Poisson StatPhys II

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Eric Poisson note II

Transcript of Poisson StatPhys II

Statistical Physics II (PHYS*4240)Lecture notes (Fall 2009)

1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3

Eric Poisson Department of Physics University of Guelph

Contents1 Review of thermodynamics 1.1 Thermodynamic variables 1.2 Equilibrium 1.3 Equation of state 1.4 Quasi-static transformations 1.5 First law of thermodynamics 1.5.1 Formulation 1.5.2 Work 1.5.3 Work depends on path 1.5.4 Heat 1.5.5 Heat capacities 1.5.6 Heat reservoir 1.6 Second law of thermodynamics 1.6.1 Two statements 1.6.2 Reversible, irreversible, and cyclic transformations 1.6.3 Clausius theorem 1.6.4 Entropy 1.6.5 Properties of the entropy 1.6.6 Example: system interacting with a heat reservoir 1.6.7 Third law of thermodynamics 1.7 Thermodynamic potentials 1.7.1 Energy 1.7.2 Enthalpy 1.7.3 Helmholtz free energy 1.7.4 Gibbs free energy 1.7.5 Landau potential 1.8 Maxwell relations 1.9 Scaling properties 1.10 Equilibrium conditions for isolated, composite systems 1.11 Equilibrium for interacting systems 1.12 Limitations of thermodynamics 1.13 Brief summary of thermodynamics 1.14 Problems 2 Statistical mechanics of isolated systems 2.1 Review of probabilities 2.1.1 Probabilities 2.1.2 Averages 2.1.3 Continuous variables 2.2 Macrostate and microstates 2.3 Statistical weight 2.3.1 Denition 2.3.2 Example: single particle in a box 2.3.3 Example: N particles in a box i 1 1 1 2 3 3 3 3 4 4 4 5 5 5 6 6 7 7 8 9 9 9 9 10 10 10 11 11 12 13 15 15 16 19 19 19 20 20 21 22 22 23 23

ii 2.4 Statistical weight in classical mechanics 2.4.1 Single particle in one dimension 2.4.2 General systems 2.4.3 Example: N particles in a box Fundamental postulates Application: ideal gas 2.6.1 Entropy 2.6.2 Gibbs paradox 2.6.3 Thermodynamic quantities Problems

Contents 24 24 26 26 27 29 29 30 30 31 35 35 35 37 38 38 40 41 44 44 45 46 47 47 48 49 50 50 51 52 55 55 55 56 57 58 58 58 59 60 60 61 63 63 63 63 64 65 65 66 66

2.5 2.6

2.7

3 Statistical mechanics of interacting systems 3.1 System in contact with a reservoir 3.1.1 Probability distributions 3.1.2 Entropy 3.2 Boltzmann distribution 3.2.1 Thermodynamic quantities 3.2.2 Energy distribution 3.2.3 Application: N simple harmonic oscillators 3.3 Gibbs distribution 3.3.1 Thermodynamic quantities 3.3.2 Fluctuations 3.4 Classical statistics 3.4.1 Partition function 3.4.2 The ideal gas again 3.4.3 Equipartition theorem 3.5 The meaning of probability 3.6 Brief summary of statistical mechanics 3.6.1 Boltzmann distribution 3.6.2 Gibbs distribution 3.7 Problems 4 Information theory 4.1 Missing information 4.1.1 Uniform probabilities 4.1.2 Assigned probabilities 4.2 Entropy 4.3 Boltzmann and Gibbs distributions 4.3.1 Maximum missing information 4.3.2 Lagrange multipliers 4.3.3 Probability distributions 4.3.4 Evaluation of the Lagrange multipliers 4.3.5 The rst law 4.4 Conclusion 5 Paramagnetism 5.1 Magnetism 5.1.1 Magnetic moment 5.1.2 Potential energy 5.1.3 Magnetic moments of atoms and molecules 5.1.4 Forms of magnetism 5.2 General theory of paramagnetism 5.2.1 The model 5.2.2 Quantum and thermal averages

Contents 5.2.3 Energy eigenvalues 5.2.4 Partition function 5.2.5 Magnetization Molecules with j = 1 2 Molecules with arbitrary j Paramagnetic solid in isolation 5.5.1 Statistical weight and entropy 5.5.2 Temperature 5.5.3 Negative temperatures? 5.5.4 Correspondence with the canonical ensemble Problems

iii 67 67 68 68 69 70 71 71 72 73 73 75 75 76 77 77 77 78 79 80 80 80 81 81 81 82 83 83 84 85 86 86 87 87 88 89 89 90 90 91 91 91 92 92 93 93 93 94 94 95 95 95 96

