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

Complex Systems Physics

Main objective: to join the methodologies of Statistical Mechanics, that consider the equilibrium states of many dimensional systems in a thermal bath, with the Theory of Stochastoc  Dynamical Systems Theory that is developed for low dimensional systems. This objective is related to one of the main goal of Complex Systems Physics, that is to develop the non-equilibrium Statistical Physics.

Contents of the course

Introduction to the Dynamical Systems theory: integrable and chaotic systems, perurbation theory, stability analysis, effect of linear and nonlinear resonances, the definition of Lyapunov exponents and the concept of attractors for dissipative systems.

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 process and Markov process. Introduction to stochastic dynamical systems, master equation and Fokker-Planck equation. Entropy rate of a stochastic process, maximum entropy principles to characterize the equilibrium states and Mimimum Entropy Production Principle for non-equilibrium stationary 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.

Elements of numerical physics: integration methods for stochastic differential equations, integration methods for Fokker-Planck equations.

Examples of complex systems models: compartmental models, Lotka Volterra models, traffic models, socio-economic models, cellular automata, nonlinear neural networks, master equations for biological systems, emergent properties, diffusion on graphs (transport networks), 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