entropy.and.PDE

entropy.and.PDE - Entropy and Partial Dierential Equations...

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Entropy and Partial Diferential Equations Lawrence C. Evans Department of Mathematics, UC Berkeley Inspiring Quotations A good many times I have been present at gatherings of people who, by the standards of traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientiFc equivalent of: Have you read a work of Shakespeare’s? –C. P. Snow, The Two Cultures and the Scientifc Revolution ... C. P. Snow relates that he occasionally became so provoked at literary colleagues who scorned the restricted reading habits of scientists that he would challenge them to explain the second law of thermodynamics. The response was invariably a cold negative silence. The test was too hard. Even a scientist would be hard-pressed to explain Carnot engines and refrigerators, reversibility and irreversibility, energy dissipation and entropy increase all in the span of a cocktail party conversation. –E. E. Daub, “Maxwell’s demon” He began then, bewilderingly, to talk about something called entropy She did gather that there were two distinct kinds of this entropy. One having to do with heat engines, the other with communication “Entropy is a Fgure of speech then” “a metaphor”. –T. Pynchon, The Crying oF Lot 49 1
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CONTENTS Introduction A. Overview B. Themes I. Entropy and equilibrium A. Thermal systems in equilibrium B. Examples 1. Simple fluids 2. Other examples C. Physical interpretations of the model 1. Equilibrium 2. Positivity of temperature 3. Extensive and intensive parameters 4. Concavity of S 5. Convexity of E 6. Entropy maximization, energy minimization D. Thermodynamic potentials 1. Review of Legendre transform 2. DeFnitions 3. Maxwell relations E. Capacities ±. More examples 1. Ideal gas 2. Van der Waals fluid II. Entropy and irreversibility A. A model material 1. DeFnitions 2. Energy and entropy a. Working and heating b. ±irst Law, existence of E c. Carnot cycles d. Second Law e. Existence of S 3. Efficiency of cycles 4. Adding dissipation, Clausius inequality B. Some general theories 1. Entropy and efficiency 1
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a. Defnitions b. Existence oF S 2. Entropy, temperature and separating hyperplanes a. Defnitions b. Second Law c. Hahn–Banach Theorem d. Existence oF S,T III. Continuum thermodynamics A. Kinematics 1. Defnitions 2. Physical quantities 3. Kinematic Formulas 4. DeFormation gradient B. Conservation laws, Clausius–Duhem inequality C. Constitutive relations 1. ±luids 2. Elastic materials D. Workless dissipation IV. Elliptic and parabolic equations A. Entropy and elliptic equations 1. Defnitions 2. Estimates For equilibrium entropy production a. A capacity estimate b. A pointwise bound 3. Harnack’s inequality B. Entropy and parabolic equations 1. Defnitions 2. Evolution oF entropy a. Entropy increase b. Second derivatives in time
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This note was uploaded on 09/30/2011 for the course MATH 222a taught by Professor Evans during the Fall '11 term at UCLA.

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entropy.and.PDE - Entropy and Partial Dierential Equations...

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