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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 the company 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 with communication ... “Entropy is a figure of speech then” ... “a metaphor”.
–T. Pynchon, The Crying of Lot 49
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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. Definitions
3. Maxwell relations
E. Capacities
F. More examples
1. Ideal gas
2. Van der Waals fluid
II. Entropy and irreversibility
A. A model material
1. Definitions
2. Energy and entropy
a. Working and heating
b. First Law, existence of E
c. Carnot cycles
d. Second Law
e. Existence of S
3. E?ciency of cycles
4. Adding dissipation, Clausius inequality
B. Some general theories
1. Entropy and e?ciency
1
a. Definitions
b. Existence of S
2. Entropy, temperature and separating hyperplanes
a. Definitions
b. Second Law
c. Hahn–Banach Theorem
d. Existence of S,T
III. Continuum thermodynamics
A. Kinematics
1. Definitions
2. Physical quantities
3. Kinematic formulas
4. Deformation gradient
B. Conservation laws, Clausius–Duhem inequality
C. Constitutive relations
1. Fluids
2. Elastic materials
D. Workless dissipation
IV. Elliptic and parabolic equations
A. Entropy and elliptic equations
1. Definitions
2. Estimates for equilibrium entropy production
a. A capacity estimate
b. A pointwise bound
3. Harnack’s inequality
B. Entropy and parabolic equations
1. Definitions
2. Evolution of entropy
a. Entropy increase
b. Second derivatives in time
c. A differential form of Harnack’s inequality
3. Clausius inequality
a. Cycles
b. Heating
c. Almost reversible cycles
V. Conservation laws and kinetic equations
A. Some physical PDE
2
1. Compressible Euler equations
a. Equations of state
b. Conservation law form
2. Boltzmann’s equation
a. A model for dilute gases
b. H -Theorem
c. H and entropy
B. Single conservation law
1. Integral solutions
2. Entropy solutions
3. Condition E
4. Kinetic formulation
5. A hydrodynamical limit
C. Systems of conservation laws
1. Entropy conditions
2. Compressible Euler equations in one dimension
a. Computing entropy/entropy flux pairs
b. Kinetic formulation
VI. Hamilton–Jacobi and related equations
A. Viscosity solutions
B. Hopf–Lax formula
C. A diffusion limit
1. Formulation
2. Construction of diffusion coe?cients
3. Passing to limits
VII. Entropy and uncertainty
A. Maxwell’s demon
B. Maximum entropy
1. A probabilistic model
2. Uncertainty
3. Maximizing uncertainty
C. Statistical mechanics
1. Microcanonical distribution
2. Canonical distribution
3. Thermodynamics
VIII. Probability and differential equations
A. Continuous time Markov chains
3
1. Generators and semigroups
2. Entropy production
3. Convergence to equilibrium
B. Large deviations
1. Thermodynamic limits
2. Basic theory
a. Rate functions
b. Asymptotic evaluation of integrals
C. Cramer’s Theorem
D. Small noise in dynamical systems
1. Stochastic differential equations
2. Ito’s formula, elliptic PDE
3. An exit problem
a. Small noise asymptotics
b. Perturbations against the flow
Appendices :
A. Units and constants
B. Physical axioms
References
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