1) Stochastic processes An example: discrete-time random walk Stochastic variables (a review) Stochastic processes: definitions, moments, stationarity, spectral density, characteristic functionals Examples: Galton branching process, Ornstein-Uhlenbeck process. Markov processes. Definition and properties. Random walk with persistence. Chapman-Kolmogorov equation. Wiener and Poisson processes. Markov chains. Example: nuclear decay. The master equation: derivation. Stochastic matrices and their properties. Matrices with specific properties. Example: pion decay. Detailed balance. Existence and unicity of stationary solution. Detailed balance in Hamiltonian systems. Macroscopic equations. Expansion of the master equation in eigenfunction. The adjoint equation. Stochastic monitoring. Death-n-birth processes. The problem of boundaries. Examples: chemical reactions. The expansion of the master equation. Markov processes of diffusion type. Problemi di epidemiologia. Gillelspie algorithm and Finite State Projection
2) The Langevin approach The heat bath of harmonic oscillators (the road to the Langevin equation) Langevin equation Stochastic differential equations Introduction to Ito and Stratonovich calculus Linear response theory Fluctuation-dissipation relations in equilibrium dynamics
3) The Fokker-Planck equation Derivation of the equation. Kramers-Moyal expansion. The backward equation. Pawula theorem. Path-integral formulation. Examples: Brownian motion. Methods of solution. Example: bistable potential. Dimensional reduction.
Prerequisites for admission
Basic statistical mechanics.
Lecture in class
Van Kampen, Stochastic Processes in Physics and Chemistry, North Holland. H. Risken, The Fokker-Planck equation, Springer R. Zwanzig, Nonequilibrium statistical mechanics, Oxford University Press
Assessment methods and Criteria
Oral colloquium of approximately 30 minutes to verify the understanding of the subjects treated in the course.