Second Annual French Complex Systems Summer School

Lyon and Paris, July 15-August 10, 2008

"Fluctuations and correlations in complex systems: An introduction to stochastic nonlinear dynamics"

Uwe Täuber
tauber AT vt DOT edu

Department of Physics
Virginia Tech
Blacksburg, VA, USA

Prof. Täuber's handwritten notes for this course can be downloaded by clicking on "files attached" at the bottom of the page.


Course Description

Stochasticity, i.e., randomly occuring propagation, interactions, or reactions, characterizes many complex systems. In addition to external noise that can influence the system, such stochastic processes generate internal noise that may crucially affect even long-time and large-scale properties. As a consequence, spatio-temporal fluctuations and correlations may play a crucial role. They influence, for example, spontaneously emerging or induced spatial patterns, the stability of structures, and front propagation. Near phase transitions or in systems with few degrees of freedom, fluctuations become especially prominent. In these situations, traditional modeling approaches in terms of rate equations or similar mean-field approximations become inadequate.

In this set of lectures, I will give an introduction to the mathematical description and modeling of the stochastic nonlinear dynamics of complex systems through master and Langevin equations. Problems and projects will be designed to reinforce the lecture material and aid students to attain a working understanding of the concepts and techniques.

My tentative lecture plan is:

Lecture 1: Introduction

• 1.1 Stochastic fluctuations and spatio-temporal correlations in complex systems
• 1.2 Stochastic processes and dynamical correlations

Literature: encyclopedia article, chapters 1 and 2
Lecture notes pp. 1-7

Lecture 2: Microscopic modeling through master equations

• 2.1 Master equation
• 2.2 Markov processes
• 2.3 Detailed balance and relaxation towards equilibrium
• 2.4 Kramers-Moyal expansion and Fokker-Planck equation

Literature: book chapter, section 2.2
Lecture Notes pp. 8-14

Lecture 3: Examples and applications

• 3.1 Biased random walks and diffusion
• 3.2 Birth-death processes
• 3.3 Diffusion-limited pair annihilation: depletion zones
• 3.4 Particle segregation in two-species annihilation

Literature: book chapter, section 2.3.1 and 2.3.2
Lecture Notes pp. 15-20


Lecture 4: Mesoscopic modeling through Langevin equations

• 4.1 Brownian motion: Einstein-Langevin theory
• 4.2 Brownian motion: Fokker-Planck equation
• 4.3 Random motion with external forces and Smoluchowski equation

Literature: book chapter, section 2.4
Lecture Notes pp. 21-25
Lecture Notes pp. 26-29

Lecture 5: Complex stochastic spatial systems

• 5.1 Chemical reactions and population dynamics
• 5.2 Active to absorbing state phase transitions
• 5.3 Cooperative relaxational dynamics: dynamic scaling and coarsening

Some literature for background and further reading:

Book chapter: Stochastic Dynamics

Encyclopedia article: Field-theoretic Methods