PHYSICS OF COMPLEX SYSTEMS
Learning outcomes of the course unit
The course includes the study of various systems displaying complex behaviors; the goal is finding the 'phenomenological' laws that govern the overall behavior of such systems.
We will describe various theoretical models and techniques, both analytical and numerical, for their characterization and we will discuss applications in the field of physics, biology, computer science and economics.
Given the interdisciplinary nature and the multiple applications of the topics covered, the course is recommended for all curricula.
Basic concepts of statistical mechanics
Courses of Analysis and Mathematical Methods
Course contents summary
- Introduction to complex systems
- Basic concepts of statistical mechanics
- Effects of disorder
- Neural networks
- Applications in sociology, biology, immunology
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.
- 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.
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).
- Lecturer Notes
- Statistical Mechanics, D. Ruelle
- An Introduction to Probability Theory and its applications, W. Feller
- Spin glass theory and beyond, M. Mezard, G. Parisi, M. Virasoro
- Introduction to Theoretical Neurobiology, H. Tuckwell
- Modeling Brain Function, D. Amit
Lectures, exercises (both in class and at home)
Assessment methods and criteria