Periodic Reporting for period 5 - HyLEF (Hydrodynamic Limits and Equilibrium Fluctuations: universality from stochastic systems)
Période du rapport: 2022-12-01 au 2023-11-30
In real-life situations, we come across many different episodes where we feel that it has similarities with something we have lived before in the past. Whether this is related to emotions or physical reactions, the similarities can be present. In nature, similarities between very different organisms also occur. As an example of this puzzling reality which somehow connects all of us, as an example we can recall a rigorous winter day in which ice particles fall from the sky. When seated in a car, we see a growing pattern of ice particles which is formed on the windscreen.
The ice particles fall, randomly, from the sky and when they hit the windscreen they form a growing pattern which can be seen in other, in principle, uncorrelated, situations such as coffee ring effects, bacterial growth like E-coli, the wake of flame, tumour growth…There are various physical systems, that when they are mathematically modelled they show identical patterns of growth. This slightly mysterious tendency for very different things to behave in very similar ways is the essence of universality. There are different shapes for these patterns and their study was the core of this project and it is related to a very active area of research in both mathematics and physics known as universality. This mysterious relationship between very distinct physical systems is encoded in some universal laws that one has to figure out how they can be characterized and how does one change from one universal law to another. They are linked by some parameter which somehow connects very different systems. This project analyzed this issue which is important for the understanding of the surrounding and mysterious world that we live in.
We have exploited this issue for a Hamiltonian system with exponential potential. The dynamics conserves the energy and the volume and the transition of the energy, has been derived and goes from an OU to SBE, and is independent of the volume evolution. We were also interested in the derivation of new PDEs from microscopic dynamics. These PDEs are of a fractional type and with several types of BC. In this regard, we have obtained several PDEs given in terms of an operator, known as the regional fractional Laplacian and with BC of Dirichlet type, Robin and Neumann. We considered models with long jumps and in contact with an infinite number of stochastic reservoirs.