PHYSICS OF COMPLEX SYSTEMS

1. INTRODUCTION TO COMPLEX SYSTEMS
2. BASIC CONCEPTS OF STATISTICAL MECHANICS
- Elements of probability theory. Random variables and their algebra. States, pure and mixed. Principles of thermodynamics by probabilistic approach
- DRL states and spontaneous symmetry breaking. Peierls argument. Ising model in the mean field approximation, the Curie-Weiss model
- Phase transitions: phenomenology, Lee-Yang theorem, Erhenfest classification. Systems at the critical point, the universality and scaling. Landau theory. Order parameters.
- Ergodicity and dynamic
- Advanced analytical methods for ferromagnetic mean-field models and analogies with mechanical systems
3. THE EFFECTS OF DISORDER
- Topological Disorder: Criterion Harris, singularity of Griffith.
- Outline of diluted models.
- Graph theory and applications. Spectral properties of finite and infinite graphs. Statistical models and dissemination of graphs.
4. SPIN-GLASS
- Frustration. Models and classes. Applications and optimization problems.
- Sherrington-Kirkpatrick model. Existence of the thermodynamic limit. Replies fictitious and real overlap between replicas.
- The Replica Trick. The replica symmetric solution. The 1RSB solution, KRSB and full-RSB.
- Method of cavities and stochastic stability.
- The theory of Parisi. Ultrametricity.
- Techniques for interpolation and mechanical approach.
- Elements of Dynamic: Ageing and the fluctuation-dissipation relations.
5. NEURONAL NETWORKS
- Neurophysiological background.
- Integrate & Fire Models.
- Cable theory.
- Perceptron. Models of neural networks with attractors. Hopfield model. Dynamic Synchronous and Asynchronous. Perception, Recognition and Recall. Synaptic connections: symmetry and Hebbian learning rule. United recall, were spurious.
- Networking at low-load: mean field approach and signal-to-noise analysis.
- Networks high-load: mean field approach, Black-out, storage capacity. Phase diagram.
- Related Patterns and pattern diluted. Soprapercolate extensive networks in parallel computing.
6. APPLICATIONS
Immune networks: idiotypic network (low-dose tolerance, bell-shaped response, ontogeny of the repertoire), the B-T network (anergy, multitasking). Social networks (Granovetter's theory).