Carissimi,
la prossima giornata di seminari sara` il 14 Settembre 2018. Ricordo che ciascuno oratore fara` una lezione introduttiva e divulgativa di 45 minuti pensata proprio per i non esperti, seguita da altri 45 minuti di tipo seminario (vedi programma).
Maggiori informazioni e aggiornamenti sulle giornate passate e future sono reperibili alla pagina web che vi invitiamo a visitare periodicamente. http://web.math.unifi.it/users/fnardi/seminari/
A SUMMER DAY IN PROBABILITY AND STATISTICAL PHYSICS
University of Florence Friday 14 September 2018
Lecturers: Stefano Olla (Paris) and Raúl Rechtman (Mexico)
Location: Aula Magna di Via S. Gallo 10, Firenze
Prof. Stefano Olla (Universite` Paris Dauphine)
Title: Hyperbolic Hydrodynamic Limits
Abstract: I will present a review of old and new results (and open problems) concerning scaling limits for conservation laws in the hyperbolic space-time scale, for a system of anharmonic oscillators with external boundary tension. The macroscopic equation is given by the compressible Euler system, with corresponding boundary conditions. The problem is particularly challenging when shockwaves are present. Some results exists when the microscopic dynamics is perturbed by a conservative stochastic viscosity. Works in Collaboration with Stefano Marchesani (GSSI) and Lu Xu (CEREMADE).
Prof. Raúl Rechtman, (Universidad Nacional Autónoma de México)
Title: Chaos and damage spreading in a probabilistic cellular automaton
Abstract: Deterministic Boolean cellular automata (CA) are discrete maps F:BN → BN, B={0,1}, x(t+1)=F(x(t)) with x ϵ BN, N large and t=0,1,… . The vector x is the state of the cellular automaton with components xi,, i=0,…,N-1 the state of cell I. Each cell is connected to others, generally in a uniform and local way, and one can define an adjacency matrix aij=1 is cell j is connected to cell i and zero otherwise. The global map F is determined by the parallel application of a local function f, such that xi(t+1) = f(vi(t)), where vi denotes the state of cells connected to cell i. Deterministic CA are thus the discrete equivalent of dynamical systems, and many concepts like trajectory (the sequence of configurations x(t)), fixed points and limit cycles can be used. There are cellular automata for which a small modification in an initial configuration propagates to the whole system, a situation similar to chaos in continuous systems, and indeed one can extend the concept of the largest Lyapunov exponent to deterministic CA using Boolean derivatives. One of the main inconvenient is that these systems do not have continuous parameters to be tuned, in order to study bifurcations.
In probabilistic cellular automata, the function f (and thus F) is defined in terms of transition probabilities so that deterministic CA can be seen as the extreme cases of probabilistic ones, when the transition probabilities are either zero or one. Probabilistic CA can be seen also as Markov chains, and one can observe interesting phase transitions after changing the transition probabilities that are therefore continuous control parameters.
A realization of a specific trajectory is determined by the extraction of one or more of random numbers for each cell. By extracting these numbers at the beginning of the simulation, for all cells and all times, one converts a probabilistic CA into a deterministic one, running over a quenched random field. One can therefore use the concepts of deterministic CA, like damage spreading and maximum Lyapunov exponent also for probabilistic CA, with the advantage of having the possibility of fine-tuning the control parameters.
In particular, we investigate a probabilistic cellular automaton which can be considered an extension of a model in the universality class of directed percolation models, but with two absorbing states. In the first part of the talk all the concepts mentioned above are defined and in the second part, the probabilistic cellular automaton is studied numerically. We show that the phase transitions when the order parameter is the average damage do not coincide with those found for the Lyapunov exponent and the reason of this is the presence of absorbing states.
Program: 11.00-11.45 Introductory lecture: Olla 11.45-12.00 Break 12.00-13.00 Seminar: Olla 13.00-14.30 Lunch 14.30-15.15 Introductory lecture: Rechtman 15.15-15.30 Break 15.30-16.30 Seminar: Rechtman
Organizers: F. Caravenna, N. Cancrini, E.N.M. Cirillo, P. Dai Pra, A. De Masi, D. Fanelli, F. Flandoli C. Giardina`, R. Livi, F. Martinelli, I.G. Minelli, F.R. Nardi, E. Presutti, B. Scoppola, E. Scoppola
Note pratiche: 1) E` importante prenotare il biglietto del treno il piu` presto possibile per poter usufruire di sconti!!!! 2) Stiamo prenotando un ristorante (diverso dal precedente) con un paio di menu fissi (di cui uno vegerariano) che possa ospitarci, ma ho bisogno del numero di persone che vogliono mangiare insieme. A coloro che fossero interessati chiedo di mandare un email a francescaromana.nardi@unifi.it Resta fermo il fatto che saremo in pieno centro di Firenze, quindi ci sono moltissime altre possibilita` per mangiare se preferite regolarvi indipendentemente.
Vi aspettiamo numerosi Francesca R. Nardi
Dipartimento di Matematica e Informatica Università degli Studi di Firenze Viale Morgagni 67, Firenze, Italy