Na física e em diversas áreas da matemática aplicada, a teoria do campo médio estuda o comportamento de modelos que descrevem muitos componentes individuais que interagem entre si, através de modelos mais simples, obtidos pela substituição das interações entre os componentes por uma interação média, o que reduz o problema envolvendo muitos componentes a um problema efetivo para um só componente. O objetivo geral é obter alguma informação sobre o comportamento do sistema complexo a um baixo custo computacional. Esta ideia surgiu pela primeira vez na mecânica estatística nos trabalhos de Pierre Curie e Pierre Weiss, para estudar transições de fase ferromagnéticas. Desde então, a teoria de campo médio tem sido aplicada a diversas áreas fora da física, incluindo a inferência estatística, os sistemas dinâmicos, a teoria de grafos, a neurociência, a inteligência artificial, os modelos epidêmicos, a teoria de filas, o desempenho das redes informáticas, a teoria dos jogos, entre outras. Apesar das diversas aplicações, a fundamentação matematicamente rigorosa da teoria de campo médio ainda representa grande desafio, também em matemática pura. O objetivo do encontro é de reunir alguns pesquisadores proeminentes de diversas áreas (em particular, sistemas dinâmicos, mecânica estatística, informática teórica e informação quântica) que têm se dedicado recentemente ao estudo da teoria do campo médio. Eles proferiram palestras de interesse geral e participarão de discussões visando estabelecer colaborações e objetivos comuns em pesquisa relacionada à teoria do campo médio.

PROGRAMA:

1) Artur Lopes (UFRGS), quinta-feira 28/11, 14:00-14:50

Título: Nonlinear Thermodynamic Formalism and Large Deviations

Resumo: We will present some new results on quadratic and non-quadratic Thermodynamic Formalism which is in some sense the generalization of the classical Curie-Weiss models to a more dynamical setting. Among the methods we applied to the problems we consider some Large Deviations properties. This is a joint work with Walter A. de Siqueira Pedra.

2) Marco Merkli (Memorial University St. John’s), quinta-feira 28/11, 15:00-15:50

Título: Quantum systems coupled to environments via mean field interactions

Resumo: We show that when a quantum system is coupled to an environment in a mean field way, then its effective dynamics is governed by a unitary group with a time-dependent Hamiltonian. The time-dependent modification of the bare system Hamiltonian is given by an explicit term involving the reservoir state. We show that entanglement within the system state is not changed during the dynamics. Our results hold for arbitrary strengths of the system-environment coupling, and for finite or infinite dimensional systems. This is joint work with Michele Fantechi, availalbe at arXiv:2409.15850.

3) Jean-Bernard Bru (Basque Center for Applied Mathematics), quinta-feira 28/11, 16:30-17:20

Título: Quantum Lattice Systems with Mean-field Interactions and the Bogoliubov Approximation

Resumo: We will discuss quantum lattice systems (fermions or quantum-spins) with mean-field interactions and explain how the so-called Bogoliubov approximation arises in this context. Roughly speaking, the Bogoliubov approximation consists in replacing specific operators appearing in the Hamiltonian of a given physical system with constants that are determined as solutions of a self-consistent equation or an associated variational problem. This method was first used successfully in Bogoliubov's superfluidity theory of Helium 4 in 1947, and a first mathematical understanding of it was not given until 1968, by Ginibre, for the grand-canonical pressure of many-boson systems. We demonstrated its validity at the level of states, i.e. for correlation functions.

4) Thomas Peron (ICMC), sexta-feira, 29/11, 9:00-9:50

Título: Mean-field theory of vector spin models on networks with arbitrary degree distributions

Resumo: In this talk we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We show that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. This is a joint work (arxiv:2110.11153) with Fernando L. Metz (UFRGS).

5) Jérôme Buzzi (Paris-Saclay), sexta-feira, 29/11, 10:00-10:50

Título: A nonlinear thermodynamical formalism for dynamical systems

Resumo: Since the 1970s and the seminal works of Sinai, Ruelle, and Bowen, the thermodynamical formalism of lattice statistical mechanics has been a powerful tool for the ergodic theory of dynamical systems. The basic object is the "pressure" of invariant probability measures m defined as P(m) = h(m) + m(V) where h(m) is the Kolmogorov-Sinai entropy and m(V) is the average of the "potential" V wrt the measure m. Motivated by basic mean-field model, we have studied "nonlinear pressures" given by Pi(m) = h(m) + E(m) where E is a continuous function of the measure m in the weak star topology. We obtain detailed results when E(m)=F(m(V_1),...,m(V_r)) where F is a smooth function and V_1,...,V_r are regular "potentials", especially when r=1 and F is analytic. This generalizes the Curie-Weiss and Potts models and allows phase transitions. The proof involves a reduction through convex analysis to the classical theory. Joint work with Benoit Kloeckner and Renaud Leplaideur.