Physics 449/451 - Statistical Mechanics - Course Notes David L. Feder September 13, 2011 Contents 1 Energy in Thermal Physics (First Law of Thermodynamics) 3 1.1 Thermal Equilibrium . 3 1.2 TheIdealGas...................................... .. 3 1.2.1 Thermodynamic Derivation . 3 1.2.2 Mechanical Derivation . 4 1.3 EquipartitionofEnergy ............................... ... 6 1.4 HeatandWork...................................... 7 1.5 CompressionWork:theAdiabat . .... 7 1.6 HeatCapacity...................................... .. 9 2 The Second Law of Thermodynamics (aka The Microcanonical Ensemble) 13 2.1 Two-State Systems (aka Flipping Coins) . ...... 13 2.1.1 Lots and lots of trials . 15 2.1.2 Digression: Statistics . 16 2.2 Flow toward equilibrium . 17 2.3 LargeSystems ..................................... .. 18 2.3.1 DiscreteRandomWalks .............................. 18 2.3.2 ContinuousRandomWalks ............................ 20 2.3.3 QuantumWalksandQuantumComputation . 22 2.4 Entropy .......................................... 25 2.4.1 Boltzmann ..................................... 25 2.4.2 ShannonEntropy.................................. 26 2.4.3 vonNeumannEntropy............................... 28 3 Equilibrium 32 3.1 Temperature ...................................... .. 32 3.2 Entropy,Heat,andWork ............................. .... 34 3.2.1 ThermodynamicApproach . .. .. .. .. .. .. .. .. .. .. .. .. 34 3.2.2 StatisticalApproach ............................... 35 3.3 Paramagnetism..................................... .. 36 3.4 Mechanical Equilibrium and Pressure . ..... 38 3.5 Diffusive Equilibrium and Chemical Potential . .... 39 4 Engines and Refrigerators 41 4.1 HeatEngines ....................................... 41 4.2 Refrigerators ..................................... ... 44 4.3 RealHeatEngines ................................... .. 44 1 PHYS 449 - Course Notes 2009 2 4.3.1 Stirling Engine . 44 4.3.2 SteamEngine.................................... 46 4.3.3 InternalCombustionEngine. .. 47 4.4 RealRefrigerators ................................. .... 49 4.4.1 HomeFridges.................................... 49 4.4.2 Liquefaction of Gases and Going to Absolute Zero . .... 50 5 Free Energy and Chemical Thermodynamics 51 5.1 FreeEnergyasWork................................. ... 51 5.1.1 Independent variables S and V .......................... 51 5.1.2 Independent variables S and P .......................... 52 5.1.3 Independent variables T and V .......................... 52 5.1.4 Independent variables T and P .......................... 53 5.1.5 ConnectiontoWork ................................ 53 5.1.6 Varying particle number . 54 5.2 Free Energy as Force toward Equilibrium . ..... 54 6 Boltzmann Statistics (aka The Canonical Ensemble) 56 6.1 TheBoltzmannFactor................................ ... 56 6.2 Z andtheCalculationofAnything . 58 6.2.1 Example: Pauli Paramagnet Again! . 60 6.2.2 Example: Particle in a Box (1D) . 62 6.2.3 Example: Particle in a Box (3D) . 63 6.2.4 Example: Harmonic Oscillator (1D) . 64 6.2.5 Example: Harmonic Oscillator (3D) . 65 6.2.6 Example:Therotor ................................ 66 6.3 The Equipartition Theorem (reprise) . ...... 68 6.3.1 DensityofStates .................................. 69 6.4 The Maxwell Speed Distribution . ... 71 6.4.1 InterludeonAverages .............................. 73 6.4.2 MolecularBeams.................................. 73 6.5 (AlreadycoveredinSec.6.2) . .. .. .. .. .. .. .. .. .. .. .. .... 75 6.6 Gibbs’Paradox..................................... .. 75 7 Grand Canonical Ensemble 77 7.1 ChemicalPotentialAgain .............................. ... 77 7.2 GrandPartitionFunction .............................. ... 78 7.3 GrandPotential.................................... ... 80 8 VirialTheoremandtheGrandCanonicalEnsemble 81 8.1 VirialTheorem ...................................... 81 8.1.1 Example:idealgas................................. 82 8.1.2 Example: Averagetemperatureofthesun . ..... 82 8.2 ChemicalPotential.................................. ... 83 8.2.1 Free energies revisited . 84 8.2.2 Example:PauliParamagnet. 85 8.3 GrandPartitionFunction .............................. ... 85 8.3.1 Examples ...................................... 87 8.4 GrandPotential.................................... ... 87 PHYS 449 - Course Notes 2009 3 9 Quantum Counting 89 9.1 Gibbs’Paradox..................................... .. 89 9.2 ChemicalPotentialAgain .............................. ... 91 9.3 Arranging Indistinguishable Particles . ...... 92 9.3.1 Bosons........................................ 92 9.3.2 Fermions ...................................... 93 9.3.3 Anyons!....................................... 95 9.4 Emergence of Classical Statistics . ...... 