FYSS7435 Stochastic Processes in Physics (5 cr)

Open the course unit brochure on Sisu
Study level:
Advanced studies
Grading scale:
0-5
Language:
English, Finnish
Responsible organisation:
Department of Physics
Curriculum periods:
2024-2025, 2025-2026, 2026-2027, 2027-2028

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

Basic skills in probability theory, basic courses in mechanics and electromagnetism 

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

Description:
Timing as agreed with teacher. In this completion method the student solves in total 50 exercises from 10 selected chapters of van Kampen's book, and presents the motivation of the problems along with their solutions to the course organiser in three meetings.
Evaluation criteria:
In the one-on-one meeting the teacher chooses 5 exercises, which the student teaches to the teacher. In each case the student starts by explaining the main point of the chapter, motivating the exercise problem, and then solving it on the whiteboard. The evaluation is based on how well the teacher understood the motivation and the solution: 3 p: well explained and understood, 2p: some points left open, or required the teacher’s relevant help, 1p: explanation with clear gaps, 0p: no proper understanding of what was presented or needed in the problem. Total number of points 3x5x3=45 p. Grade 1 requires 25 p, 2: 29 p, 3: 33 p, 4: 37 p, 5 40 p.
Select all marked parts

Method 2

Description:
Timing as agreed with teacher. In this method the student presents three lectures on methods dealing with stochastic processes. The first lecture is on classical stochastic processes, the second on quantum stochastic processes, and the third is on a small project where the student studies a mutually agreed theme.
Evaluation criteria:
After each lecture, a self-evaluation with grade 0-3 p. After each lecture, the teacher prepares a short evaluation report, and evaluates the lecture with grading 0-3 p (as in method 1). Total points (max 18 p) yield the course grade: 9 p – 1, 11 p – 2, 13 p – 3, 15 p – 4, 17 p – 5.
Select all marked parts
Parts of the completion methods
x

Independent study (5 cr)

Type:
Independent study
Grading scale:
0-5
Evaluation criteria:
In the one-on-one meeting the teacher chooses 5 exercises, which the student teaches to the teacher. In each case the student starts by explaining the main point of the chapter, motivating the exercise problem, and then solving it on the whiteboard. The evaluation is based on how well the teacher understood the motivation and the solution: 3 p: well explained and understood, 2p: some points left open, or required the teacher’s relevant help, 1p: explanation with clear gaps, 0p: no proper understanding of what was presented or needed in the problem. Total number of points 3x5x3=45 p. Grade 1 requires 25 p, 2: 29 p, 3: 33 p, 4: 37 p, 5 40 p.
Language:
English, Finnish
Study methods:

  • 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

x

Independent study (5 cr)

Type:
Independent study
Grading scale:
0-5
Evaluation criteria:
After each lecture, a self-evaluation with grade 0-3 p. After each lecture, the teacher prepares a short evaluation report, and evaluates the lecture with grading 0-3 p (as in method 1). Total points (max 18 p) yield the course grade: 9 p – 1, 11 p – 2, 13 p – 3, 15 p – 4, 17 p – 5.
Language:
English, Finnish
Study methods:

  • 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. 

Teaching