Theory of Stochastic Processes

The module develops the theory of stochastic processes and the methods for their analysis. Examples to practice the acquired methods will be chosen primarily from the area of biophysics, but the major teaching and learning content (see below) is relevant for all areas of physics.

Table of Content:

• Probability Theory
• Stochastic Processes
• Master Equation
• Stochastic Thermodynamics
• Observables on Trajectories
• Focker-Planck Equations
• Langevin Equations
• Approximation Methods and Limits

At the end of the module students know the basic methods to handle physical systems that can be described by stochastic processes as well as the basic assumptions necessary to apply them. They are able to

1. set up and solve Master equations, stochastic differential equations and Fokker Planck equations and know simple simulation methods.
2. understand and apply the basic principles of stochastic thermodynamics.
3. apply and adapt approximation methods for the analysis of complex stochastic processes

A solid basis in statistical physics (e.g. PH0008) is assumed.

The modul consists of a lecture and exercise classes.

In the thematically structured lecture the theoretical contents are presented and discussed. The theoretical models for the description of stochastic processes are developed on the blackboard together with the students. Concrete physical examples are studied in-depth and compared to experimental results.

In the exercises additional understanding questions are answered together with the students. Furthermore, specific topics of the lecture are discussed in-depth and relevant aspects are reviewed in regular intervals. Questions of students are given a large space.

In the homeworks the students have the opportunity to apply the presented techniques to concrete problem examples and to analyze the results. In the course of this analytic calculation exercises, simple numerical simulations and conceptual questions with answers in the form of continuous text are chosen as task form. The solution proposals of the students are corrected to give the students feedback for their modeling and solution abilities and to detect and correct misconceptions as early as possible. 

Lecture notes, problem sheets (containing homework problems and questions for exercises), web page

• Crispin Gardiner: "Stochastic Methods: A Handbook for the Natural and Social Sciences" (Springer)
• N.G. van Kampen: "Stochastic Processes in Physics and Chemistry" (North-Holland)

There will be a written exam of 90 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.

For example an assignment in the exam might be:

• Modelling of a concrete example with a Master equation and solving the equation
• Approximation of a given Master equation through a Fokker-Planck equation with the Kramers-Moyal or Van Kampen Expansion, Solution of the Fokker-Planck equation
• Transformation of a Langevin Equation into a Fokker-Planck equation
• Calculating the entropy production of a system obeying a Master equation

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.

There will be a bonus (one intermediate stepping of "0,3" to the better grade) on passed module exams (4,3 is not upgraded to 4,0). The bonus is applicable to the exam period directly following the lecture period (not to the exam repetition) and subject to the condition that the student passes the mid-term of At least 70% of the problems in the homeworks and the mock exam need to be worked on. A problem counts as worked on if at least an ansatz leading to the solution is found.