lieb.qxp 4/9/98 11:08 AM Page 571 A Guide to Entropy and the Second Law of Thermodynamics Elliott H. Lieb and Jakob Yngvason his article is intended for readers who, low-temperature physics and is not immutable— like us, were told that the second law of as are the others). In brief, these are: thermodynamics is one of the major The Zeroth Law, which expresses the transitiv- achievements of the nineteenth cen- ity of thermal equilibrium and which is often said tury—that it is a logical, perfect, and un- to imply the existence of temperature as a pa- Tbreakable law—but who were unsatisfied with the rametrization of equilibrium states. We use it below “derivations” of the entropy principle as found in but formulate it without mentioning temperature. textbooks and in popular writings. In fact, temperature makes no appearance here A glance at the books will inform the reader that until almost the very end. the law has “various formulations” (which is a bit The First Law, which is conservation of energy. odd, as if to say the Ten Commandments have It is a concept from mechanics and provides the various formulations), but they all lead to the ex- connection between mechanics (and things like istence of an entropy function whose reason for falling weights) and thermodynamics. We discuss existence is to tell us which processes can occur this later on when we introduce simple systems; and which cannot. We shall abuse language (or re- the crucial usage of this law is that it allows en- formulate it) by referring to the existence of en- ergy to be used as one of the parameters describ- tropy as the second law. This, at least, is unam- ing the states of a simple system. biguous. The entropy we are talking about is that The Second Law. Three popular formulations of defined by thermodynamics (and not some analytic this law are: quantity, usually involving expressions such as p ln p, that appears in information theory, prob- Clausius: No process is possible, the − ability theory, and statistical mechanical models). sole result of which is that heat is trans- There are three laws of thermodynamics (plus ferred from a body to a hotter one. one more, due to Nernst, which is mainly used in Kelvin (and Planck): No process is pos- Elliott H. Lieb is professor of mathematics and physics at sible, the sole result of which is that a Princeton University. His e-mail address is lieb@math. body is cooled and work is done. princeton.edu. Work partially supported by U.S. Na- tional Science Foundation grant PHY95-13072A01. Carathéodory: In any neighborhood of Jakob Yngvason is professor of theoretical physics at Vi- any state there are states that cannot enna University. His e-mail address is yngvason@ be reached from it by an adiabatic thor.thp.univie.ac.at. Work partially supported by process. the Adalsteinn Kristjansson Foundation, University of Ice- land. All three formulations are supposed to lead to ©1997 by the authors. Reproduction of this article, by any the entropy principle (defined below). These steps means, is permitted for noncommercial purposes. can be found in many books and will not be trod- MAY 1998 NOTICES OF THE AMS 571 lieb.qxp 4/9/98 11:08 AM Page 572 den again here. Let us note in passing, however, lated previous work on the foundations of classi- that the first two use concepts such as hot, cold, cal thermodynamics are given in [7]. The literature heat, cool that are intuitive but have to be made on this subject is extensive, and it is not possible precise before the statements are truly meaning- to give even a brief account of it here, except for ful. No one has seen “heat”, for example. The last mentioning that the previous work closest to ours (which uses the term “adiabatic process”, to be de- is that of [6] and [2] (see also [4], [5], and [9]). fined below) presupposes some kind of parame- These other approaches are also based on an in- trization of states by points in Rn, and the usual vestigation of the relation , but the overlap with ≺ derivation of entropy from it assumes some sort our work is only partial. In fact, a major part of our of differentiability; such assumptions are beside work is the derivation of a certain property (the the point as far as understanding the meaning of “comparison hypothesis” below), which is taken as entropy goes. an axiom in the other approaches. It was a re- Why, one might ask, should a mathematician be markable and largely unsung achievement of Giles interested in this matter, which historically had [6] to realize the full power of this