Return to equilibrium and nucleation in the BCS model

• Grant number: 24/16826-0
• Support Opportunities: Scholarships in Brazil - Master
• Start date: April 01, 2025
• End date: August 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Walter Alberto de Siqueira Pedra
• Grantee: Caio Diniz Santana
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

We know from everyday experience that thermodynamic equilibrium is stable in the sense that systems in equilibrium spontaneously return to it after suffering small disturbances. This phenomenological aspect is so common that it makes up the postulates of thermodynamics. In fact, the "zero law" of thermodynamics refers precisely to the return to equilibrium. However, from the point of view of statistical mechanics, i.e. the reduction of thermodynamics to the mechanics of the microscopic elements (typically atoms or itinerant subatomic particles such as electrons in crystals) that make up matter, this property is highly non-trivial. In fact, to our knowledge, there are rigorous proofs of the return to equilibrium only for fermion and boson ideal gases, which are systems of non-interacting quantum particles, as well as for local perturbations of such systems. In the present master's project we propose to give a complete mathematically rigorous proof of the return to equilibrium for a fully interacting quantum model, that is, a model that describes particles that interact homogeneously in space, and not only in finite regions, as in the cases already studied. The particular model we intend to study is the BCS model of superconductor theory. This is a model with a mean-field interaction term, which represents (effective) forces that have a small intensity but act over large distances. It is important to note here that we will be considering the usual BCS model and not, as has been done in several papers, its large coupling limit. In fact, this particular limit corresponds to an exactly integrable model and has already been exhaustively studied in the literature. We believe that recent advances in the analysis of the dynamics of fermionic models with mean-field interactions will make such a proof possible. To our knowledge, this would be the first mathematically rigorous proof of the return to equilibrium in a spatially homogeneous model for interacting quantum particles that is not exactly integrable. During the master's studies, a research internship lasting two to three months will be carried out in the quantum mechanics group at the Basque Center for Applied Mathematics (BCAM) in Bilbao, under the supervision of Prof. Jean-Bernard Bru, co-author of the works on which the master thesis will mainly be based. If the objective is met as expected, the result on the return to equilibrium should be published in a good international journal, most likely in collaboration with Prof. Bru. Finally, the master's project will enable the student to carry out research, for example in a possible PhD, on topics of current interest related to the mathematically rigorous study of many-fermion systems.