Diffusive and Stochastic Processes

Course content

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.

Education

Master programme in Physics

Learning outcome

Skills
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.

Knowledge
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.

Competences
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).

ECTS
7,5 ECTS
Type of assessment
Oral examination, 30 min
No preparation time
Aid
Without aids
Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners
Criteria for exam assessment

See Skills.

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 24
  • Theory exercises
  • 35
  • Exam
  • 0,5
  • Preparation
  • 146,5
  • English
  • 206,0