property. something to do with attempts to understand and Let us begin the story with some basic con- improve the efficiency of steam engines? The an- cepts. swer, as we perceive it, is that the law is really an 1. Thermodynamic system: Physically this con- interesting mathematical theorem about an or- sists of certain specified amounts of certain dering on a set, with profound physical implica- kinds of matter, e.g., a gram of hydrogen in a tions. The axioms that constitute this ordering are container with a piston, or a gram of hydro- somewhat peculiar from the mathematical point gen and a gram of oxygen in two separate con- of view and might not arise in the ordinary rumi- tainers, or a gram of hydrogen and two grams nations of abstract thought. They are special but of hydrogen in separate containers. The sys- important, and they are driven by considerations tem can be in various states which, physically, about the world, which is what makes them so in- are equilibrium states. The space of states of teresting. Maybe an ingenious reader will find an the system is usually denoted by a symbol application of this same logical structure to another such as and states in by X,Y,Z, etc. field of science. Physical motivation aside, a state-space, math- The basic input in our analysis is a certain kind ematically, isΓ just a set to beginΓ with; later on we of ordering on a set, and denoted by will be interested in embedding state-spaces in some convex subset of some Rn+1; i.e., we will in- ≺ troduce coordinates. As we said earlier, however, (pronounced “precedes”). It is transitive and re- the entropy principle is quite independent of co- flexive, as in A1, A2 below, but X Y and Y X ordinatization, Carathéodory’s principle notwith- ≺ ≺ do not imply X = Y, so it is a “preorder”. The big standing. question is whether can be encoded in an ordi- 2. Composition and scaling of states: The notion ≺ nary, real-valued function, denoted by S, on the set, of Cartesian product, 1 2, corresponds sim- × such that if X and Y are related by , then ply to the two (or more) systems being side by ≺ S(X) S(Y ) if and only if X Y. The function S side on the laboratoryΓ table;Γ mathematically it ≤ ≺ is also required to be additive and extensive in a is just another system (called a compound sys- sense that will soon be made precise. tem), and we regard the state-space 1 2 as × A helpful analogy is the question: When can a being the same as 2 1. Points in 1 2 are × × vector-field, V(x), on R3 be encoded in an ordinary denoted by pairs (X,Y), as usual. ΓThe Γsub- function, f (x), whose gradient is V? The well- systems comprisingΓ a compoundΓ systemΓ Γ are known answer is that a necessary and sufficient physically independent systems, but they are condition is that curl V =0. Once V is observed to allowed to interact with each other for a pe- have this property, one thing becomes evident and riod of time and thereby to alter each other’s important: it is necessary to measure the integral state. of V only along some curves—not all curves—in The concept of scaling is crucial. It is this con- order to deduce the integral along all curves. The cept that makes our thermodynamics inap- encoding then has enormous predictive power propriate for microscopic objects like atoms about the nature of future measurements of V. In or cosmic objects like stars. For each state- the same way, knowledge of the function S has space and number λ>0 there is another enormous predictive power in the hands of state-space, denoted by (λ), with points de- chemists, engineers, and others concerned with the noted byΓ λX. This space is called a scaled copy ways of the physical world. of . Of course we identifyΓ (1) = and Our concern will be the existence and proper- 1X = X. We also require ( (λ))(µ) = (λµ) and ties of S, starting from certain natural axioms µ(λXΓ )=(µλ)X. The physical interpretationΓ Γ of about the relation . We present our results with- (λ) when is the space ofΓ one gramΓ of hy- ≺ out proofs, but full details and a discussion of re- drogen is simply the state-space of λ grams Γ Γ 572 NOTICES OF THE AMS VOLUME 45, NUMBER 5 lieb.qxp 4/9/98 11:10 AM Page 573 of hydrogen. The state λX is the state of λ vice consisting of some auxiliary system grams of hydrogen with the same “intensive” and a weight in such a way that the properties as X, e.g., pressure, while “exten- auxiliary system returns to its initial sive” properties like energy, volume, etc., are state at the end of the process, whereas scaled by a factor λ (by definition).