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. Efficiency of cycles
4. Adding dissipation, Clausius inequality
B. Some general theories
1. Entropy and efficiency
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 coefficients
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. Itˆo’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
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. Efficiency of cycles
4. Adding dissipation, Clausius inequality
B. Some general theories
1. Entropy and efficiency
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 coefficients
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. Itˆo’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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