510820 - KINETIC MODELS AND APPLICATIONS

courses

ID:

510820

Duration (hours):

48

CFU:

SSD:

FISICA MATEMATICA

Year:

2025

Date/time interval

Secondo Semestre (02/03/2026 - 05/06/2026)

Course Objectives

The aim of the course is to provide an introduction to the study of the principal kinetic equations in the context of gas dynamics and for the mathematical description of multiagent systems. Part of the course will focus on the development of analytical methods for the study of the convergence to equilibrium. The student will know how to study several classes of differential problems, adapting to the context tools and demonstrations developed and used during the course.

Course Prerequisites

Calculus in high dimension. Elements of probability and statistics. Fourier transform.

Teaching Methods

The course consists of blackboard lectures, during which the theoretical part of the course will be covered, and a laboratory part, in which numerical methods useful for understanding the theory will be developed.

Assessment Methods

The exam will consist of an oral test. This test is aimed at verifying the degree of understanding of the theoretical topics carried out in class, of the clarity of exposition but also of the ability to apply these notions in concrete situations. For these reasons, the student should have a substantial understanding of all the theory presented, which can be verified either through questions on specific topics or through the proposal of problems concerning the course topics and solvable using the tools introduced during the lectures. The questions will be articulated on variable difficulty so as to establish the degree of depth in the acquisition of these skills The formulation of the grade will be obtained by considering the overall breadth and depth of learning, as well as the clarity of exposition and the skills demonstrated in problem solving.

Texts

L. Pareschi, G. Toscani. INTERACTING MULTI-AGENT SYSTEMS. KINETIC EQUATIONS AND MONTE CARLO METHODS. Oxford University Press, Oxford, 2014. C. Villani. A review of mathematical topics in collisional kinetic theory. Handbook of Mathematical Fluid Dynamics (Vol. 1), edited by S. Friedlander and D. Serre, published by Elsevier Science (2002).

The course aims to introduce, from a physical and mathematical perspective, the main results of kinetic theory and their application to emerging phenomena in multi-agent systems, life sciences, and optimization and control problems. Below are the main sections characterizing the course: - Collisional particle systems and conserved quantities, H-theorem - Simplified collisional equations of BGK and Fokker-Planck type - The Boltzmann equation and H-theorem - Interacting systems and mesoscopic equations - Quantitative characterization of agent-based models and emerging phenomena, with applications in social sciences, finance, and life sciences - Derivation of mean-field and hydrodynamic models, Chapman-Enskog expansion - Control problems for mean-field equations - Application of kinetic methods in optimization - Monte Carlo methods for kinetic equations - Uncertainty quantification and data-oriented approaches