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Lieb & Yngvason - A Guide to Entropy and the Second Law of Thermodynamics [1998]

Lieb & Yngvason - A Guide to Entropy and the Second Law of Thermodynamics [1998]

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arXiv:math-ph/9805005 v1 6 May 1998 EHLJY 27/Feb/98 A GUIDE TO ENTROPY AND THE SECOND LAW OF THERMODYNAMICS Elliott H. Lieb Departments of Mathematics and Physics, Princeton University Jadwin Hall, P.O. Box 708, Princeton, NJ 08544, USA Jakob Yngvason ∗∗ Institut f¨ur Theoretische Physik, Universit¨ at Wien, Boltzmanngasse 5, A 1090 Vienna, Austria 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 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 ln p , 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 low temperature physics and is not immutable like the others). In brief, these are: The Zeroth Law, which expresses the transitivity of 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 ther- modynamics. We discuss this later on when we introduce simple systems; the crucial Work partially supported by U.S. National Science Foundation grant PHY95-13072A01. ∗∗ Work partially supported by the Adalsteinn Kristjansson Foundation, University of Iceland. c circlecopyrt 1997 by the authors. Reproduction of this article, by any means, is permitted for non-commercial purposes. 1
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EHLJY 27/Feb/98 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 possible, the sole result of which is that a body is cooled and work is done.
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