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Lecture Notes on Nonequilibrium Statistical Physics (A Work in Progress)

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Lecture Notes on Nonequilibrium Statistical Physics (A Work in Progress) Lecture Notes on Nonequilibrium Statistical Physics (A Work in Progress) Daniel Arovas Department of Physics University of California, San Diego June 13, 2021 Contents 1 Fundamentals of Probability 1 1.1 References...................................... ........ 1 1.2 StatisticalPropertiesofRandomWalks . ............... 2 1.2.1 One-dimensionalrandomwalk . ........ 2 1.2.2 Thermodynamiclimit. ....... 4 1.2.3 Entropyandenergy.............................. ...... 5 1.3 BasicConceptsinProbabilityTheory . .............. 6 1.3.1 Fundamentaldefinitions . ....... 6 1.3.2 Bayesianstatistics . ......... 7 1.3.3 Randomvariablesandtheiraverages . ........... 8 1.4 EntropyandProbability . ........... 9 1.4.1 Entropyandinformationtheory. .......... 9 1.4.2 Probability distributions from maximum entropy . ............... 11 1.4.3 Continuousprobabilitydistributions . .............. 15 1.5 GeneralAspectsofProbabilityDistributions . .................. 16 1.5.1 Discreteandcontinuousdistributions . ............. 16 1.5.2 Centrallimittheorem . ........ 18 1.5.3 Momentsandcumulants .. ..... ...... ..... ...... .... ..... 19 1.5.4 MultidimensionalGaussianintegral . ........... 20 1.6 BayesianStatisticalInference . ............... 21 1.6.1 FrequentistsandBayesians. .......... 21 i ii CONTENTS 1.6.2 UpdatingBayesianpriors. ......... 22 1.6.3 Hyperparametersandconjugatepriors . ............ 23 1.6.4 Theproblemwithpriors . ....... 25 2 Stochastic Processes 27 2.1 References...................................... ........ 27 2.2 IntroductiontoStochasticProcesses . ................ 28 2.2.1 DiffusionandBrownianmotion . ......... 28 2.2.2 Langevinequation.............................. ....... 29 2.3 DistributionsandFunctionals. .............. 32 2.3.1 Basicdefinitions ................................ ...... 32 2.3.2 CorrelationsfortheLangevinequation . ............. 35 2.3.3 GeneralODEswithrandomforcing. ......... 37 2.4 TheFokker-PlanckEquation . ........... 39 2.4.1 Basicderivation ............................... ....... 39 2.4.2 Brownianmotionredux. ....... 40 2.4.3 Ornstein-Uhlenbeckprocess . .......... 41 2.5 TheMasterEquation................................ ........ 43 2.5.1 Equilibrium distribution and detailed balance . ................ 43 2.5.2 Boltzmann’s H-theorem ................................. 44 2.5.3 FormalsolutiontotheMasterequation. ........... 45 2.6 FormalTheoryofStochasticProcesses . ............... 48 2.6.1 Markovprocesses ................................ ..... 49 2.6.2 Martingales.................................... ..... 51 2.6.3 Differential Chapman-Kolmogorov equations . ............. 52 2.6.4 Stationary Markov processes and ergodic properties . ............... 55 2.6.5 Approachtostationarysolution . ........... 56 2.7 Appendix:Nonlineardiffusion . ............ 57 2.7.1 PDEswithinfinitepropagationspeed . ......... 57 CONTENTS iii 2.7.2 Theporousmedium and p-Laplacianequations. 59 2.7.3 Illustrativesolutions . .......... 60 2.8 Appendix : Langevin equation for a particle in a harmonic well............... 62 2.9 Appendix : General Linear Autonomous Inhomogeneous ODEs............... 64 2.9.1 SolutionbyFouriertransform . .......... 64 2.9.2 HigherorderODEs ............................... ..... 66 2.9.3 Kramers-Kr¨onigrelations. ......... 69 2.10 Appendix:MethodofCharacteristics . .............. 71 2.10.1 Quasilinear partial differential equations . .................. 71 2.10.2 Example ...................................... 72 3 Stochastic Calculus 73 3.1 References...................................... ........ 73 3.2 GaussianWhiteNoise .............................. ......... 74 3.3 StochasticIntegration . ............ 75 3.3.1 Langevinequationindifferentialform . ............. 75 3.3.2 Definingthestochasticintegral . .......... 75 3.3.3 Summary of properties of the Itˆostochastic integral ................. 77 3.3.4 Fokker-Planckequation. ......... 79 3.4 StochasticDifferentialEquations . ................ 80 3.4.1 Itˆochangeofvariablesformula . ........... 80 3.4.2 Solvability by change of variables . ............ 81 3.4.3 MulticomponentSDE.............................. ..... 82 3.4.4 SDEswithgeneral α expressed as ItˆoSDEs (α = 0).................. 83 3.4.5 Change of variables in the Stratonovich case . .............. 84 3.5 Applications.................................... ......... 85 3.5.1 Ornstein-Uhlenbeckredux . ......... 85 3.5.2 Time-dependence ............................... ...... 86 3.5.3 Colorednoise .................................. ..... 86 iv CONTENTS 3.5.4 Remarksaboutfinancialmarkets . ........ 89 4 The Fokker-Planck and Master Equations 93 4.1 References...................................... ........ 