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.