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1 Table of Contents [ntc] 1. Introduction: Contents and Maps Table of contents [ntc] Equilibrium thermodynamics overview [nln6] Thermal equilibrium and nonequilibrium [nln1] Levels of description in statistical physics [nln2] Contraction - memory - time scales [nln15] Markov process: map of specifications [nln16] Brownian motion: panoramic view [nln23] Linear response and equilibrium dynamics [nln24] 2. Probability: Intuition - Ambiguity - Absurdity - Puzzles Two bus companies: regular versus random schedules Pick the winning die [nex2] Educated guess [nex4] Coincident birthdays [nex82] Win the new car or take the goat! [nex11] Three-cornered duel [nex13] Bad luck: waiting for the worst [nex74] Bertrand s paradox Random quadratic equations [nex12] Crossing a river [nex84] Combinatorics of poker hands [nex124] Know your odds [nex125] 3. Elements of Probability Theory with Applications Elements of set theory [nln4] Set identities [nex88]
2 Sample space and events Probability axioms and simple theorems [nex94] Joint probability and conditional probability [nex90] Symmetry and elementary events Bayes theorem Statistical independence Statistical uncertainty and information [nln5], [tex47] Event or complement? That is the question [nex9] Successive random picks [nex91] Heads or tails [nex93] Quantity and quality [nex76] Diagnosis of a rare disease [nex77] Subtlety of statistical independence [nex1] Random train connections [nex92] Random inkjet printer [nex10] Information and the reduction of ignorance [tex48] Information of sequenced messages [tex61] 4. Random Variables: Concepts Probability distributions Moments, variance, standard deviation Moment expansion and characteristic function Cumulant expansion Factorial moments and cumulants, generating function Multivariate distributions [nln7] Transformation of random variables Propagation of statistical uncertainty [nex24] Chebyshev s inequality [nex6] Law of large numbers [nex7]
3 Binomial, Poisson, and Gaussian distribution [nln8] Binomial to Poisson distribution [nex15] De Moivre - Laplace limit theorem [nex21] Central limit theorem [nln9] Multivariate Gaussian distribution Robust probability distributions [nex19] Stable probability distributions [nex81] Exponential distribution [nln10] Waiting time problem [nln11] Pascal distribution [nex22] 5. Random Variables: Applications Reconstructing probability distributions [nex14] Probability distribution with no mean value [nex95] Variances and covariances [nex20] Statistically independent or merely uncorrelated? [nex23] Sum and product of uniform distribution [nex96] Exponential integral distribution [nex79] Generating exponential and Lorentzian random numbers [nex80] Random chords (Bertrand s paradox) [nex5] From Gaussian to exponential distribution [nex8] Transforming a pair of random variables [nex78] Gaussian shootist versus Lorentzian shootist [nex3] Moments and cumulants of the Poisson distribution [nex16] Maxwell velocity distribution [nex17] Random bus schedules [nex18] Life expectancy of the young and the old [nex106] Life expectancy of the ever young [nex38] Random frequency oscillator [nex35]
4 6. Stochastic Processes: Concepts Time-dependent probability distributions Correlation functions and characteristic functions Equilibrium - nonequilibrium - stationarity Classification of processes (factorizing/markov/non-markov) Deterministic versus stochastic time evolution Contraction - memory - time scales [nln15] General specification of Markov process Chapman-Kolmogorov equation Diffusion process and Cauchy process Stationarity, normalization, consistency, Markovian nature [nex26] Computer generated sample paths [nsl1] Continuous versus discontinuous processes (Lindeberg condition) [nex97] Differential Chapman-Kolmogorov equation Fokker-Planck equation (drift and diffusion processes) Drift equation (deterministic processes) [nex29] Master equation (jump processes) [nex28] Non-differentiability of sample paths [nex99] Master equation with finite jump moments [nex32] Equations of motion for mean and variance [nex30] Markov process: map of specifications [nln16] Approach to a stationary state (detailed balance) [nex85] Markov chains (discrete variables, discrete time) Transition matrix, left and right eigenvectors, stationary states Regularity, ergodicity, detailed balance, absorbing states Master equation with detailed balance (discrete variables, continuous time) [nln12]
5 Regression theorem for autocorrelation functions [nex39] Birth death processes (specifications, models, levels of description) [nln18] Birth and death of single species [nln19] Birth-death master equation: stationary state [nln17] Nonlinear birth-death process 7. Stochastic Processes: Applications Diffusion process [nex27] Cauchy process [nex98] Random walk in one dimension: unit steps at unit times [nex34] Random walk in one dimension: unit steps at random times [nex33] Random walk in one dimension: [nex100] tiny steps at frequent times Random walk in Las Vegas: chance and necessity [nex40] Poisson process [nex25] Free particle with uncertain position and velocity [nex36] Fokker-Planck equation with constant coefficients [nex101] House of the mouse: two-way doors only [nex102] House of the mouse: some one-way doors [nex103] House of the mouse: one-way doors only [nex104] House of the mouse: mouse with inertia [nex105] House of the mouse: mouse with memory [nex43] Mixing marbles red and white [nex42] Random traffic around city block [nex86] Modeling a Markov chain [nex87] Ornstein-Uhlenbeck process [nex31] [nex41] Predator-prey system: deterministic, stochastic, observational [nsl3] Populations with linear birth and death rates I [nex44]
