!DOCTYPE html> JV: Stochastic Processes

SoSe2020. Stochastic Processes


2020 04 12      I provide now a first draft of a skript with exercises in Sage
that are packed with the skript in a gzip-compressed tar bundle.
2020 04 07      Rather than with lectures we will start the course on Stochastic Processes by a software project
where we develop a suite of numerical tool to simulate stochastic processes and display the results.
This page is the opening call for the project. Please contact me if you are interested to participate.


Probabilities.      distributions, moments, cumulants
characteristic functions, Laplace-Legendre transformations
Random Processes. random walks, Lévy walks, anomalous transport
central limit theorem, Lévy distribution, large deviation theory
Markov processes. master equation, Kramers-Moyal expansion, Fokker-Planck equation
ergodicity, transport theory, first passage problems
Langevin dynamics. stochastic differential equations, Stratonovic and Ito calculus
Kramers problem, reactions coordinates, path of least action
stochastic thermodynamics, fluctuation relations, linear-response theory
Stochastic Thermodynamics.     fluctuation relations, thermodynamic efficiency, linear-response theory

Background and Supporting Material for Lectures

I will use Python and Sage for numerical studies and visualization. The script files are notebooks that run interactively in Jupyter. Please store them on your computer, start Jupyter, and then open the notebook. The files contain fairly minimalistic explanations and comments. Feel free to add more comments and explanations, and return them to me.

As introductions to Python and Sage I recommend the books Steward: "Python for Scientists" and Zimmermann u.a.: "Computational Mathematics with SageMath".

I am working on a skript with more information on the course. The PDF-version of the first draft can be downloaded here. A gzip-compressed tar bundle that contains the PDF-file and Sage-Skripts with further illustrations and first homework assignments is available here.


Set    Topic        Assignment       

00. Foundations study Sect. 1–2 of J. L. Garcia-Palacios: Intro to the theory of stoch processes and Brownian motion, arXiv:cond-mat/0701242 »
01. Probabilities study Chapter 1 of my lecture notes and complete the open tasks in the Sage-files provided in this bundle.
02. Probability Distributions study Chapter 2 of Haken’s book and complete the tasks of exercise set 2.
03. Markov Processes 1 study Chapter 4 of Haken’s book and complete the tasks of exercise set 3.
04. Markov Processes 2 study the paper Wachtel et al: Fluctuating currents in stochastic thermodynamics. I, Phys. Rev. E 92, 042132 (2015) and complete the tasks of exercise set 4.
05. Brownian Motion, Noise,
and Fluctuations
consult Chapter 4 of Haken’s book,
study the paper Lebowitz and Spohn, J. Stat. Phys. 96 (1999) 333, and complete the tasks of exercise set 5.


Course.      after lectures are resumed
          Vollmer          Wed 11:15 – 12:45      ITP, Brüderstr    SR 114
          Thu 13:30 – 15:00      ITP, Brüderstr    SR 114
Exercise Sessions. Vollmer      by appointment


There will be oral exams at the earliest opportunity after the end of the term.


     H. Haken      Synergetics. An Introduction      Springer 1983.
     an excellent introduction to key ideas from a physics perspective
     Feller An Introduction to Probability Theory and its Applications, Vol 1 Wiley 1968.
     an excellent introduction to key ideas from a mathematics perspective
     van Kampen Stochastic Processes in Physics and Chemistry Elsevier 1981
     a rich source for applications in physics, in particular of Master equations
specialized sources worth considering:
     Risken The Fokker-Planck Equation Springer 1989
     Gardiner Handbook of Stochastic Methods Springer 1985
     Redner A Guide to First-Passage Processes Cambridge 2001
visionary classical texts:
     Einstein      Investigations on the Theory of the Brownian Motion      Dover 1956     
     Hill      Free Energy Transduction and Biochemical Cycle Kinetics      Dover 2005     

U. Behn      Literature List (PDF)      Links of Interest