Diffusive and Stochastic Processes

Course content

Stochastic descriptions offer powerful ways to understand fluctuating and noisy phenomena, and are widely used in many disciplines including physics, chemistry, biology, and economics. In this course, basic analytical and numerical tools to analyze stochastic phenomena are introduced and will be demonstrated on several important examples. Students will learn to master stochastic descriptions for analyzing non-equilibrium complex phenomena.


MSc Programme in Physics
MSc Programme in Physics with a minor subject

Learning outcome

At the conclusion of the course students are expected to be able to:

  • Describe diffusion process using Langevin equation and Fokker-Plank equation.
  • Solve  several examples of the first passage time problems.
  • Explain basic concepts in stochastic integrals, and use it to describe geometric Brownian motions.
  • Explain the Poisson process and the birth and death process. Use master equations to describe time evolution and steady state of the processes.
  • Explain the relationship between master equations and Fokker-Plank equations using approximation methods such as Kramers-Moyal expansions.
  • Explain asymmetric simple exclusion process and related models to describe traffic flow and jamming transition in one-dimensional flows.
  • Apply the concepts and techniques to various examples of stochastic phenomena.


In this course, the basic tools to analyse stochastic phenomena are introduced by using the diffusion process as one of the most useful examples of stochastic process. The topics include Langevin equations, Fokker-Planck equations, first passage problems, and master equations. The tools are then used to analyze selected stochastic models that have wide applications to various real phenomena. The topics are chosen from non-equilibrium stochastic phenomena, including geometric Brownian motion (used in e.g. modeling finance), birth and death process (used in e.g. chemical reactions and population dynamics), and asymmetric simple exclusion process (used in e.g. traffic jam formation). Throughout the course, exercises for analytical calculations and numerical simulations are provided to improve the students' skills.

This course will provide the students with mathematical tools that have application in range of fields within and beyond physics. Examples of the fields include non-equilibrium statistical physics, biophysics, soft-matter physics, complex systems, econophysics, social physics, chemistry, molecular biology, ecology, etc. This course will provide the students with a competent background for further studies within the research field, i.e. a M.Sc. project.

Lectures and exercise sessions. Computer exercise included.

Equilibrium statistical physics, physics bachelor level mathematics (Especially: differential and integral calculus, differential equations, Taylor expansions). Basic programming skills.

Academic qualifications equivalent to a BSc degree is recommended.

It is expected that the student brings a laptop with a programming environment (e.g. Python) installed.


Written feedback to assignments

7,5 ECTS
Type of assessment
Written assignment, 8 days
Oral examination, 25 minutes with no preparation time
Type of assessment details
20% of the grade from a programming assignment given at least two weeks before the final exam.
80% of the grade from a 25-minute oral examination with no aids allowed and no preparation time.

The two parts of the exam do not need to be passed separately.
Only certain aids allowed

All aids allowed for the written assignment.

No aids allowed for the oral examination.

Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners

Same as ordinary exam.

It is possible to arrange a new programming assignment (20% of the grade) two weeks before the re-exam date. Please contact the course responsible to arrange this.

Criteria for exam assessment

See Learning outcome

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 24
  • Preparation
  • 142
  • Theory exercises
  • 35
  • Exam
  • 5
  • English
  • 206


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

Block 4
The number of places might be reduced if you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
Study Board of Physics, Chemistry and Nanoscience
Contracting department
  • The Niels Bohr Institute
Contracting faculty
  • Faculty of Science
Course Coordinator
  • Namiko Mitarai   (7-736f7a6778676f4674686f34717b346a71)
Saved on the 15-02-2024

Are you BA- or KA-student?

Are you bachelor- or kandidat-student, then find the course in the course catalog for students:

Courseinformation of students