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Notes on STATISTICAL MECHANICS K.P.N. Murthy Notes on STATISTICAL MECHANICS August 28, 2017 DRAFT It was certainly not by design that the particles fell into order They did not work out what they were going to do, but because many of them by many chances struck one another in the course of infinite time and encountered every possible form and movement, that they found at last the disposition they have, and that is how the universe was created. Titus Lucretius Carus (94 BC - 55BC) de Rerum Natura Everything existing in the universe is the fruit of chance and ne- cessity. Democritus (370 BC) The moving finger writes; and, having writ, moves on : nor all your piety nor wit shall lure it back to cancel half a line nor all your tears wash out a word of it Omar Khayyam (1048 - 1131) Whatever happened, happened for good. Whatever is happening, is happening for good. Whatever will happen, will happen for good. Bhagavat Gita ” Ludwig Boltzmann, who spent much of his life studying sta- tistical··· mechanics, died in 1906, byDRAFT his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn .... to study statistical mechanics. Perhaps it will be wise to approach the subject rather cautiously. ” David Goodstein, States··· Matter, Dover (1975) (opening lines) All models are wrong, some are useful. George E P Box DRAFT Contents Quotes ............................................. i 1. Micro-Macro Synthesis ................................... 3 1.1 Aim of Statistical Mechanics . 3 1.2 Micro- MacroConnections ............................. 4 1.2.1 BoltzmannEntropy.............................. 4 1.2.2 Boltzmann-Gibbs-Shannon Entropy................ 4 1.2.3 Heat........................................... 5 1.2.4 Work .......................................... 5 1.2.5 HelmholtzFreeEnergy........................... 5 1.2.6 Energy Fluctuations and Heat Capacity............ 6 1.3 Micro World : Determinism and Time-Reversal Invariance.............................. 6 1.4 MacroWorld:Thermodynamics ........................ 6 1.5 Books............................................ .... 9 1.6 ExtraReading:Books................................ 13 1.7 ExtraReading:Papers ............................... 13 2. Maxwell’s Mischief ....................................... 15 2.1 ExperimentandOutcomes............................. 15 2.2 Samplespaceandevents ..............................DRAFT . 15 2.3 Probabilities . .. 16 2.4 Rules of probability . 16 2.5 Randomvariable ..................................... 17 2.6 Maxwell’s mischief : Ensemble . 18 2.7 Calculation of probabilities from an ensemble . 19 2.8 Construction of ensemble from probabilities . 19 2.9 Counting of the elements in events of the sample space : Cointossing .......................................... 19 2.10 Gibbsensemble ..................................... .. 22 2.11 Why should a Gibbs ensemble be large? ................. 22 3. Binomial, Poisson, and Gaussian .......................... 27 3.1 Binomial Distribution . 27 3.2 Moment Generating Function ........................... 29 iv Contents 3.3 Binomial Poisson ................................... 31 3.4 Poisson Distribution→ . 32 3.4.1 Binomial Poisson `ala Feller . 33 3.5 CharacteristicFunction→ .............................. .. 34 3.6 Cumulant Generating Function.......................... 35 3.7 TheCentralLimitTheorem ............................ 35 3.8 Poisson Gaussian ................................... 38 3.9 Gaussian→ .......................................... ... 38 4. Isolated System: Micro canonical Ensemble ............... 41 4.1 Preliminaries. .. 41 4.2 ConfigurationalEntropy............................. ... 42 4.3 IdealGasLaw:Derivation ............................. 43 4.4 Boltzmann Entropy Clausius’ Entropy ............... 44 4.5 Some Issues on Extensitivity−→ of Entropy.................. 44 4.6 BoltzmannCounting.................................. 44 4.7 MicrocanonicalEnsemble .............................. 45 4.8 Heaviside and his Θ Function ........................... 46 4.9 Dirac and his δ Function ............................... 46 4.10 AreaofaCircle ..................................... .. 48 4.11 Volume of an N-Dimensional Sphere ..................... 50 4.12 Classical Counting of Micro states . 52 4.12.1 Counting of the Volume.......................... 52 4.13 DensityofStates .................................. .... 52 4.13.1 A Sphere Lives on its Outer Shell : Power Law can beIntriguing.................................... 53 4.14 Entropy of an IsolatedSystem ........................ .. 53 4.15 Propertiesofan IdealGas ........................... ... 54 4.15.1 Temperature.................................... 54 4.15.2 Equipartition Theorem........................... 54 4.15.3 Pressure.......................................DRAFT . 