CLASSICAL AND QUANTUM MECHANICAL COMPLEX SYSTEMS
Learning outcomes of the course unit
At the end of the course, the student will know different models of equilibrium and non-equilibrium statistical mechanics, learning both analytical and numerical techniques. He will be able to understand how such models can be applied to different systems both in physical fields but also in interdisciplinary applications. In particular, the large scale behaviour of the models is used to describe the global phenomenology of complex systems in the fields of biology, sociology, economics and informatics
Course contents summary
The course is devoted to the study of different kinds of systems showing typical complex behaviours due to the presence of a large number of degree of freedom. We will discuss several theoretical models adopting both analytical and numerical techniques; our aim is to find the phenomenological laws describing the global behaviour of such systems. First, we will treat purely statistical models, then we will focus on stochastic dynamics, finally we will consider graphs and complex networks.
We will discuss applications in the fields of physics, biology, epidemics, informatics and economy. Due to the interdisciplinary nature of the subjects and to the different possible applications, the course is recommended to students in different fields.
1 Equilibrium statistical mechanics
Ensemble theory, mean field and phase transitions. Focus on some statistical models relevant for their phenomenology and for their applications: interdisciplinary applications of the Ising model, p-spin, Hopfield model, XY model (Kosterlitz Thouless transition), polymeric chains, percolation.
2 Dynamics, Montecarlo method, detailed balance. Master equations and random walks. Brownian motion, Langevin and Fokker-Plank equations.
Out of equilibrium systems. Transport and Einstein equation. Entropy prodictions in time dependent dynamics. Slow dynamics: coarsening of magnetic domains, Arrhenius law. Superdiffusion and sub diffusion.
Purely dynamical models: SIS and SIR models in epidemics. Voter model. Sand-pile model and self-organized criticality. Synchronization and Kuramoto model. Neural networks dynamics.
3 Graphs and complex networks
Definition of graph: degree, radius, adjacency matrix.
Linear models on graphs: harmonic oscillators, electric networks and random walks. Fractal dimension and spectral dimension. Anomalous diffusion on fractals.
Complex networks, small world and scale free network (Watts-Strogatz e di preferential attachment). Study of some statistical models on complex networks: percolation and epidemic models.
4 Applications. In the different subjects we discuss applications in physics but also in interdisciplinary fields: biology, epidemics, informatics and sociology
Assessment methods and criteria
Oral exam on the contents of the lecture course.