93 4.2 Fokker-PlanckEquation . ........... 94 4.2.1 Forwardandbackwardtimeequations . .......... 94 4.2.2 Surfacesandboundaryconditions . .......... 94 4.2.3 One-dimensionalFokker-Planckequation . ............. 95 4.2.4 EigenfunctionexpansionsforFokker-Planck . ............... 97 4.2.5 Firstpassageproblems . ........ 101 4.2.6 Escape from a metastable potential minimum . ............ 106 4.2.7 Detailedbalance............................... ....... 108 4.2.8 MulticomponentOrnstein-Uhlenbeckprocess . ............. 110 4.2.9 Nyquist’stheorem.............................. ....... 112 4.3 MasterEquation ................................... ....... 113 4.3.1 Birth-deathprocesses . ......... 114 4.3.2 Examples:reactionkinetics. ........... 115 4.3.3 Forward and reverse equations and boundary conditions .............. 118 4.3.4 Firstpassagetimes ............................. ....... 120 4.3.5 FromMasterequationtoFokker-Planck . .......... 122 4.3.6 Extinctiontimesinbirth-deathprocesses . ............... 126 5 The Boltzmann Equation 131 5.1 References...................................... ........ 131 5.2 Equilibrium, Nonequilibrium and Local Equilibrium . .................. 132 5.3 BoltzmannTransportTheory . ........... 134 5.3.1 DerivationoftheBoltzmannequation . ........... 134 5.3.2 CollisionlessBoltzmannequation . ............ 135 5.3.3 Collisionalinvariants . .......... 137 CONTENTS v 5.3.4 Scatteringprocesses. ......... 137 5.3.5 Detailedbalance............................... ....... 139 5.3.6 Kinematicsandcrosssection. ......... 140 5.3.7 H-theorem ......................................... 141 5.4 WeaklyInhomogeneousGas . ......... 143 5.5 RelaxationTimeApproximation . ............ 145 5.5.1 Approximationofcollisionintegral . ............. 145 5.5.2 Computationofthescatteringtime . ........... 145 5.5.3 Thermalconductivity . ........ 146 5.5.4 Viscosity ..................................... 148 5.5.5 Oscillatingexternalforce . ........... 150 5.5.6 QuickandDirtyTreatmentofTransport . ........... 151 5.5.7 Thermal diffusivity, kinematic viscosity, and Prandtlnumber . 152 5.6 DiffusionandtheLorentzmodel . ............ 153 5.6.1 Failure of the relaxation time approximation . ............... 153 5.6.2 ModifiedBoltzmannequationanditssolution . .......... 154 5.7 LinearizedBoltzmannEquation . ............ 156 5.7.1 Linearizingthecollisionintegral. .............. 156 5.7.2 Linear algebraic properties of Lˆ ............................. 157 5.7.3 Steady state solution to the linearized Boltzmann equation ............. 158 5.7.4 Variationalapproach . ........ 159 5.8 TheEquationsofHydrodynamics . ........... 162 5.9 NonequilibriumQuantumTransport . ............ 163 5.9.1 Boltzmannequationforquantumsystems . ........... 163 5.9.2 TheHeatEquation............................... 167 5.9.3 CalculationofTransportCoefficients . ............ 168 5.9.4 OnsagerRelations .............................. ....... 169 5.10 Appendix : Boltzmann Equation and Collisional Invariants ................. 171 vi CONTENTS 6 Applications 175 6.1 References...................................... ........ 175 6.2 Diffusion....................................... ........ 176 6.2.1 Returnstatistics .............................. ........ 176 6.2.2 Exitproblems .................................. 178 6.2.3 Viciousrandomwalks ............................ 181 6.2.4 Reactionrateproblems . ........ 182 6.2.5 Polymers ...................................... 183 6.2.6 Surfacegrowth................................. 194 6.2.7 L´evyflights................................... 201 6.2.8 Holtsmarkdistribution . ......... 204 6.3 Aggregation..................................... ........ 206 6.3.1 Masterequationdynamics . ....... 206 6.3.2 Momentsofthemassdistribution . ......... 208 6.3.3 Constantkernelmodel . ....... 208 6.3.4 Aggregationwithsourceterms . ......... 212 6.3.5 Gelation...................................... 214 Chapter 1 Fundamentals of Probability 1.1 References – C. Gardiner, Stochastic Methods (4th edition, Springer-Verlag, 2010) Very clear and complete text on stochastic methods with many applications. – J. M. Bernardo and A. F. M. Smith, Bayesian Theory (Wiley, 2000) A thorough textbook on Bayesian methods. – D. Williams, Weighing the Odds: A Course in Probability and Statistics (Cambridge, 2001) A good overall statistics textbook, according to a mathematician colleague. – E. T. Jaynes, Probability Theory (Cambridge, 2007) An extensive, descriptive, and highly opinionated presentation, with a strongly Bayesian ap- proach. – A. N. Kolmogorov, Foundations of the Theory of Probability (Chelsea, 1956) The Urtext of mathematical probability theory. 1 2 CHAPTER 1. FUNDAMENTALS OF PROBABILITY 1.2 Statistical Properties of Random Walks 1.2.1 One-dimensional random walk Consider the mechanical system depicted in Fig. 1.1, a version of which is often sold in novelty shops.
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