Periodic Reporting for period 3 - HyLEF (Hydrodynamic Limits and Equilibrium Fluctuations: universality from stochastic systems)
Reporting period: 2019-12-01 to 2021-05-31
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. Either this is related to emotions or to 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 on a car, we see a growing pattern of ice particles which is formed in 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 as coffee ring effects, bacterial growth like E-coli, the wake of a flame, tumor growth…There are various different physical systems, that when they are mathematically modeled 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 is 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 intends to analyze this issue which will be important for the understanding of the surrounding and mysterious world that we live in.
We are at the moment, exploiting this problem for a Hamiltonian system with exponential potential. The dynamics conserve 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, but the volume depends on the evolution of the energy. In the latter case, we obtain several SPDEs where terms from the energy are present. We also want to complete the scenario of the fluctuations of this model since there are regimes still open. We are also interested in the derivation of new PDEs from microscopic dynamics. These PDEs are of a fractional type and with several types of BC. Recently we have obtained several PDEs given in terms of an operator, known in the PDE literature 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. We also plan to obtain other types of fractional PDEs from microscopic dynamics and to derive the porous media equation (fractional) with several BC out of those systems. Finally, we also plan to derive hyperbolic laws with several BC out of the hydrodynamic limit of particle systems and we also plan to derive solutions to other SPDEs (fractional) with different types of BC out of the fluctuations of particle systems.