Shaken Dynamics on the 3-D Cubic Lattice

On the space of $\pm 1$ spin configurations on the 3d-square lattice, we consider the \emph{shaken dynamics}, a parallel Markovian dynamics that can be interpreted in terms of Probabilistic Cellular Automata, whose transition probabilities are defined in terms of a pair ferromagnetic Ising-type Hamiltonian with nearest neighbor interaction $J$, depending on an additional parameter $q$, measuring the tendency of the system to remain locally in the same state. We compute the stationary measure of the shaken dynamics and we investigate its relation with the Gibbs measure for the Ising model. It turns out that the two parameters $J$ and $q$ tune the geometry of the underlying lattice. By a judicious use of perturbative methods we show rigorously that our model exhibits a line of critical points in $J-q$ plane that separates the ordered phase from the disordered one, and we perform numerical simulation to determine the phase transition curve. Our method allows us to find in a unified way the critical values of $J$ for Ising model with first neighbors interaction, defined on a whole class of lattice, intermediate between the two-dimensional hexagonal and the three-dimensional cubic one, such as, for example, the tetrahedral lattice. Finally we estimate the critical exponents of the magnetic susceptibility and show that our model captures a phase transition in the geometry of the system at $q = 0$

Tides and dumbbell dynamics

We discuss a model describing the effects of tidal dissipation on satellite's orbits. Tidal bulges are described in terms of a dumbbell, coupled to the rotation by a dissipative interaction. The assumptions on this dissipative coupling turns out to be crucial in the evolution of the system.Comment: 2 figure

Il fascino della matematica e delle sue applicazioni

Il fascino della matematica e delle sue applicazioni e` un progetto del Dipartimento di Scienze di Base e Applicate per l'Ingegneria finanziato nell'ambito de bando terza missione per l'anno 2020 di Sapienza Universita` di Roma. L'iniziativa ha avuto come scopo quello di diffondere non solo gli aspetti piu` intriganti della matematica, ma anche le sue ricadute sulle scienze applicate, attraverso una serie di seminari, tenuti da docenti universitari e docenti di scuola superiore, rivolti principalmente a un pubblico di studenti della scuola superiore. Il testo raccoglie il contributo di alcuni degli oratori coinvoti nei seminari