Diffusive and Stochastic Processes

Skills
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.

Knowledge
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.

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.