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

The main objective of the Complex Systems Physics course is to demonstrate how the theory of Stochastic Dynamical Systems enables the development of a non-equilibrium statistical mechanics, which constitutes one of the key research areas within the physics of complex systems.

Contents of the course

Dynamical systems theory: concepts of integrable and chaotic systems, Liouville's equation and the evolution of an observable, stability analysis of an orbit, introduction to perturbation theory, effects of linear and nonlinear resonances, definition of Lyapunov exponents, concepts of predictability, and properties of dissipative systems (existence of attractors).

The probabilistic approach to describe chaotic dynamics, Gibbs Entropy and Kolmogorov-Sinai entropy for a dynamical system, conditional entropy. Relation between entropy and the concept of predictability for dynamical systems.

Definition of stochastic processes and Markov processes (both discrete and continuous). Introduction to stochastic dynamical systems and the Fokker-Planck equation. Entropy production in stochastic processes, maximum entropy principles for characterizing equilibrium states, and the principle of minimum entropy production for non-equilibrium steady states.

Wiener process, Ito integral and stochastic differential equations. Stochastic dynamical systems and stochastically perturbed dynamical systems, properties of the Fokker Planck equation for diffusion processes, transition rate theory (Kramers' theory)and concept of stochastic resonance.

Introduction to Stochastic Thermodyamics

The preceding theoretical concepts will be applied to examples of complex system models: compartmental models, Lotka-Volterra models, traffic models, socioeconomic models, cellular automata, nonlinear neural networks, master equations for biological systems, emergent properties, diffusion on graphs (transport networks), and reaction-diffusion models.

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

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

Angelo Vulpiani Chaos: From Simple Models to Complex Systems Volume 17 di Series on advances in statistical mechanics, 2010

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