Stochastic Processes
A.Y. 2024/2025
Learning objectives
The course will teach the methods of non-equilibrium statistical mechanics, like Markov chains, stochastic equations, Moyal expansion, Fokker-Planck equation, Mori-Zwanzig dimensional reduction.
Expected learning outcomes
The student will know the theory and the methods of non-equilibrium statistical mechanics, which lie at the basis of physical modelling of kinetic phenomena and of several tools of data analysis.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
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.
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: Mori-Zwanzig equation.
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.
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: Mori-Zwanzig equation.
Prerequisites for admission
Basic statistical mechanics, mathematical analysis, linear algebra and operators.
Teaching methods
Lecture in class
Teaching Resources
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
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.
FIS/03 - PHYSICS OF MATTER - University credits: 3
FIS/04 - NUCLEAR AND SUBNUCLEAR PHYSICS - University credits: 3
FIS/04 - NUCLEAR AND SUBNUCLEAR PHYSICS - University credits: 3
Lessons: 42 hours
Professor:
Tiana Guido
Professor(s)