Lectures on the Mechanical Foundations of Thermodynamics

Foreword
Preface
	References
Contents
1 The Constitutive Statements of Thermodynamics
	1.1 First and Second Law
		1.1.1 First Law of Thermodynamics
		1.1.2 Second Law of Thermodynamics
	1.2 Massieu Potentials
		1.2.1 The β,V Potential ΨF (Free Entropy)
		1.2.2 The E,π Potential ΨH
		1.2.3 The β,π Potential ΨG
		1.2.4 The β,V,α Massieu Grand Potential
	1.3 Common Thermodynamic Potentials
		1.3.1 The S, V Internal Energy E
		1.3.2 The T, V Helmholtz Free Energy F
		1.3.3 The S, P Enthalpy H
		1.3.4 The T, P Gibbs Free Energy G
		1.3.5 The T,V,µ, Grand Potential
	References
2 Minimal Mechanical Model of Thermodynamics
	2.1 A Particle in a 1D Box
		2.1.1 Remarks
	2.2 Helmholtz Theorem
	2.3 First Encounter with Ergodicity and the Microcanonical Ensemble
	2.4 Non-uniqueness of Integrating Factors
	2.5 Examples and Applications
		2.5.1 Thermodynamics of a Harmonic Oscillator
		2.5.2 The Peculiar Thermodynamics of a Log-Oscillator
		2.5.3 Particle in a Double Well Potential
	References
3 The Microcanonical Ensemble
	3.1 Many-Body Model of Thermodynamics
	3.2 Remarks on Ergodicity
		3.2.1 Metric Indecomposability
		3.2.2 The Invariant Measure
	3.3 Properties of the Structure Function
	3.4 Ideal Gas
		3.4.1 Remarks
	3.5 Some History
		3.5.1 Boltzmann Expressions of Microcanonical Entropy
		3.5.2 Boltzmann Expression of Microcanonical Ensemble
	References
4 The Canonical Ensemble
	4.1 A System in Weak Interaction with a Large Ideal Gas
	4.2 Validity of the Heat Theorem within the Canonical Ensemble
	4.3 Properties of the Partition Function
	4.4 Ideal Gas
	4.5 Linear Response Coefficients and Fluctuations
	4.6 Finite Bath Ensemble
	4.7 A System in Weak Contact with a Logarithmic Oscillator
	References
5 The TP Ensemble
	5.1 A System in Contact with a Thermal Reservoir and Subject to a Constant Pressure
	5.2 Validity of the Heat Theorem within the TP Ensemble
	5.3 Fluctuations
	5.4 Ideal Gas
	5.5 Thermally Insulated System Subject to a Constant Pressure
	References
6 The Grandcanonical Ensemble
	6.1 Relaxing the Constraint on N
	6.2 A Slight Detour: Gibbs “Correct” Counting
	6.3 Validity of the Heat Theorem within the Grandcanonical Ensemble
	6.4 Ideal Gas
		6.4.1 Fluctuations
	6.5 Remarks
	References
7 Ensemble (in)-Equivalence
	7.1 The Problem
	7.2 Full Equivalence
	7.3 Equivalence
		7.3.1 Hilbert and Dunkel Model of Evaporation
		7.3.2 The Saddle Point Approximation
	7.4 Partial Equivalence
	7.5 Inequivalence
		7.5.1 Thirring Artificial Model of a Star
	7.6 (In)-equivalent Expressions of the Microcanonical Entropy
	References
8 Statistical Mechanical Foundation of the Law of Entropy Increase
	8.1 The Main Question
	8.2 Entropy Increase for Initial Passive Statistics
	8.3 Entropy Increase in Large Ergodic Systems with Initial Microcanonical Statistics
		8.3.1 The Hard-Sphere Gas
		8.3.2 From the Hard-Sphere Gas to a Generic Ergodic System
	8.4 Passive Statistics and Kelvin Formulation of the Second Law
		8.4.1 Geometric Proof
	8.5 Entropy Increase in a Forced Harmonic Oscillator
	8.6 Remarks
	References