Mathematical Models in Statistical Physics

Research Areas

Presentation

Random effects in statistical physics have become experimentally accessible and a central point of mathematical modelling. For many models, conjectures and predictions can be formulated through numerical simulations and approximations, but a mathematically satisfactory analysis is not yet available. This project focuses on several research directions, including fluctuations in kinetic models, random growth models, quantum-mechanical transport, non dissipative transport in topological materials.

    Coordinators: Giada Basile, Alessandra Faggionato, Sergio Simonella, Vittoria Silvestri, Lorenzo Taggi


    Coordinators: Elena Agliari, Giuseppe Visconti

Research Themes

The Eccellenza project promotes exchanges between different research groups, within the Department of Mathematics "Guido Castelnuovo". These groups gather multiple competences and are active in several lines of investigation.

Regular Seminars

Upcoming Events

  • Thematic period, Winter-Spring 2025: Statistical Mechanics and Nonequilibrium Processes (see dedicated link)
  • Thematic period, Fall 2025: Data Science (see dedicated link)

Past Events

  • Workshop: Singular Interactions and Effective Models in Mathematical Physics
Date: July 15 - 17  2024Timetable: Monday 15  09:30 -18:00 | Tuesday 16  09:30 - 18:00 | Wednesday 17  09:30 - 12:30Venue: room -Lecturer: R. Adami (Politecnico di Torino),  P. Antonelli (Gran Sasso Science Institute), L. Barletti * (Università degli Studi di Firenze), R. Carlone (Università Federico II), M. Correggi (Politecnico di Milano), K. Datchev (Purdue University), M. Falconi (Politecnico di Milano), D. Fermi (Politecnico di Milano), M. Gallone (Scuola Internazionale Superiore di Studi Avanzati), S. Gustafson (University of British Columbia), H. Kovařík (Università degli studi di Brescia), M. Ikeda (Rikagaku Kenkyusho), M. Olivieri (University of Copenhagen), V. Romano (Università di Catania), L. Tentarelli (Politecnico di Torino). [ *  To be confirmed ]Abstract: The workshop is part of the activities of the PRIN project. The unifying theme of the research project is the analysis of partial differential equations ruling relevant classical or quantum models in domains where spatial singularities can be present. Singularities can develop as a consequence of the dynamics (blow-up), or they can be present from the beginning in the domain where the model is set. In this second case singularities can describe spatial inhomogeneities of the propagation medium (described for example by point or delta interactions with various co-dimension or by quantum graphs) or they can serve as effective approximations (zero range models).Web Site: link
  • Doctoral course: Introduction to stochastic partial differential equations
Date: June 2024  (for any further information please contact Lorenzo Bertini: bertini@mat.uniroma1.it)Timetable: -Venue: room BLecturer: Sandra Cerrai (UMD - University of Maryland)Abstract: Stochastic partial differential equations have becomen a basic modelling tools in natural sciences. Their rigourous analysis have revelaved a rich mathematical structure which has not yet been fully uncovered.The proposed course aims to introduce the students to the main problems and techniques in the field.Tentative program:
  • Existence and uniqueness
  • Kolmogorov equations
  • Reaction diffusions equations
  • Long time behavior
  • Averaging phenomena
  • Small noise asymptotics
References: G. Da Prato, J. Zabczyck, Stochastic equations in  infinite dimensions, 2nd edition Cambridge University Press, 2014.
  • Doctoral course: Introduction to Random Geometry
Date: March 18 / 20 / 22 / 25 / 27  &  April 2 / 3 / 8 / 10 / 12  2024Timetable: h. 16:00 - 18:00Venue: room GLecturer: Vittoria Silvestri (Università di Roma La Sapienza) Abstract: This course wants to give an overview of active research topics in the field of Random Geometry, with a focus on growth models. We will start by discussing discrete growth models such as the Eden model, Diffusion Limited Aggregation and Internal DLA. We will then move to the continuum for the remaining part of the course. After discussing conformal invariance of Brownian motion, we will focus on the class of randomly growing domains on the complex plane which can be described via Loewner dynamics. We will introduce several random aggregation models on the complex plane, which go under the name of Hastings-Levitov models and Aggregate Loewner Evolutions, of which we will study the large-scale features, presenting existing results and several open questions.Keywords: Random aggregation, Diffusion Limited Aggregation, Schramm-Loewner Evolutions, Hastings-Levitov, Aggregate Loewner Evolutions.
Prerequisites: The course will be as self-contained as possible. However, basic notions of Probability and Analysis are necessary. Some knowledge of complex analysis, martingale theory and stochastic calculus is desirable but not required.
  • Doctoral course: An introduction to the theory of complex networks
