87997 - PHYSICS OF COMPLEX SYSTEMS

Conoscenze e abilità da conseguire

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.

Contenuti

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.

First part (Bazzani) 10h: examples of complex systems models, logistic epidemic models, Lotka Volterra, traffic models, nonlinear oscillators and neuronal models (bifurcation), master equation for chemical reactions and diffusion on graphs (mobility network), concept of attractors, emergent properties (concept).

Second part (Lenci) 15h: Dynamical systems. Deterministic chaos. Degrees of chaos. The probabilistic approach to dynamics. Invariant measures. Introduction to ergodic theory: ergodicity, mixing. Decay of correlations and its physical implications. Osedelets Theorem and Lyapunov exponents. Time scales. Poincare’ Recurrence Theorem vs chaos. Microscopic reversibility vs macroscopic irreversibility.

Third part (Lenci) 9h: Introduction to Information 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.
Kolmogorov-Sinai entropy and Lyapunov exponents.

Fourth part (Bazzani) 14h: Stochastic dynamical systems, Markov processes, stochastic differential equations, stochastically perturbed dynamical systems and Fokker Planck equation for diffusion processes, transition rate theory (Kramers' theory), Phase Transitions (Hopfield neural network), Stochastic resonance, Stochastic Thermodynamics, dynamical systems on graphs (definition), examples relevant for complex systems.

 

Testi/Bibliografia

Module Prof. A. Bazzani

Materials and notes provided by the professor

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.

Reference texts for the module taught by prof. Lenci:

V. I. Arnold, A. Avez, Ergodic Problems of Classical Mechanics, Addison-Wesley
P. Walters, An Introduction to Ergodic Theory, Springer
T. M. Cover, J. A. Thomas, Elements of Information Theory, Wiley

Metodi didattici

Frontal lessons and exercises

Modalità di verifica e valutazione dell'apprendimento

Presentation of a short project/essay on a topic related to the themes of the course, with possible questions on the material covered in class

Strumenti a supporto della didattica

use of computer for model simulations

Orario di ricevimento

Consulta il sito web di Armando Bazzani

Consulta il sito web di Marco Lenci