At the end of the course, students will acquire modelling and analytical techniques to understand and treat, from a mathematical-physical perspective, problems in the realm of complex systems.

Basic mathematics and physics. Elements of ODEs and PDEs and of numerical analysis.

Part 1: Elements of modern physics Special relativity: postulates, Lorentz-Poincaré transformations, 4-impulse, relativistic electromagnetism. Elements of quantum mechanics: quantum states and physical observables. Schroedinger equation and some of its applications. Statistical physics: microcanonical, canonical and grand canonical ensemble. Black-body spectrum. Ideal quantum gases (bosons and fermions). 1D Ising model. Part 2: Boltzmann-type kinetic theory for multi-agent systems Introduction to multi-agent systems with reference to crowd dynamics models: Helbing-Molnár microscopic model. Kinetic description of interacting particle systems. Stochastic microscopic algorithms of binary interactions and derivation of a Boltzmann-type equation in weak form. Boltzmann equation in strong form: collision operator, gain and loss terms. Evolution equations for the moments of the distribution function and conservation properties. Quasi-invariant interaction limit, Fokker-Planck equation, study of the asymptotic kinetic distribution function. Monte Carlo algorithms for the numerical approximation of Boltzmann-type collisional kinetic equations. Application of the theory to some examples of multi-agent systems chosen among: opinion dynamics, wealth redistribution, vehicular traffic, crowd dynamics. Part 3:Principles of transport and kinetic theories Elements of thermodynamics of non-equilibrium processes. Brownian motion: Langevin equation, Fokker-Planck equation. Equipartition of energy and virial theorem. Fluctuation-dissipation theorem. Probability measures in the phase space and their time evolution for dynamical systems. Birkhoss-Khinchin ergodic theorem. BBGKY hierachy. Derivation of the Boltzmann equation and H-theorem. Perturbations and linear response. Isomorphisms between dynamical systems and stochastic processes. Complexity and information entropy. Legendre transform and large deviations theory. Jarzynski identity and fluctuation theorems.

The course consists of three parts: 1. elements of modern physics (40 hours); 2. Boltzmann-type kinetic theory for multi-agent systems (20 hours). 3. principles of transport and kinetic theories (40 hours); In particular, in the third part students will be introduced to the use of the software "Mass Motion" for the simulation of crowd dynamics.

The course consists of three parts: 1. elements of modern physics (40 hours); 2. Boltzmann-type kinetic theory for multi-agent systems (20 hours). 3. principles of transport and kinetic theories (40 hours); In particular, in the third part students will be introduced to the use of the software "Mass Motion" for the simulation of crowd dynamics.

For parts 1 and 3 lecture notes will be provided. For part 2: L. Pareschi, G. Toscani. Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2013.

Lecture notes; Exercise with solutions ; Video lectures (previous years);

Exam: Written test;

Students are required to attend at least 80% of the total hours of parts 2 and 3. Consistently with the expected learning outcomes recalled above, the examination aims to test the student ability to understand the mathematical formalisation of physical theories in the realm of complex systems, using the logical-deductive approach to formulate hypotheses and to draw rigorous conclusions about the way in which these systems behave. The examination is organised in two steps: A) mandatory written test on the contents of part 1, with maximum grade 26/30. The duration of the written test is 90 minutes. During the written test, students are not allowed to consult textbooks and notes; B) optionally, in order to increase the evaluation obtained in step A, students may choose to either sit an oral examination on the contents of both parts 2 and 3 or produce a report on a research topic concerning part 2 or part 3. The oral examination consists in theoretical questions, including proofs of the main results presented during the lectures. The report consists instead in an in-depth analysis in written form of a research topic agreed with the teacher on the basis of the interests of the student. The student will be provided with reading material (such as e.g., scientific papers, book chapters and the like) to prepare the report. The student will be required to present and discuss orally his/her report using either the blackboard or some slides prepared on purpose. Both the oral examination and the report allow the students to obtain the maximum grade 6/30. The final grade of the exam is the algebraic sum of the grades obtained in step A and possibly in step B. The grade 31 is converted into 30 whereas the grade 32 is converted into 30 with merit (30L).

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.