Stochastic Equations and Processes in Physics and Biology

Lecture course in summer-semester 2012 consisting of 8 lectures in total

Lecturer:

Dr. Andrey Pototsky

(University of Cape Town)

Period:

June 18 -  July 12, 2012

Day, time, and place

Mondays,  16-18 h,  and

Thursdays,  14-16 h

in room ER 164 at TU Berlin.

Abstract

Fluctuations and noise are ubiquitous in nature. Our experience shows that if a classical system, no matter how large or small, is subjected to a process of a measurement, the outcome cannot be predicted with the absolute certainty. Thus, the position of pollen grains, suspended in the water, changes randomly in time as they jiggle due to Brownian motion; neural cells in any living organism generate a random series of electric pulses in the response to the synaptic input from other cells; weather forecasts are unreliable because of the chaotic nature of the global weather. In all these examples, the quantum effects can be neglected, and the origin of randomness is traced down to the complexity of the system or chaos.

This course of lectures is designed to give an introduction into the phenomenological theory of stochastic equations and processes. The mathematical formalism of the stochastic calculus is explained on the examples, borrowed from physics and biology. Covered topics include:

• stochastic ODEs
• Ito and Stratonovich interpretation
• numerical integration of the stochastic equations
• power spectral density and correlation function
• the Fokker-Planck equation
• dynamical phase transitions in coupled stochastic networks

The course is designed to be accessible for the non-experts in the field.

Lecture pdfs

Introduction

Lectures part 1

Lectures part 2

Lectures part 3

Lectures part 4