spii - Statistical Physics II(PHYS*4240 Lecture notes(Fall...

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Statistical Physics II (PHYS*4240) Lecture notes (Fall 2009) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 η Eric Poisson Department of Physics University of Guelph
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Contents 1 Review of thermodynamics 1 1.1 Thermodynamic variables 1 1.2 Equilibrium 1 1.3 Equation of state 2 1.4 Quasi-static transformations 3 1.5 First law of thermodynamics 3 1.5.1 Formulation 3 1.5.2 Work 3 1.5.3 Work depends on path 4 1.5.4 Heat 4 1.5.5 Heat capacities 4 1.5.6 Heat reservoir 5 1.6 Second law of thermodynamics 5 1.6.1 Two statements 5 1.6.2 Reversible, irreversible, and cyclic transformations 6 1.6.3 Clausius’ theorem 6 1.6.4 Entropy 7 1.6.5 Properties of the entropy 7 1.6.6 Example: system interacting with a heat reservoir 8 1.6.7 Third law of thermodynamics 9 1.7 Thermodynamic potentials 9 1.7.1 Energy 9 1.7.2 Enthalpy 9 1.7.3 Helmholtz free energy 10 1.7.4 Gibbs free energy 10 1.7.5 Landau potential 10 1.8 Maxwell relations 11 1.9 Scaling properties 11 1.10 Equilibrium conditions for isolated, composite systems 12 1.11 Equilibrium for interacting systems 13 1.12 Limitations of thermodynamics 15 1.13 Brief summary of thermodynamics 15 1.14 Problems 16 2 Statistical mechanics of isolated systems 19 2.1 Review of probabilities 19 2.1.1 Probabilities 19 2.1.2 Averages 20 2.1.3 Continuous variables 20 2.2 Macrostate and microstates 21 2.3 Statistical weight 22 2.3.1 De±nition 22 2.3.2 Example: single particle in a box 23 2.3.3 Example: N particles in a box 23 i
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ii Contents 2.4 Statistical weight in classical mechanics 24 2.4.1 Single particle in one dimension 24 2.4.2 General systems 26 2.4.3 Example: N particles in a box 26 2.5 Fundamental postulates 27 2.6 Application: ideal gas 29 2.6.1 Entropy 29 2.6.2 Gibbs’ paradox 30 2.6.3 Thermodynamic quantities 30 2.7 Problems 31 3 Statistical mechanics of interacting systems 35 3.1 System in contact with a reservoir 35 3.1.1 Probability distributions 35 3.1.2 Entropy 37 3.2 Boltzmann distribution 38 3.2.1 Thermodynamic quantities 38 3.2.2 Energy distribution 40 3.2.3 Application: N simple harmonic oscillators 41 3.3 Gibbs distribution 44 3.3.1 Thermodynamic quantities 44 3.3.2 Fluctuations 45 3.4 Classical statistics 46 3.4.1 Partition function 47 3.4.2 The ideal gas — again 47 3.4.3 Equipartition theorem 48 3.5 The meaning of probability 49 3.6 Brief summary of statistical mechanics 50 3.6.1 Boltzmann distribution 50 3.6.2 Gibbs distribution 51 3.7 Problems 52 4 Information theory 55 4.1 Missing information 55 4.1.1 Uniform probabilities 55 4.1.2 Assigned probabilities 56 4.2 Entropy 57 4.3 Boltzmann and Gibbs distributions 58 4.3.1 Maximum missing information 58 4.3.2 Lagrange multipliers 58 4.3.3 Probability distributions 59 4.3.4 Evaluation of the Lagrange multipliers 60 4.3.5 The ±rst law 60 4.4 Conclusion 61 5 Paramagnetism 63 5.1 Magnetism 63 5.1.1 Magnetic moment 63 5.1.2 Potential energy 63 5.1.3 Magnetic moments of atoms and molecules 64 5.1.4 Forms of magnetism 65 5.2 General theory of paramagnetism 65 5.2.1
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spii - Statistical Physics II(PHYS*4240 Lecture notes(Fall...

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