5.3 5.4 5.5

5.6

6 Quantum statistics of ideal gases 6.1 Quantum statistics 6.1.1 Microstates 6.1.2 Bosons and fermions 6.1.3 Partition function 6.1.4 Grand partition function 6.1.5 Mean occupation numbers 6.1.6 Thermodynamic quantities 6.2 Bose-Einstein statistics 6.2.1 Grand partition function 6.2.2 Mean occupation numbers 6.3 Fermi-Dirac statistics 6.3.1 Grand partition function 6.3.2 Mean occupation numbers 6.4 Summary 6.5 Classical limit 6.5.1 Classical approximation 6.5.2 Evaluation of 6.5.3 Other thermodynamic quantities 6.5.4 Conclusion 6.6 Slightly degenerate ideal gases 6.6.1 Evaluation of 6.6.2 Energy and pressure 6.6.3 Higher-order corrections 6.7 Highly degenerate Fermi gas. I: Zero temperature 6.7.1 Fermi gas at T = 0 6.7.2 Evaluation of the Fermi energy 6.7.3 Energy and pressure 6.7.4 Entropy 6.8 Highly degenerate Fermi gas. II: Low temperature 6.8.1 Fermi-Dirac integrals 6.8.2 Chemical potential 6.8.3 Energy and pressure 6.8.4 Heat capacity 6.9 Highly degenerate Fermi gas. III: Applications 6.9.1 Conduction electrons in metals 6.9.2 White dwarfs 6.9.3 Neutron stars 6.10 Highly degenerate Bose gas. I: Zero temperature 6.11 Highly degenerate Bose gas. II: Low temperatures 6.11.1 Bose-Einstein integrals 6.11.2 Number of particles in ground state

iv 6.11.3 Energy and pressure 6.11.4 Heat capacity 6.12 Highly degenerate Bose gas. III: Bose-Einstein condensation 6.12.1 Superuidity 6.12.2 Superconductivity 6.12.3 Holy Grail 6.13 Problems 7 Black-body radiation 7.1 Photon statistics 7.2 Energy eigenvalues 7.3 Density of states 7.4 Energy density 7.5 Other thermodynamic quantities 7.6 The cosmic microwave background radiation 7.6.1 Recombination in the early universe 7.6.2 Dipole anisotropy 7.6.3 Black-body radiation as seen by a moving observer 7.6.4 Proper motion of the galaxy 7.6.5 Quadrupole anisotropy 7.7 Problems

Contents 97 97 97 98 98 98 99 101 101 102 103 103 104 105 105 106 106 107 107 107

1Review of thermodynamics1.1 Thermodynamic variablesThe purpose of thermodynamics is to describe the properties of various macroscopic systems at, or near, equilibrium. This is done with the help of several state variables, such as the internal energy E , the volume V , the pressure P , the number of particles N , the temperature T , the entropy S , the chemical potential , and others. These state variables depend only on what the state of the system is, and not on how the system was brought to that state. The variables are not all independent, and thermodynamics provides general relations among them. Some of these relations are not specic to particular systems, but are valid for all macroscopic systems.

1.2

Equilibrium

A system is in equilibrium if its physical properties do not change with time. The state variables must therefore be constant in time. More precisely, a system is in equilibrium if there are no macroscopic motions (mechanical equilibrium) macroscopic energy uxes (thermal equilibrium) unbalanced phase transitions or chemical reactions (chemical equilibrium) occurring within the system. We will see in Sec. 10 that the requirement for mechanical equilibrium is that P must be constant in time and uniform throughout the system. This is easily understood: a pressure gradient produces an unbalanced macroscopic force which, in turn, produces bulk motion within the system; such a situation does not represent equilibrium. The requirement for thermal equilibrium is that T must be constant in time and uniform throughout the system. This also is easily understood: If one part of the system is hotter than the other, energy (in the form of heat) will ow from that part to the other; again such a situation does not represent equilibrium. Finally, the requirement for chemical equilibrium is that must be constant in time and uniform throughout the system. The chemical potential is not a very 1

2

Review of thermodynamics

familiar quantity, and this requirement can perhaps not be understood easily. But at least it is clear that unbalanced chemical reactions (we take this to include nuclear reactions as well) cannot occur in a system at equilibrium. Consider for example the reaction of -decay: n p + e + . If this reaction is unbalanced (the reversed reaction can occur at suciently high densities), the number of neutrons in the system will keep on decreasing, while the number of protons, electrons, and anti-neutrinos will keep on increasing. Such a situation does not represent equilibrium. A system which is at once in mechanical, thermal, and chemical equilibrium is said to be in thermodynamic equilibrium.

1.3

Equation of state

The equation of state of a thermodynamic system expresses the fact that not all of its state variables are independent. Consider a system described by two state variables X and Y , apart from the temperature T . (For example, an ideal gas is described by the variables X V , Y P .) We suppose that the system is initially in equilibrium at temperature T . We ask the question: Can X and Y be varied independently, so that the system remains in equilibrium after the transformation? The answer, which must be determined empirically, is: No! For the system to remain in equilibrium, a given variation of X must be accompanied by a specic variation in Y . If Y is not varied by this amount, the system goes out of equilibrium. There must therefore exist an equation of state, f (X, Y, T ) = 0 , (1.1)

relating X , Y , and T for all equilibrium congurations. For non-equilibrium states, f = 0. [For example, the equation of state of an ideal gas is f (V, P, T ) = P V N kT = 0.] The equation of state cannot be derived within the framework of thermodynamics. (This is one of the goals of statistical mechanics.) It must be provided as input, and determined empirically. Suppose the system goes from one equilibrium conguration (X, Y, T ) to a neighbouring one (X + X, Y + Y, T ). The temperature is kept constant for simplicity; it could also be varied. We can use the equation of state to calculate Y in terms of X . We have f (X, Y, T ) = 0 and f (X + X, Y + Y, T ) = 0, since both these congurations are at equilibrium. So f (X + X, Y + Y, T ) f (X, Y, T ) = 0. Expressing the rst term as a Taylor series about (X, Y, T ), up to rst order in the displacements X and Y