96 10 Quantum Statistics 99 10.1 BoseandFermiDistributions . ..... 99 10.1.1 Fermions ...................................... 100 10.1.2 Bosons........................................ 102 10.1.3 Entropy....................................... 104 10.2 Quantum-Classical Transition . ...... 106 10.3 EntropyandEquationsofState. ....... 107 11 Fermions 110 11.1 3DBoxatzerotemperature. ...... 110 11.23DBoxatlowtemperature .. .. .. .. .. .. .. .. .. .. .. .. .... 111 11.3 3Disotropicharmonictrap .. .. .. .. .. .. .. .. .. .. .. .. .... 113 11.3.1 DensityofStates ................................. 113 11.3.2 LowTemperatures ................................ 114 11.3.3 SpatialProfile ................................... 115 11.4AFewExamples ..................................... 117 11.4.1 Electrons in Metals . 117 11.4.2 ElectronsintheSun ................................ 117 11.4.3 Ultracold Fermionic Atoms in a Harmonic Trap . 118 12 Bosons 119 12.1QuantumOscillators ................................. .. 119 12.2Phonons......................................... 120 12.3BlackbodyRadiation................................. .. 123 12.4 Bose-EinsteinCondensation . ...... 126 12.4.1 BECin3D ..................................... 126 12.4.2 BEC in Lower Dimensions . 127 12.4.3 BECinHarmonicTraps.............................. 129 PHYS 449 - Course Notes 2009 2 Introduction The purpose of these course notes is mainly to give your writing hand a break. I tend to write lots of equations on the board, because I want to be rigorous with the material. But I write very quickly and my handwriting isn’t pretty (this is probably a huge understatement). So these course notes contain (hopefully) all the equations that I will be writing on the board, so that when you take notes during class you can focus on the concepts and my mistakes, rather than furiously trying to scribble down everything I am writing, which will probably contain mistakes anyhow. Not to say that these notes don’t contain mistakes! These notes also have occasional non-mathematical expressions (i.e. sentences). Chapter 1 Energy in Thermal Physics (First Law of Thermodynamics) This chapter deals with very fundamental concepts in thermodynamics, many of which you can intuit from your experience. 1.1 Thermal Equilibrium Some questions to ponder: What are the ways that you measure room temperature? • What are the ways to measure temperatures that are much hotter? Colder? • What exactly is temperature? • What is absolute zero? • How do systems reach a given temperature? • What does it means to say a system is in equilibrium? • What is thermal equilibrium? • 1.2 The Ideal Gas 1.2.1 Thermodynamic Derivation Robert Boyle (1627-1691) was an Irish alchemist (!) who helped to establish chemistry as a legitimate field. After much observing, he found in 1662 that gases tended to obey the follow equation: P V = k, (1.1) where P is the pressure, V is the volume, and k is some constant that depends on the specific gas. This equation was known as Boyle’s Law. In 1738 Daniel Bernoulli derived it using Newton’s 3 PHYS 449 - Course Notes 2009 4 equations of motion (see more about this in the next section), under the assumption that the gas was made up of particles too tiny to see, but no one paid any attention because these particles were not believed to actually exist. Later, Joseph-Louis Gay-Lussac (1778-1850) observed that at constant pressure, one always has V T , or V1T2 = V2T1. This is known as Charles’ Law after some guy named Charles. BenoˆıtPaul∝Emile´ Clapeyron (1799-1864) put the two laws together to obtain P V = P0V0 (267 + T ) , (1.2) in which the temperature is measured in degrees Celcius. The number 267 came from observations of Gay-Lussac. This was pretty impressive, since absolute zero is known today to be -273.15◦C. Lorenzo Romano Amedeo Carlo Avogadro di Quaregna (Quaregga) e di Cerreto (1776-1856), otherwise known as Avogadro, showed in 1811 that the P0V0 out front of Clapeyron’s equation was related to the ‘amount of substance’ of the gas, and wrote: P V = nR (267 + T ) , (1.3) where n is the number of moles of the gas, and R =8.31 J/mol/K is a universal constant (indepen- dent of the type of gas). It’s easier to think of the number of particles N (atoms or molecules) rather 23 than the number of moles, so one can write N = nNA, where NA =6.02214179(30) 10 is known as Avogadro’s number and corresponds to the number of atoms in a mole of gas.× Finally, if we measure temperature in units of Kelvin (K = 273.15+◦ C) and make the substitution R = NAkB, 23 where k =1.381 10− J/K is Boltzmann’s constant, we finally obtain the ideal gas law: B × P V = NkBT. (1.4) This is nice because we don’t have to worry about moles. What’s a mole anyhow?? That’s about all the history you’re going to get! The ideal gas law