Lectures on the Mechanical Foundations of Thermodynamics

Description

This brief provides a modern pedagogical exposition of the mechanical approach to statistical mechanics initiated by Boltzmann with his early works (1866-1871). Despite the later contribution by Helmholtz, Boltzmann himself (1884-1887), Gibbs, P. Hertz, and Einstein, the mechanical approach remained almost unknown to the modern reader, in favour of the celebrated combinatorial approach, developed by Boltzmann himself during his probabilistic turn (1876-1884). The brief constitutes an ideal continuation of a graduate course of classical mechanics and requires knowledge of basic calculus in many dimension (including differential forms), thermodynamics, probability theory, besides Hamiltonian mechanics. The cornerstone of the whole presentation is the ergodic hypothesis. Special attention is devoted to Massieu potentials (the Legendre transforms of the entropy) which are most natural in statistical mechanics, and also allow for a more direct treatment of the topic of ensemble equivalence. 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 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