Diffusive and Stochastic Processes  PhD course
PhD School at the Faculty of SCIENCE at University of Copenhagen
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 nonequilibrium complex phenomena.
Formel requirements
For students outside of Denmark:
To apply for participation in this course, it is required that you send an email to the course organizer with your information and motivation for joining the course. Do not use the online application. Thank you.
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 FokkerPlank equation.
 Explain the first passage time and Kramers escape problem
 Explain the fluctuationdissipation 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 FokkerPlank equations using KramasMoyal expansion and the linear noise approximation.
 Explain asymmetric simple exclusion process and some related models to describe traffic flow and jamming transition in onedimensional flows.
 Apply the concepts and techniques to various examples from nonequilibrium 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, FokkerPlanck equations, Kramars escape, and the fluctuationdissipation theorem. Then selected stochastic models that have wide applications to various real phenomena are introduced and analysed. The topics are chosen from nonequilibrium 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 nonequilibrium statistical physics, biophysics, softmatter 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.
Remarks
Academic qualifications:
Equilibrium statistical physics, physics bachelor level mathematics (Especially: differential and integral calculus, differential equations, Taylor expansions).
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