Final Report Summary - MALADY (MACROSCOPIC LAWS AND DYNAMICAL SYSTEMS)
The basic goal of the MALADY project was to contribute to the rigorous understanding of how the macroscopic world might arise from microscopic evolution laws.
This is an enormously vast problem in the field of Non-Equilibrium Statistical Mechanics. We restricted ourselves to situations in which the microscopic description is classical (either deterministic or stochastic) and we aimed at performing a hydrodynamics limit, that is a rescaling of space and times that reflects the space-time scale separation between the microscopic and the macroscopic world. As this is well known to be extremely hard, especially when the microscopic dynamics is purely deterministic, we planned to explore also the possibility to achieve this goal via a two step process: first a weak coupling limit yielding an effective mesoscopic equation and then a hydrodynamics limit applied to the latter.
In a series of papers we successfully proved that such a two step strategy can be carried out both for weakly stochastic and purely determinist dynamics. In addition, we showed that the Green-Kubo formula of the microscopic models is strictly related to the Green-Kubo formula for the derived mesoscopic equations, whereby providing a physical justification to our mathematical strategy and illustrating the physical meaning of the derived mesoscopic equations.
When the microscopic noise acts in sufficiently many directions or when it perturbs a linear dynamics, we instead succeeded in carrying out the hydrodynamics limit directly. In addition, in the latter case (for one dimensional models that preserve momentum) we showed that the limit macroscopic law is not the Fourier law but rather yields a heat superdiffusion governed by a fractional heat equation.
We also investigated the use of these hydrodynamic limits to describe isothermal or adiabatic thermodynamic irreversible transformations between equilibrium states and extended this approach for transformation between stationary non equilibrium states. Then we introduced and proved direct quasi-static limits where reversible quasi-static thermodynamic transformations are obtained directly by space-time scaling of microscopic dynamics. This gives finally a precise mathematical connection between quasi-static transformations, that are basic concepts in classical thermodynamics, and the microscopic dynamics.
As already mentioned, to achieve the hydrodynamics limit directly for purely deterministic system is a formidable challenge, yet our work further clarified which kind of results must first be established in order to make this task conceivable. Of course, the first step is establishing a Green-Kubo formula for the macroscopic diffusivity.
Toward this is necessary a very precise understanding of the statistical properties of the microscopic models. In this direction we made substantial progresses by proving exponential decay of correlations for the billiard flows and an analogous of the Freidlin--Wentzell theory for some classes of partially hyperbolic fast--slow systems. These results, in particular, open the door to the study of the statistical properties of weakly coupled hyperbolic flows (a class of systems to which all our deterministic microscopic models belong).
Our work also had some unexpected spin-off, not envisioned in the original project. Notably, we established very precise error bounds for the counting problem of closed prime geodesics in compact manifolds of strictly negative curvature.
This is an enormously vast problem in the field of Non-Equilibrium Statistical Mechanics. We restricted ourselves to situations in which the microscopic description is classical (either deterministic or stochastic) and we aimed at performing a hydrodynamics limit, that is a rescaling of space and times that reflects the space-time scale separation between the microscopic and the macroscopic world. As this is well known to be extremely hard, especially when the microscopic dynamics is purely deterministic, we planned to explore also the possibility to achieve this goal via a two step process: first a weak coupling limit yielding an effective mesoscopic equation and then a hydrodynamics limit applied to the latter.
In a series of papers we successfully proved that such a two step strategy can be carried out both for weakly stochastic and purely determinist dynamics. In addition, we showed that the Green-Kubo formula of the microscopic models is strictly related to the Green-Kubo formula for the derived mesoscopic equations, whereby providing a physical justification to our mathematical strategy and illustrating the physical meaning of the derived mesoscopic equations.
When the microscopic noise acts in sufficiently many directions or when it perturbs a linear dynamics, we instead succeeded in carrying out the hydrodynamics limit directly. In addition, in the latter case (for one dimensional models that preserve momentum) we showed that the limit macroscopic law is not the Fourier law but rather yields a heat superdiffusion governed by a fractional heat equation.
We also investigated the use of these hydrodynamic limits to describe isothermal or adiabatic thermodynamic irreversible transformations between equilibrium states and extended this approach for transformation between stationary non equilibrium states. Then we introduced and proved direct quasi-static limits where reversible quasi-static thermodynamic transformations are obtained directly by space-time scaling of microscopic dynamics. This gives finally a precise mathematical connection between quasi-static transformations, that are basic concepts in classical thermodynamics, and the microscopic dynamics.
As already mentioned, to achieve the hydrodynamics limit directly for purely deterministic system is a formidable challenge, yet our work further clarified which kind of results must first be established in order to make this task conceivable. Of course, the first step is establishing a Green-Kubo formula for the macroscopic diffusivity.
Toward this is necessary a very precise understanding of the statistical properties of the microscopic models. In this direction we made substantial progresses by proving exponential decay of correlations for the billiard flows and an analogous of the Freidlin--Wentzell theory for some classes of partially hyperbolic fast--slow systems. These results, in particular, open the door to the study of the statistical properties of weakly coupled hyperbolic flows (a class of systems to which all our deterministic microscopic models belong).
Our work also had some unexpected spin-off, not envisioned in the original project. Notably, we established very precise error bounds for the counting problem of closed prime geodesics in compact manifolds of strictly negative curvature.