arXiv subjects
Publications and source records attributed to Errico Presutti.
In this paper we study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic field which vanishes as $|x|^{-α}$, $α>0$, as $|x|\to \infty$. We prove that in dimensions $d\ge 2$ for all $β$ large enough if $α>1$ there is a phase transition while if $α<1$ there is a unique DLR state.
We study a free boundary problem which arises as the continuum version of a stochastic particles system in the context of Fourier law. Local existence and uniqueness of the classical solution are well known in the literature of free boundary problems. We introduce the notion of generalized solutions (which extends that of classical solutions when the latter exist) and prove global existence and uniqueness of generalized solutions for a large class of initial data. The proof is obtained by characterizing a generalized solution as the unique element which separates suitably defined lower and upper barriers in the sense of mass transport inequalities.
We consider the one dimensional symmetric simple exclusion process (SSEP) with additional births and deaths restricted to a subset of configurations where there is a leftmost hole and a rightmost particle. At a fixed rate birth of particles occur at the position of the leftmost hole and at the same rate, independently, the rightmost particle dies. We prove convergence to a hydrodynamic limit and discuss its relation with a free boundary problem.
This paper is a follow-up of the work initiated in [3], where it has been investigated the hydrodynamic limit of symmetric independent random walkers with birth at the origin and death at the rightmost occupied site. Here we obtain two further results: first we characterize the stationary states on the hydrodynamic time scale and show that they are given by a family of linear macroscopic profiles whose parameters are determined by the current reservoirs and the system mass. Then we prove the existence of a super-hyrdrodynamic time scale, beyond the hydrodynamic one. On this larger time scale the system mass fluctuates and correspondingly the macroscopic profile of the system randomly moves within the family of linear profiles, with the randomness of a Brownian motion.
We study a system of particles in the interval $[0,ε^{-1}] \cap \mathbb Z$, $ε^{-1}$ a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously new particles are injected at site 0 at rate $jε$ ($j>0$) and removed at same rate from the rightmost occupied site. The removal mechanism is therefore of topological rather than metric nature. The determination of the rightmost occupied site requires a knowledge of the entire configuration and prevents from using correlation functions techniques. We prove using stochastic inequalities that the system has a hydrodynamic limit, namely that under suitable assumptions on the initial configurations, the law of the density fields $ε\sum ϕ(εx) ξ_{ε^{-2}t}(x)$ ($ϕ$ a test function, $ξ_t(x)$ the number of particles at site $x$ at time $t$) concentrates in the limit $ε\to 0$ on the deterministic value $\int ϕρ_t$, $ρ_t$ interpreted as the limit density at time $t$. We characterize the limit $ρ_t$ as a weak solution in terms of barriers of a limit free boundary problem.
We consider the symmetric simple exclusion process in the interval $\La_N:=[-N,N]\cap\mathbb Z$ with births and deaths taking place respectively on suitable boundary intervals $I_+$ and $I_-$, as introduced in De Masi et al. (J. Stat. Phys. 2011). We study the stationary measure density profile in the limit $N\to\infty$
We consider the symmetric simple exclusion process in the interval $[-N,N]$ with additional birth and death processes respectively on $(N-K,N]$, $K>0$, and $[-N,-N+K)$. The exclusion is speeded up by a factor $N^2$, births and deaths by a factor $N$. Assuming propagation of chaos (a property proved in a companion paper "Truncated correlations in the stirring process with births and deaths") we prove convergence in the limit $N\to \infty$ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.
We consider the stirring process in the interval $\La_N:=[-N,N]$ of $\mathbb Z$ with births and deaths taking place in the intervals $I_+:=(N-K,N]$, $K>0$, and respectively $I_-:=[-N,-N+K)$. We prove bounds on the truncated moments uniform in $N$ which yield strong factorization properties.
We study the one-dimensional stationary solutions of an integro-differential equation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising systems with Kac potentials, \cite{GiacominLebowitz}. We construct stationary solutions with non zero current and prove the validity of the Fourier law in the thermodynamic limit showing that below the critical temperature the limit equilibrium profile has a discontinuity (which defines the position of the interface) and satisfies a stationary free boundary Stefan problem. Under-cooling and over-heating effects are also studied. We show that if metastable values are imposed at the boundaries then the mesoscopic stationary profile is no longer monotone and therefore the Fourier law is not satisfied. It regains however its validity in the thermodynamic limit where the limit profile is again monotone away from the interface.
The relation between thermostats of "isoenergetic" and "frictionless" kind is studied and their equivalence in the thermodynamic limit is proved in space dimension $d=1,2$ and, for special geometries, $d=3$.
This is the second of two papers on a continuum version of the Potts model, where particles are points in $\mathbb R^d$, $d\ge 2$, with a spin which may take $S\ge 3$ possible values. Particles with different spins repel each other via a Kac pair potential of range $\ga^{-1}$, $\ga>0$. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist $S+1$ mutually distinct DLR measures.