Date: March 5 / 12 / 19 / 26  2024Timetable: h. -Venue: room -Lecturer: Massimo Ostilli (UFBA)Abstract: The theory of complex networks provides a pivotal tool for understanding and analyzing a wide variety of complex systems, both artificial and natural, like the Internet, the World Wide Web, the brain, the food-webs and many others. This compact (four-days) course provides an introduction to this very interdisciplinary theory via simple probabilistic arguments explaining both the structure (in terms of degree distribution, clustering, shortest-path, etc...) and the functionality (in terms of percolation, resilience, spreading of virus, etc...) of complex networks. Except for the last section (day 4), the course does not assume any prerequisites, rendering it suitable for a vast spectra of students and researchers, especially for those interested in applied math. The day 4, however, is more suitable for an audience having a background in statistical mechanics.
  • Doctoral course: Topological quantum transport: Chern numbers in the lab
Date: February 29  &  March 1 / 4 / 5 / 7 / 8 / 14 / 15  2024Timetable: h. 16:00-18:00Venue: room GLecturer: Domenico Monaco (Università di Roma La Sapienza)Abstract: The course aims at illustrating the mathematical theory behind the quantum Hall effect, in which a 2-dimensional electron gas immersed in a magnetic field responds to an external electric potential difference with a current flowing in the perpendicular direction, whose conductivity appears to be quantized to an integer value (in appropriate physical units). This quantization phenomenon has been explained by relating the conductivity to a topological invariant, the Chern number, of a vector bundle arising from the quantum states in the occupied energy levels. The lectures will then show how certain techniques from vector bundle theory (holonomy, parallel transport, obstruction theory) can be employed to characterize the physical properties of this quantum transport phenomenon. These techniques will be illustrated with a “hands-on” approach, and geometric objects will be presented as naturally arising from the underlying quantum theory.
  • Reading course: Mathematical Methods for Quantum Mechanics
Date: February 2 / 14  &  March 6 / 20 2024Timetable: h. 14:00-16:00Venue: room GLecturer: Domenico Monaco* - Gianluca Panati* - Alessandro Teta* ( * Università di Roma La Sapienza)Course 1: Domenico MonacoAbstract: The course will be activated upon request from PhD students. A reading plan will be decided based on the students’ interests, and will be focused on topics related to the mathematical theory of quantum mechanics like: Spectral analysis, Schroedinger operators with zero-range interactions, nonlinear Schroedinger equation, semi-classical and adiabatic limit of quantum dynamics, decoherence, geometric and variational methods for periodic Schroedinger operators, historical aspects of Quantum Mechanics.
  • Doctoral course: Elementi di storia della Meccanica Quantistica
Date: February 6 / 8 / 13 / 15 / 20 / 27 (conclusive seminar)  2024Timetable: h. 14:00Venue: room ELecturer: Alessandro Teta (Università di Roma La Sapienza)Lecture 1:
  1. Introduzione
  2. Richiami di Meccanica:
    • formulazione assiomatica
    • struttura algebrica della meccanica
    • metodo dei fronti d’onda
  3. Richiami di Elettromagnetismo:
    • equazione di Maxwell nel vuoto
    • conservazione dell'energia
    • campi di una carica in moto assegnato
Lecture 2:
  1. Corpo nero (Planck 1900)
  2. Natura corpuscolare della luce (Einstein 1905)
  3. Costituzione atomica della materia
  4. Modello atomico di Bohr (1913)
  5. Vecchia teoria dei quanti (Sommerfeld 1916)
Lecture 3:
  1. Formulazione della Meccanica delle Matrici (Born 1924, Heisenberg 1925, Born-Jordan 1925, Dirac 1926)
Lecture 4:
  1. Analogia ottica-meccanica:
  • limite di piccola lunghezza d'onda dell'ottica ondulatoria
  • formulazione della Meccanica Ondulatoria (De Broglie 1924, Schroedinger 1926)
  • interpretazione statistica  (Born 1926)
Lecture 5:
  1. Equivalenza degli approcci di Heisenberg e Schroedinger:
    • formulazione della teoria
    • interferenza
    • entanglement
    • cenni al problema dei fondamenti
    • interpretazione di Copenhagen
    • problema della misura in MQ
Lecture 6  (conclusive seminar):
  1. Il problema del limite classico della MQ
  2. Decoerenza,  un caso di studio: il problema delle tracce in una camera a nebbia (Mott 1929)


Dates: January 29-31, 2024

Venue: room Conversi, Physics Dep., Sapienza University

Organizers: Stefano Lupi, Salvatore Macis, Gianluca Panati, Andrea Perali

Co-sponsored by: PNRR PE0000023-NQSTI and PRIN 2022AKRC5P

  • Doctoral course: Branching Random Walks in d>2
Dates: May 29  &  June 1 / 8 / 9 / 12 / 15 / 19 / 22, 2023Timetable: 11:00-13:00Venue: room BLecturer: Amine Asselah (Paris Creteil)Abstract: Simple Random Walk is the first example of a Markovian process, whereas Branching Random Walk has a time indexed by a random tree, and loses the SRW’s sequential way of visiting space.The highlights of the course would be two phenomena in high dimensions linked with folding:
    • The chances that two independent walks meet for a long time
    • The chances that two independent branching walks both visit a large region, and this is to be discovered during the course
We plan to have self-contained lectures (with little prerequisites), and will start with reviewing useful facts (classical and not) on random walks.

Preprints and publications