PHYSICS OF COMPLEX SYSTEMS

The aim of the course is to provide the students with a general overview of the theory of complex systems. The course of study will start from the description of critical phenomena and following the recent development of the field, it will describe self-similar systems in space, time and topology. The methods of measurement and modeling of these systems will be presented starting from the calculation of the fractal dimension, to the scaling and to the renormalization group. Various multidisciplinary applications will be presented during the course. Many of these are of direct interest to the course of study in any case all of them are interesting for students as they show the correct way to apply the theoretical tools defined in the first part of the courses.
The final part of the course will present various research topics related to the various curricula of the master's course in order to inform students on the choice of the path that best meets their expectations.

At the end of the course, students are expected to be able to read the current literature in this area, identify the optimal technique (experimental, theoretical or computational), and tackle a specific problem, by collecting and analyzing the data and modeling the phenomenon

Fundamental tools of mathematics and physics, calculus, linear algebra, probability distributions, entropy.

Introduction (partly covered in other courses)
*) The Statistical Description of Physical Systems
*) The Interpretation of Statistical Quantities
*) Phase transitions
*) Ising Model

Complex Systems
*) Friction and Fluctuations
*) Evolution of Phase Space Probabilities
*) Theory of Critical Phenomena
*) Montecarlo Simulations and Computational Instruments
*) Fractals
*) Scaling Theory
*) Renormalization Group
*) Universality
*) Self-Organised Criticality
*) Complex Networks

State of the art
*) Complexity and Quantum Physics
*) Complexity in the Physics of the Brain
*) Complexity in the Physics of Financial and Economic Systems

All activities will be accompanied by numerical and computer programming exercises.

* H. E. Stanley "Introduction to Phase Transitions and Critical Phenomena" OUP (1971)
* G. Caldarelli "Scale-Free Networks" OUP (2007)
* D. Easley and J. Kleinberg “Networks Crowds and Markets” CUP (2010) (http://www.cs.cornell.edu/home/kleinber/networks-book/ )
* A-L Barabási "Network Science" CUP (2016) (http://networksciencebook.com/ )
* R. Bauerschmidt, D.C. Brydges, and G. Slade "Introduction to renormalisation group method" (https://arxiv.org/pdf/1907.05474.pdf )

The final exam will be based on a report and a presentation by the students on a specific topic agreed with the instructor.

oral

Grade out of thirtieths based on the thesis and the oral presentation and exam.

Traditional interacting methods, on-line teaching, or a combination of the two will be used, depending on students logistic and situations.