A Guide to Entropy and the Second Law of Thermodynamics
This article is intended for readers who, like us, were told that the second law of thermodynamics is one of the major achievements of the nineteenth cenwry—that it is a logical, perfect, and unbreakable law—but who were unsatisfied with the “derivations”
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A Guide to Entropy and the Second Law of
Thermodynamics
Elliott H. Lieb and Jakob Yngvason
T
his article is intended for readers who, like us, were told that the second law of thermodynamics is one of the major achievements of the nineteenth century-that it is a logical, perfect, and unbreakable law-but who were unsatisfied with the "derivations' of the entropy principle as found in textbooks and in popular writings. A glance at the books will inform the reader that the law has "various formulatiOns " (which is a bit odd, as if to say the Ten Commandments have various formulations), but they all lead to the existence of an entropy function whose reason for existence is to tell us which processes can occur and which cannot. We shall abuse language (or reformulate it) by referring to the existence of entropy as the second law. This, at least, is unambiguous. The entropy we are talking about is that defined by thermodynamics (and not some analytic quantity, usually involving expressions such as -p lnp, that appears In information theory, probability theory, and statistical mechanical models). There are three laws of thermodynamics (plus one more, due to Nernst, which is mainly used in Elliott H. U eb is professor of mathematics and physics at Princeton UniverSity. His e·mail address is 1 i eb@math. pri nee ton . edu. Work partially supported by US. Na· tional Science Foundation grant PHY95-13072AOI. Jakob Yngvason is professor of theoretical physics at Vienna University. His e-mail address is yngvason@ thor . thp.univie . ae . n. Work partially supported by the Adalsteinn Kristjansson Foundation, University ofIce· land. ©1997 by the authors_ Reproduction of this artie/e, by airy means, is permitted for noncommercial purposes.
MAy 1998
low·temperature physics and is not immutableas are the others). In brief, these are: The Zeroth Law, which expresses the transitivity of thermal equilibrium and which is often said to imply the existence of temperature as a parametrization of equilibrium states. We use it below but formulate it without mentioning temperature. In fact, temperature makes no appearance here until almost the very end. The First Law, which is conservation of energy. It is a concept from mechanics and provides the connection between mechanics (and things like falling weights) and thermodynamics. We discuss this later on when we introduce simple systems; the crucial usage of this law is that it allows energy to be used as one of the parameters describing the states of a simple system. The Second Law. Three popular formulations of this law are: Clausius: No process is possible, the sole result of which is that heat Is transferred from a body to a hotter one.
Kelvin (and Planck): No process is pos· sible, the sole result of which is that a body is cooled and work is done. Caratheodory: In any neighborhood of any state there are states that cannot be reached from it by an adiabatic process.
All three formulations are supposed to lead to the entropy prinCiple (defined below). These steps can be found in man
