Thermodyn

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Unformatted text preview: Statistical Mechanics Uwe-Jens Wiese Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics Bern University December 23, 2010 2 Contents 1 Introduction 9 2 Kinetic Theory of the Classical Ideal Gas 13 2.1 Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Pressure and Temperature of the Ideal Gas . . . . . . . . . . . . . 15 3 Microcanonical and Canonical Ensemble 19 3.1 The Hamilton Function . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 The Concept of an Ensemble . . . . . . . . . . . . . . . . . . . . . 21 3.3 The Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . 22 3.4 The Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 Particle on an Energy Ladder . . . . . . . . . . . . . . . . . . . . . 25 3.6 Model for a Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.7 Canonical Ensemble for Particles on a Ladder . . . . . . . . . . . . 28 3.8 Microcanonical Ensemble for Particles on a Ladder . . . . . . . . . 29 4 Information and Entropy 33 4.1 Information and Information Deficit . . . . . . . . . . . . . . . . . 33 4.2 The Concept of Entropy . . . . . . . . . . . . . . . . . . . . . . . . 35 3 4 CONTENTS 4.3 Entropy and Free Energy in the Canonical Ensemble . . . . . . . . 36 4.4 Entropy of Particles on a Ladder . . . . . . . . . . . . . . . . . . . 36 4.5 The Principle of Maximum Entropy . . . . . . . . . . . . . . . . . 38 4.6 The Arrow of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Canonical Ensemble for the Ideal Gas 45 5.1 The Maxwell-Boltzmann Distribution . . . . . . . . . . . . . . . . 45 5.2 Ideal Gas in a Gravitational Field . . . . . . . . . . . . . . . . . . 46 5.3 Distinguishability of Classical Particles . . . . . . . . . . . . . . . . 48 5.4 The Entropy of the Classical Ideal Gas . . . . . . . . . . . . . . . . 49 5.5 Gibbs’ Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.6 Mixing Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6 Grand Canonical Ensemble 55 6.1 Introduction of the Grand Canonical Ensemble . . . . . . . . . . . 55 6.2 Grand Canonical Ensemble of Particles on a Ladder . . . . . . . . 57 6.3 Chemical Potential of Particles on a Ladder . . . . . . . . . . . . . 58 6.4 Chemical Potential of the Classical Ideal Gas . . . . . . . . . . . . 60 6.5 Grand Canonical Ensemble for the Ideal Gas . . . . . . . . . . . . 61 7 Pressure Ensemble 63 7.1 Introduction of the Pressure Ensemble . . . . . . . . . . . . . . . . 63 7.2 The Pressure of the Classical Ideal Gas . . . . . . . . . . . . . . . 64 7.3 The Pressure Ensemble for the Classical Ideal Gas . . . . . . . . . 65 7.4 Overview of Different Ensembles . . . . . . . . . . . . . . . . . . . 66 CONTENTS 5 8 Equilibrium Thermodynamics 69 8.1 The First Law of Thermodynamics . . . . . . . . . . . . . . . . . . 69 8.2 Expansion of a Classical Ideal Gas . . . . . . . . . . . . . . . . . . 70 8.3 Heat and Entropy Change . . . . . . . . . . . . . . . . . . . . . . . 71Heat and Entropy Change ....
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