Anna De Masi

Coexistence of Ordered and Disordered Phases in Potts Models in the Continuum

This is the second of two papers on a continuum version of the Potts model, where particles are points in ℝd, d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ−1, γ>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.

Quasi-Static Hydrodynamic Limits

We consider hydrodynamic limits of interacting particles systems with open boundaries, where the exterior parameters change in a time scale slower than the typical relaxation time scale. The limit deterministic profiles evolve quasi-statically. These limits define rigorously the thermodynamic quasi static transformations also for transitions between non-equilibrium stationary states. We study first the case of the symmetric simple exclusion, where duality can be used, and then we use relative entropy methods to extend to other models like zero range systems. Finally we consider a chain of anharmonic oscillators in contact with a thermal Langevin bath with a temperature gradient and a slowly varying tension applied to one end.

Potts Models in the Continuum. Uniqueness and Exponential Decay in the Restricted Ensembles

In this paper we study a continuum version of the Potts model, where particles are points in ℝd, d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ−1, γ>0. In mean field, for any inverse temperature β there is a value of the chemical potential λβ at which S+1 distinct phases coexist. We introduce a restricted ensemble for each mean field pure phase which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin-Shlosman theory (Dobrushin and Shlosman in J. Stat. Phys. 46(5–6):983–1014, 1987), we prove that while the Dobrushin high-temperatures uniqueness condition does not hold, yet a finite size condition is verified for γ small enough which implies uniqueness and exponential decay of correlations. In a second paper (De Masi et al. in Coexistence of ordered and disordered phases in Potts models in the continuum, 2008), we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures.

Fourier Law, Phase Transitions and the Stationary Stefan Problem

We study the one-dimensional stationary solutions of the integro-differential equation which, as proved in Giacomin and Lebowitz (J Stat Phys 87:37–61, 1997; SIAM J Appl Math 58:1707–1729, 1998), describes the limit behavior of the Kawasaki dynamics in Ising systems with Kac potentials. 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 its validity however in the thermodynamic limit where the limit profile is again monotone away from the interface.

Super-hydrodynamic limit in interacting particle systems

This paper is a follow-up of the work initiated in (Arab J Math, 2014), where we 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 as a family of linear macroscopic profiles parameterized by their mass. Then we prove that beyond hydrodynamics there exists a longer time scale where the evolution becomes random. On such a super-hydrodynamic scale the particle system is at each time close to the stationary state with same mass and the mass fluctuates performing a Brownian motion reflected at the origin.

Tunnelling in Nonlocal Evolution Equations

Abstract We study “tunnelling” in a one-dimensional, nonlocal evolution equation by assigning a penalty functional to orbits which deviate from solutions of the evolution equation. We discuss the variational problem of computing the minimal penalty for orbits which connect two stable, stationary solutions.

Slow Motion and Metastability for a Nonlocal Evolution Equation

In this paper we consider a nonlocal evolution equation in one dimension, which describes the dynamics of a ferromagnetic system in the mean field approximation. In the presence of a small magnetic field, it admits two stationary and homogeneous solutions, representing the stable and metastable phases of the physical system. We prove the existence of an invariant, one dimensional manifold connecting the stable and metastable phases. This is the unstable manifold of a distinguished, spatially nonhomogeneous, stationary solution, called the critical droplet.(4, 10) We show that the points on the manifold are droplets longer or shorter than the critical one, and that their motion is very slow in agreement with the theory of metastable patterns. We also obtain a new proof of the existence of the critical droplet, which is supplied with a local uniqueness result.

Non-equilibrium Stationary States in the Symmetric Simple Exclusion with Births and Deaths

This paper is a follow-up of the study initiated in [1], [2], where current reservoirs in the context of stochastic interacting particle systems have been proposed as a method to investigate stationary non-equilibrium states with steady currents produced by action at the boundary. Due to the particular difficulties in implementing this new method, we consider the simplest possible particle system. The bulk dynamics is the symmetric simple exclusion process (SSEP) in the interval ΛN = [−N,N ] ∩ Z (N a positive integer and N → ∞ eventually), namely the state space is {0, 1}ΛN (at most one particle per site): independently each particle tries to jump at rate N/2 to each one of its nearest neighbor (n.n.) sites, the jump then takes place if and only if the chosen site is empty, jumps outside ΛN are suppressed. To induce a current we send in particles from the right and take them out from the left, and would like this to happen at rate Nj/2, j > 0 a fixed parameter independent of N . Due to the restrictions imposed by the configurational space, we have to be more precise when defining this dynamics. For this we fix a parameter K ≥ 1 (an integer) and two intervals I± of length K at the boundaries: I+ ≡ [N −K + 1, N ] and I− ≡ [−N,−N +K − 1]. At rate Nj/2, when I+ is not totally occupied, we create a particle at its rightmost empty site; with the same rate, unless I− is empty, we take out a particle from its leftmost occupied site. In case I+ is already full, or I− empty, the corresponding mechanism aborts. In [1], [2] we have proved that at any time t > 0 propagation of chaos holds and that in the limit N → ∞ the hydrodynamical equation is the linear heat equation:

Extinction time for a random walk in a random environment

We consider a random walk with death in [−N, N] moving in a time dependent environment. The environment is a system of particles which describes a current flux from N to −N. Its evolution is influenced by the presence of the random walk and in turns it affects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in N) for the survival probability up to time t which goes as c exp{−bN−2 t}, with c and b positive constants.

Exponential rate of convergence in current reservoirs

In this paper we consider a family of interacting particle systems on [−N,N] that arises as a natural model for current reservoirs and Fick’s law. We study the exponential rate of convergence to the stationary measure, which we prove to be of the order N −2 .