54 4.15.4 IdealGasLaw .................................. 55 4.15.5 ChemicalPotential .............................. 55 4.16 Quantum Counting of Micro states ...................... 57 4.16.1 Energy Eigenvalues : Integer Number of Half Wave Lengths in L .................................... 58 4.17 Chemical Potential : Toy Model ........................ 60 5. Closed System : Canonical Ensemble ..................... 63 5.1 WhatisaClosedSystem? ............................. 63 5.2 ToyModel`alaHBCallen ............................. 63 5.3 Canonical Partition Function . 64 5.3.1 Derivation `ala Balescu . 64 5.4 HelmholtzFreeEnergy................................ 66 5.5 Energy Fluctuations and Heat Capacity .................. 68 Contents v 5.6 Canonical Partition Function : Ideal Gas . 69 5.7 Method of Most Probable Distribution . 70 5.8 Lagrangeandhis Method.............................. 73 5.9 Generalisation to N Variables........................... 75 5.10 Derivation of Boltzmann Weight. 76 5.11 Mechanical and Thermal Properties.................... .. 77 5.12 Entropyof a Closed System........................... .. 78 5.13 FreeEnergyto Entropy ............................. ... 79 5.14 MicroscopicView :Heat andWork ...................... 80 5.14.1 Work in Statistical Mechanics : W = pi dEi ...... 81 i X 5.14.2 Heat in Statistical Mechanics : q = Ei dpi ...... 82 i 5.15 Adiabatic Process - a Microscopic View . .X . 82 5.16 Q(T,V,N)foranIdealGas............................. 83 6. Open System : Grand Canonical Ensemble ............... 87 6.1 WhatisanOpenSystem? ............................. 87 6.2 Micro-Macro Synthesis : and ........................ 90 6.3 Statistics of Number of ParticlesQ G . 92 6.3.1 EulerandhisTheorem........................... 92 6.3.2 : Connection to Thermodynamics ............... 92 6.3.3 Gibbs-DuhemQ Relation........................... 92 6.3.4 Average number of particles in an open system, N . 93 6.3.5 Probability P (N), that there are N particles inh ani opensystem .................................... 93 6.3.6 Number Fluctuations ............................ 94 6.3.7 Number Fluctuations and Isothermal Compressibility 95 6.3.8 Alternate Derivation of the Relation : σ2 / N 2 = N h i kB T kT /V ..................................... 97 6.4 EnergyFluctuations .................................DRAFT .. 99 7. Quantum Statistics ....................................... 105 7.1 Occupation Number Representation..................... 105 7.2 Open System and (T,V,µ)............................ 105 7.3 Fermi-Dirac StatisticsQ . .. 108 7.4 Bose-Einstein Statistics . .. 108 7.5 Classical Distinguishable Particles . 109 7.6 Maxwell-Boltzmann Statistics . 109 7.6.1 Q (T,V,N) (T,V,µ) .................. 110 MB → QMB 7.6.2 MB(T,V,µ) QMB(T,V,N) .................. 111 7.7 ThermodynamicsofQ an→ open system ..................... 111 7.8 Average number of particles, N ........................ 112 7.8.1 Maxwell-Boltzmann Statisticsh i . 112 7.8.2 Bose-Einstein Statistics . 113 vi Contents 7.8.3 Fermi-Dirac Statistics . 113 7.8.4 Study of a System with fixed N Employing Grand Canonical Formalism ............................ 113 7.9 Fermi-Dirac, Maxwell-Boltzmann and Bose-Einstein Statis- tics are the same at High Temperature and/or Low Densities 114 7.9.1 Easy Method : ρΛ3 0.......................... 114 7.9.2 Easier Method : λ →0........................... 116 → 7.9.3 Easiest Method Ω(n1,n2, )=1 ................. 117 7.10 Mean Occupation Number.............................··· . 117 7.10.1 IdealFermions ..................................b 118 7.10.2 IdealBosons.................................... 118 7.10.3 Classical Indistinguishable Ideal Particles . 118 7.11 Mean Ocupation : Some Remarks........................ 119 7.11.1 Fermi-Dirac Statistics . 119 7.11.2 Bose-Einstein Statistics . 119 7.11.3 Maxwell-Boltzmann Statistics . 120 7.11.4 At High T and/or Low ρ all Statistics give the same nk ........................................... 120 7.12 Occupationh i Number : Distribution and Variance . 121 7.12.1 Fermi-Dirac Statistics and Binomial Distribution . 122 7.12.2 Bose-Einstein Statistics and Geometric Distribution . 122 7.12.3 Maxwell-Boltzmann Statistics and Poisson Distribution124 8. Bose-Einstein Condensation .............................. 127 8.1 Introduction ...................................... .... 127 8.2 N = n ........................................ 127 h i h ki k 8.3 SummationtoX Integration ............................. 128 8.4 Graphical Inversion and Fugacity....................... 130 8.5 Treatment of the Singular Behaviour..................... 131 8.6 Bose-Einstein CondensationDRAFT Temperature................ 133 8.7 GrandPotentialfor bosons ........................... .. 134 8.8 EnergyofBosonicSystem
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