Syllabus
1. Probability and Random Variables Probability paradigms and paradoxes. Random variables. Their role in physics. Distribution function, characteristic function. Bernoulli, Gauss and Poisson distributions, other examples. Two random variables, joint and conditional distribution, correlations. Sequences of random variables. Limit theorems. Lévy distributions. Information entropy.
2. Stochastic Processes Discrete-time Markov chains (random walks, Ehrenfest model, branching process). Continuous-time Markov processes (Pauli master equation, Poisson process, dichotomic process, continuous-time random walks). Markov processes with continuous realizations (Wiener process, Ornstein-Uhlenbeck process). Point processes and renewal processes. Shot noises. Stationary processes, their spectra. Self-averaging. Stochastic differential equations, multiplicative and additive noise. Noise-induced transitions.
3. Diffusion Theory Thermal noise, Brownian motion. Langevin Equation and Fokker-Planck Equation. Biased diffusion. Boundaries. Dwelling time and first passage time. Methods of solution (Laplace transform, eigenfunction expansion, propagators, path integrals). Diffusion vs. trapping. Kramers equation. Fractal diffusion. Dichotomic diffusion, telegrapher's equation. Extremal properties of trajectories. Physics of random polymer chains.
4. Selected Applications in Physics Static (quenched) and dynamical disorder. Stochastic Schroedinger (Liouville) equation, optical Bloch equation, absorption. Harmonic oscillator with randomly modulated frequency, motional narrowing. Laser equation. Statistical properties of light. Percolation. Stochasticity versus chaotic behaviour of deterministic systems. Random-growth models. Molecular motors. Stochastic resonance.
5. Selected Applications in Other Fields Growth of populations (predator-prey systems, Verhulst model). Genetic models. Chemical systems (diffusion-controled reactions, coagulation). Migration of population. Spreading of diseases. Queueing models (systems of service). Hazardous plays. Assurance models.
Annotation
The lecture starts with an accesible introductory formulation of the probability theory as suited for the students of physics. New consepts are demonstrated on a selected set of examples from physics. Subsequently, advanced physical problems are treated using Markov chains (Ehrenfest model, branching process), continuous-time Markov processes
(Poisson process, random walks, dichotomic process), Markov processes with continuous realizations (Wiener process,
Ornstein-Uhlenbeck process), point processes and renewal processes, Langevin Equation and Fokker-Planck Equation,
Brownian motion, path-integral description of stochastic processes.