Entropy and partial differential equations - Evans L.C..pdf

Entropy and Partial Differential Equations
Lawrence C. Evans
Department of Mathematics, UC Berkeley
Inspiring Quotations
A good many times Ihave 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 Ihave been
provoked and have asked pany how many of them could describe the Second Law of
Thermodynamics. The response was cold: it was also negative. Yet Iwas asking something
which is about the scientific equivalent of: Have you read a work of Shakespeare’s?
–C. P. Snow, The Two Cultures and the Scientific 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 munication... “Entropy is a figure of speech then”... “a metaphor”.
–T. Pynchon, The Crying of Lot 49
1
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.