PHYSICS OF COMPLEX SYSTEMS
OBJECTIVES OF THE COURSE
The course will be devoted to the conceptual and quantitative study of several systems featuring complex behavior, in the physics of matter, in graphs and network theory, in biological systems, in computer science and in other interdisciplinary applications. We will consider several aspects of the most advanced theoretical approaches and of their application, providing the basis for theoretical modelling and for the application of methods in statistical mechanics and in dynamics to the study of complex systems.
COURSE CONTENTS SUMMARY
Advanced methods in statistical mechanics for complex systems, in physics and in interdisciplinary domains.
- Advanced methods in statistical mechanics for complex systems: singularities, phase transitions and ergodicity breaking. Systems at a critical point, scaling and universality. Order parameters and spontaneous symmetry breaking. Mean field, Laundau-Ginzburg theory. Critical dimensions. Static and dynamical scaling. Spin models. Variational methods for free energies.
- The effects of disorder: Harris criterion, Griffith singularities. Spin glasses, SK model and p-spin modes. Random fields models, supersymmetry and dimensional reduction. Spin models and optimization problems, phase transitions in complexity classes. Polymers in equilibrium. Percolation.
- Dynamics: stochastic differential equations and supersymmetric models. Asymptotic out of equilibrium regimes. Growth models, coarsening. KPZ equation for surface growth. Polimers, statics and dynamics. Fluctuation dissipation theorem and its estension.
- Complex networks: graph theory, topological characterization of graphs and complex networks. Spectral properties on finite and infinite graphs. Statistical models and diffusion on graphs. Syncronization. Biological networks: neural networks, immune networks. Harmonic oscillations and normal modes in proteins.
- Quantum phase transitions: an introduction.
Chapters from the following books:
G. Parisi, Statistical Field Theory, Addison-Wesley (1988)
N. Goldenfeld, Lectures On Phase Transitions And The Renormalization Group Frontiers in physics (1992)
J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Oxford University Press (2007)
Course Material and Articles
ASSESSMENT METHODS AND CRITERIA
Oral and practical lessons