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NFYK10006U  Diffusive and Stochastic Processes Volume 2015/2016

Course information

Credit7,5 ECTS
LevelFull Degree Master
Duration1 block
Block 4
Course capacityNo restriction to number of participants
Continuing and further education
Study boardStudy Board of Physics, Chemistry and Nanoscience
Contracting department
  • The Niels Bohr Institute
Course responsible
  • Namiko Mitarai (7-716d786576656d4472666d326f7932686f)
Saved on the 10-09-2015

Master programme in Physics


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

Learning Outcome

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

  • Describe diffusion process using random walk, Langevin equation, and Fokker-Plank equation.
  • Explain the first passage time and Kramers escape problem
  • Explain the fluctuation-dissipation theorem.
  • Explain basic concepts in stochastic integrals.
  • 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 Kramas-Moyal expansion and the linear noise approximation.
  •  Explain asymmetric simple exclusion process and some related models to describe traffic flow and jamming transition in one-dimensional flows.
  • Apply the concepts and techniques to various examples from non-equilibrium complex phenomena.

In this course, first 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 random walks, Langevin equations, Fokker-Planck equations, Kramars escape, and the fluctuation-dissipation theorem. Then selected stochastic models that have wide applications to various real phenomena are introduced and analysed. The topics are chosen from non-equilibrium stochastic phenomena, including birth and death process and Master equation, and asymmetric simple exclusion process. 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.

Teaching and learning methods
Lectures and exercise sessions. Computer exercise included.
Academic qualifications
Equilibrium statistical physics, physics bachelor level mathematics (Especially: differential and integral calculus, differential equations, Taylor expansions).
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Credit7,5 ECTS
Type of assessment
Oral examination, 30 min
No preparation time
AidWithout aids
Marking scale7-point grading scale
Censorship formNo external censorship
Several internal examiners
Criteria for exam assesment

See Skills.

Theory exercises35
Total 206,0
Saved on the 10-09-2015