Symplectic Thermodynamics from Maximum Entropy
The following sections are included:
Previous Approaches to Geometric Thermodynamics
Seven Approaches to the Maximum Entropy Formalism
Axiomatic Subjective Approach
Counting Sequences of Trials
Via Steepest Descents in Two Ways
Via Probability in Three Ways
The Thermodynamic Limit
The Density of States
The Partition Function
Maximum Entropy Applied to Statistical Mechanics
Some Symplectic and Contact Geometry
Hypersurfaces Determined by a Function
The Conormal Bundle
The Wavefront Set
The Space of Tangent Contact Elements
Legendre Transforms and Linear State Spaces
The Origin of the Lagrangian Submanifolds in Physics
Constrained Integration and Extremization
Paths Constrained on Surfaces
The Wavevector as a Kind of Force
Distributions Constrained on Subsystems
Thermodynamic Forces
Lagrange Multipliers and Legendre Maps
Lagrangian Submanifolds and Constrained Extremization
Theorem on the Pushforward of Legendre Submanifolds
The Contact Structure for Thermodynamics
Legendre Transforms and Thermodynamic Potentials
Phase Transitions and the Geometry of the Equation of State
Caustics and Phase Transitions
Convexity and First Order Phase Transitions
A Generalization of Maxwell’s Equal Area Rule
Relations Between Symplectic Thermodynamics and Mechanics
A) Eikonal Waves and Stationary Phase
B) Thermodynamic Limit and Steepest Descents
A) Waves and the Feynman Path Integral
B) Probability and the Maximum Entropy Formalism
A) Wave Path Integrals over a Subspace
B) Probability Distribution Averages over a Subspace
A) Lagrange Multipliers and Canonical Conjugacy
B) Lagrange Multipliers and Thermodynamic Conjugacy
A) Fourier Transforms and Legendre Transforms
B) Laplace Transforms and Legendre Transforms