6 Populations with linear birth and death rates II [nex112] Catalyst-driven chemical reaction: stationary state [nex46] Catalyst driven chemical reaction: dynamics [nex107] Catalyst driven chemical reaction: total rate of reactions [nex108] Air in leaky tank I: generating function [nex48] Air in leaky tank II: probability distribution [nex109] Air in leaky tank III: detailed balance [nex49] Air in leaky tank IV: evolution of mean and variance [nex110] Pascal distribution and Planck radiation law [nex50] Effects of nonlinear death rate I: Malthus-Verhulst equation [nex111] Effects of nonlinear death rate II: stationarity and fluctuations [nex51] Modified linear birth rate I: stationarity [nex113] Modified linear birth rate II: evolution of mean and variance [nex114] Modified linear birth rate III: generating function [nex115] Modified linear birth rate IV: probability distribution [nex116] Bistable chemical system [nex52] Ultracold neutrons in an ideal Steyerl bottle [nex47] Random light switch [nex45] 8. Brownian Motion Relevant time scales (collisions, relaxation, observations) Einstein s theory Smoluchovski equation with link to Fokker-Planck equation Einstein relation (example of fluctuation-dissipation relation) Fick s law for particle current Fourier s law for heat current Thermal diffusivity [nex117] Shot noise (e.g. electric current in vacuum tube)
7 Campbell s theorem [nex37] Critically damped ballistic galvanometer [nex70] Langevin s theory (on most contracted level of description) White noise Brownian motion and Gaussian white noise [nln20] Wiener process [nsl4] Autocorrelation function of Wiener process [nex54] Ballistic and diffusive regimes of Langevin solution Langevin s equation: attenuation without memory [nln21] Formal solution of Langevin equation [nex53] Velocity correlation function of Brownian particle I [nex55] Mean-square displacement of Brownian particle [nex56], [nex57], [nex118] Ergodicity [nln13] Intensity spectrum and spectral density (Wiener-Khintchine theorem) [nln14] Fourier analysis of Langevin equation Velocity correlation function of Bownian particle II [nex119] Generalized Langevin equation: attenuation with memory [nln22] Fluctuation-dissipation theorem Velocity correlation function of Brownian particle III [nex120] Brownian harmonic oscillator I: Fourier analysis [nex121] Brownian harmonic oscillator II: position correlation function [nex122] Brownian harmonic oscillator III: contour integrals [nex123] Brownian harmonic oscillator IV: velocity correlations [nex58] Brownian harmonic oscillator V: formal solution for velocity [nex59] Brownian harmonic oscillator VI: nonequilibrium correlations [nex60] Generalized Langevin equation inferred from microscopic dynamics
8 Brownian motion: levels of contraction and modes of description [nln23] 9. Linear Response and Equilibrium Dynamics Overview [nln24] Many-body system perturbed by radiation field [nln25] Response function and generalized susceptibility [nln26] Kubo formula for response function [nln27] Symmetry properties [nln30] Kramers-Kronig dispersion relations [nln37] Causality property of response function [nex63] Energy transfer between system and radiation field [nln38] Reactive and absorptive part of response function [nex64] Fluctuation-dissipation theorem (quantum and classical) [nln39] Spectral representations [nex65] Linear response of classical realxator [nex66] Linear response of classical oscillator [nex67] 10. Zwanzig-Mori Formalism Introduction [nln28] Time-dependence of expectation values (quantum and classical) Zwanzig s kinetic equation: generalized master equation [nln29] [nex68] Projection operator method (Mori formalism) [nln31] Kubo inner product [nln32] Projection operators [nln33] First and second projections [nln34] [nln35] Continued-fraction representation of relaxation function [nln36] Recursion method (algorithmic implementation of Mori formalism)
9 Relaxation function with uniform continued-fraction coefficients [nex69] Continued-fraction expansion and moment expansion Generalized Langevin equation n-pole approximation Green s function formalism Structure function of harmonic oscillator [nex71], [nex72], [nex73] Scattering process and dynamic structure factor Electron scattering, neutron scattering, light scattering Scattering from a free atom Scattering from an atom bound in a harmonic potential Scattering from a harmonic crystal
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Contents of this Document [ntc5] 5. Random Variables: Applications Reconstructing probability distributions [nex14] Probability distribution with no mean value [nex95] Variances and covariances [nex20]
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