87997 - Physics of Complex Systems

Learning outcomes

At the end of the course the student will have the basic knowledge of Complex Systems Physics with application to biological and social systems. He/she will acquire theoretical tools to analyze, predict and control the evolution of models, including: - statistical physics and dynamical system theory of complex systems; - dynamics of systems on network structures; - stochastic thermodynamics; - stochastic dynamical systems.

Course contents

Main objective: to join Statistical Mechanics approach, that studies the equilibrium states of many dimensional systems, with the theory of Dynamical Systems, which is especially developed for low dimensional systems. The Physics of Complex Systems aims to develop a theory of non-equilibrium Statistical Physics.

Dynamical Systems theory and introduction to Information Theory. Deterministic chaos and Lyapunov exponents. The probabilistic approach to dynamics. and Time scales. Microscopic reversibility vs macroscopic irreversibility. Theory. Shannon entropy, conditional entropy, joint entropy, mutual information. Entropy rate of a stochastic process. Entropy as a measure of complexity/unpredictability. Kolmogorov-Sinai entropy rate for a dynamical system.

Stochastic dynamical systems, Markov processes, stochastic differential equations, stochastically perturbed dynamical systems and Fokker Planck equation for diffusion processes, transition rate theory (Kramers' theory), stochastic resonance.

Examples of complex systems models, compartmental models, Lotka Volterra models, traffic models, nonlinear neuronal models (bifurcation), master equation for chemical reactions and diffusion on graphs (mobility network), concept of attractors, emergent properties.

Readings/Bibliography

Materials and notes provided during the lessons

Gregoire Nicolis, Catherine Nicolis Foundations of Complex Systems Nonlinear Dynamics, Statistical Physics, Information and Prediction World Scientific, 3 set 2007

Yaneer Bar-yam Dynamics Of Complex Systems Perseus Books Cambridge, MA, USA ©1997

Nino Boccara "Modeling Complex Systems" Graduate Text in Contemporary Physics, Springer, 2004
Per Bak "How Nature Works: The Science of Self-Organised Criticality" New York, NY: Copernicus Press, 1996

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry. Elsevier, 2007.

V. I. Arnold, A. Avez, Ergodic Problems of Classical Mechanics, Addison-Wesley

T. M. Cover, J. A. Thomas, Elements of Information Theory, Wiley

Teaching methods

Frontal lessons and use of computational models

Assessment methods

Presentation of a project/essay on a topic related to the topics discussed during the course, with possible questions on the course program

Teaching tools

use of computer for model simulations

Office hours

See the website of Armando Bazzani