FYSS7435 Stochastic Processes in Physics (5 cr)
Description
Mostly themes selected by the student, but primarily
Describing stochasticity: probability distributions
Random events
Stochastic processes and their description
Markov processes
Langevin equation, noise
Master equation
Fokker-Planck equation
Possibly methods to describe open quantum systems: quantum Langevin equation along with the input/output formalism, density matrix and the quantum master equation (Lindblad equation), quantum jump method
Learning outcomes
At the end of the course, the students will be able to
Explain the occurrence of randomness in physical processes and their measurements
Explain the starting point of describing stochastic processes, and some of the ways to describe them
Explain and use some of the main methods for treating stochastic processes: master equation, Langevin equation, Fokker-Planck equation
Explain some properties or approximations related to stochastic processes: Markovianity, stationarity, one-step processes, macroscopic equation
Evaluate their own understanding (and possible gaps in it) related to the studied topic
Additional information
Given on autumn semester, every year.
Description of prerequisites
Literature
- van Kampen: "Stochastic Processes in Physics and Chemistry"
- Gardiner & Zoller, A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics
Completion methods
Method 1
Method 2
Independent study (5 cr)
Student chooses 10 chapters from van Kampen’s book (or equivalent material) and solves 5 problems from each chapter. Then they return the written solutions to the teacher, and on a designated time teaches to the teacher the main points of the chapters and along with the few selected exercises.
Four meetings: In the beginning of the course (in common, if many students) a short meeting outlining the basic principles. Then three 2-hour meetings one-on-one with the teacher going through the chapters. The first meeting involves 4 chapters, then 3 chapters for the second and third meeting.
Teaching
9/2/2024–7/31/2025 Independent project
Independent study (5 cr)
Self-study aiming towards three one-on-one meetings with the teacher.
1st Meeting: Student prepares and presents to the teacher a 2-hour lecture on classical stochastic methods (on a whiteboard). The lecture contains at least (a) explaining the overall stochastic processes, (b) Langevin equation, (c) master equation, (d) Fokker-Planck equation, and (e) Markov approximation. Lecture includes at least two calculational examples of these themes.
2nd Meeting: Student prepares and presents to the teacher a 2-hour whiteboard lecture on stochastic methods in open quantum systems explaining at least the following concepts: (i) general formulation of the problem, (ii) density matrix and quantum master equation, (iii) input/output relation and quantum Langevin equation, (iv) quantum jump method. The lecture includes at least two calculational examples of these themes. The examples can be agreed with the lecturer beforehand.
3rd meeting: Small project using the above methods. 5-10 page notes, lecturing the theme to the teacher.