Dr. Luca Donati
Zuse Insitute Berlin
Personal Information
- Department: Modeling and Simulation of Complex Processes
- Research group: Computational Molecular Design
- Email: donati /at\ zib.de
- Office: 4033 (ZIB)
- Publications
- Past courses
- Stochastics IV WS2224
- Small Data Analysis 2024
- Student research group 2023
- Stochastics IV WS2223
Stochastics IV WS2324
- Topic Stochastic and Diffusive Processes
- Location Room 046 at Takustr. 9 (FU Berlin) on Tuesday from 12 to 14;
- Room 009 at Arnimallee. 6 (FU Berlin) on Tuesday from 14 to 16.
- The course will start on 17 October.
- Whiteboard page
- Github repository which collects the jupyter notebooks and the notes presented in class. The notebooks can be opened and used in the browser by means of binder (no need to download/install anything).
-
Content Stochastic processes are mathematical models used to describe the dynamics of random phenomena and are widely applied in many disciplines ranging from physics, chemistry, biology, and economics. During the course, students will learn both the theory underlying stochastic processes and advanced numerical methods to solve problems with real applications. - The course is highly interdisciplinary and students from physics, chemistry, and mathematics are encouraged to participate.
- Topics for the exam.
- Overview
- Lecture notes 12
- Lecture 12, jupyter notebooks Suggested readings
- Deuflard P. and Weber M., 2004
- Lecture notes 11
- Lecture 11, jupyter notebooks Suggested readings
- Donati L., 2018
- Donati L., 2021
- Lecture notes 10
- Lecture 10, jupyter notebooks Suggested readings
- Klus S. 1918
- Lecture notes 9 (Work in progress...)
- Lecture 9, jupyter notebooks Suggested readings
- Baron P., Reaction Rate Theory and Rare Events Simulations, chapter 16
- Kramers H.A. 1940
- Lecture notes 8 (Work in progress...)
- Lecture 8, jupyter notebooks
- Exercise 8 Suggested readings
- Baron P., Reaction Rate Theory and Rare Events Simulations, chapter 16
- Kramers H.A. 1940
- Lecture notes 7
- Lecture 7, jupyter notebooks
- Exercise 7
- Exercise 7, jupyter notebooks Suggested readings
- Baron P., Reaction Rate Theory and Rare Events Simulations, chapter 18
- Lecture notes 6
- Lecture 6, jupyter notebooks Suggested readings
- Kubo R. 1966
- Van Kampen N.G., The Expansion of the Master Equation 1976, chapters 1-5
- Lecture notes 5
- Notes Hamiltonian Dynamics
- Notes Statistical Mechanics
- Notes integrators
- Lecture 5, jupyter notebooks Suggested readings
- Kupferman R. 2002
- Lecture notes 4a
- Lecture notes 4b
- Lecture 4, jupyter notebooks
- Exercise 4
- Exercise 4, jupyter notebooks Suggested readings
- Tuckerman M.E., Statistical Mechanincs: Theory and Molecular Simulation 2010, Chapter 15
- Gillespie D.T., Markov Processes 1992, Chapter 5
- Shorack G.R., Probability for statisticians 2000, Chapter 7
- Lecture notes 3
- Exercise 3
- Exercise 3, jupyter notebooks Suggested readings
- Gardiner W., Handbook of Stochastic Methods 1994, chapter 3
- Pawula R.,1967
- Lecture notes 2
- Exercise 2
- Exercise 2, jupyter notebooks Suggested readings
- Gardiner W., Handbook of Stochastic Methods 1994, chapter 2
- Introduction to the course
- Lecture notes 1
- Exercise 1
- Exercise 1, jupyter notebooks Suggested readings
- Gardiner W., Handbook of Stochastic Methods 1994, chapter 1
- Brown R. 1828
- Einstein A. 1905
- Langevin P. 1908
- Simulation of Brownian motion (video)
- Figure.
(a) Brownian motion; (b) Solution of the SIR model generated by Gillespie algorithm; (c) Solution of the Fokker-Planck equation generated by SqRA [ref. 2].
- References
- L. Donati, M. Weber. and B. G. Keller In: J. Math. Phys. 63 (2022), p. 123306, DOI.
Lecture 13: Conclusion and overview of the course
Lecture 12: Fuzzy clustering and PCCA+
Lecture 11: Square Root Approximation (SqRA) of the infinitesimal generator
Lecture 10: Transfer operator formalism
Lecture 9: Kramers rate theory for low friction regime
Lecture 8: Kramers rate theory for moderate and high friction regime
Lecture 7: Introduction to escape rate problem, backward Kolmogorov equation, Mean First Passage Time, Pontryagin's formula, Ornstein_Uhlenbeck process, integration schemes for SDEs
Lecture 6: Fluctuation-Dissipation Theorem; overview of Fourier analysis; from Generalized Langevin Equation to Langevin Dynamics; introduction to Stochastic Calculus; System Size expansion method for Master equations
Lecture 5: Overview of Hamiltonian dynamics and Statistical Mechanics; The Generalized Langevin Equation: the memory kernel and the noise term
Lecture 4: Derivation of the Generalized Langevin Equation from the Kac-Zwanzig model (4a); method of generating function and Gillespie's algorithm to solve the master equation (4b)
Lecture 3: Markov processes, derivation of Chapman-Kolmogorv equation, Kramers-Moyal expansion, master equation, Fokker-Planck equation, Pawula theorem
Lecture 2: Overview of probability theory and statistics
Lecture 1: Brownian motion, Einstein's theory, Langevin's theory