in \Gamma ^\prime\), such that:
\[ (X,\,X_0) \prec Y_1, \quad X_i ≺ Y_{i+1} \ (i = 1,...,N − 1), \quad X_N \prec (Y ,\,Y_0). \] If \(\Gamma\) is connected to \(\Gamma ^\prime\) we write \(\Gamma \prec \Gamma ^\prime\). The last axiom is:
- (M) Absence of sinks. \(\Gamma \prec \Gamma ^\prime \implies \Gamma ^\prime \prec \Gamma\).
With this additional axiom, one can prove a “weak” form of the entropy principle: that the multiplicative and additive constants in the definition of \(S\) for each simple system can be chosen in such a way that \(X\prec Y\) implies \(S(X) \leq S(Y)\) for any two comparable states (not necessarily of the same system).
The strong form of the entropy principle consists in the statement that all states in connected spaces are comparable, and that adiabatic accessibility is completely characterized by \(S\), in the sense that \(X \prec Y\) if and only if \(S(X)\leq S(Y)\). This result depends on a particular assumption (Eq. (6.34) of the reference) that the authors do not axiomatize, but that is subject to experimental verification.
References
Footnotes
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Citation
@online{gherardi2024,
author = {Gherardi, Valerio},
title = {“{The} {Physics} and {Mathematics} of the {Second} {Law} of
{Thermodynamics}” by {E.H.} {Lieb} and {J.} {Yngvason}},
date = {2024-06-25},
url = {https://vgherard.github.io/posts/2024-06-25-the-physics-and-mathematics-of-the-second-law-of-thermodynamics-by-eh-lieb-and-j-yngvason/},
langid = {en}
}