Livio Triolo

Stability of the interface in a model of phase separation.

The paper is concerned with the asymptotic behaviour of the solutions to a nonlocal evolution equation which arises in models of phase separation. As in the Allen–Cahn equations, stationary spatially nonhomogeneous solutions exist, which represent the interface profile between stable phases. Local stability of these interface profiles is proved.

Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics

This is the first of three papers on the Glauber evolution of Ising spin systems with Kac potentials. We begin with the analysis of the mesoscopic limit, where space scales like the diverging range, gamma -1, of the interaction while time is kept finite: we prove that in this limit the magnetization density converges to the solution of a deterministic, nonlinear, nonlocal evolution equation. We also show that the long time behaviour of this equation describes correctly the evolution of the spin system till times which diverge as gamma to 0 but are small in units log gamma -1. In this time regime we can give a very precise description of the evolution and a sharp characterization of the spin trajectories. As an application of the general theory, we then prove that for ferromagnetic interactions, in the absence of external magnetic fields and below the critical temperature, on a suitable macroscopic limit, an interface between two stable phases moves by mean curvature. All the proofs are consequence of sharp estimates on special correlation functions, the v-functions, whose analysis is reminiscent of the cluster expansion in equilibrium statistical mechanics.

Motion by curvature by scaling nonlocal evolution equations

We prove convergence to a motion by mean curvature by scaling diffusively a nonlinear, nonlocal evolution equation. This equation was introduced earlier to describe the macroscopic behavior of a ferromagnetic spin system with Kac interaction which evolves with Glauber dynamics. The convergence is proven in any time interval in which the limiting motion is regular.

Glauber evolution with Kac potentials: III. Spinodal decomposition

This is the last of a series of papers on the Glauber dynamics of spin systems in with Kac potentials. It deals with phase separation, studying the evolution of an initial state which is a Bernoulli measure with zero average while the temperature of the Glauber dynamics is below the critical value. The state with 0 magnetization is then thermodynamically unstable and we prove that it is so also dynamically. In fact the stable phases, that have magnetization , develop into non-trivial patterns after times proportional to , the range of the Kac interaction. We characterize the typical spin configurations, both during the separation and when this is completed. In particular, we study the magnetization pattern at the boundaries of the clusters and the development of the interfaces.

Travelling fronts in nonlocal models for phase separation in an external field

We consider the one-dimensional, nonlocal, evolution equation derived by De Masi et al. (1995) for Ising systems with Glauber dynamics, Kac potentials and magnetic field. We prove the existence of travelling fronts, their uniqueness modulo translations among the monotone profiles and their linear stability for all the admissible values of the magnetic field for which the underlying spin system exhibits a stable and metastable phase.

Glauber evolution with Kac potentials: II. Fluctuations

In this paper we continue the analysis of the Glauber evolution in Ising systems with Kac interactions. In the first paper, we have proved that in a continuum limit, called the mesoscopic limit, the magnetization density converges to the solution of a non-local deterministic equation. Here we study the fluctuations around the limit proving convergence to a generalized Ornstein - Uhlenbeck process. We also prove asymptotic formulae for the correlation functions that improve those established in the previous paper and that will be used in a successive paper to study phase separation.

Drug user dynamics: a compartmental model of drug users for scenario analyses

A six-typology compartment model of trends in the use of illicit drugs in Italy is developed to evaluate policies and interventions through scenario analyses. Inside the big compartments of just-cannabis users and multiple drug users, three subpopulations are taken into account: the occasional, regular and intensive user. Moreover, compartments for health care, assisted persons together with a constant source (susceptible) are added; removed people (i.e. those who leave drugs for any reason) are put in a compartment which does not participate in the dynamics. Flows from the susceptible reservoir among the drug-user compartments and to the removed or the health care compartments are studied in a deterministic linear ODE framework, where parameters are estimated using Italian data. This system shows an evolution towards a steady state, the speed of convergence being dependent on parameters in a fairly visible way. This model represents a theoretical development in drug policy analysis, as it shows the relevance of flux parameters, which are in principle subject to modifications due to institutional efforts, together with a quantitative evaluation of their role in the evolution of the whole system.

Molecular dynamics simulation of vascular network formation

Endothelial cells are responsible for the formation of the capillary blood vessel network. We describe a system of endothelial cells by means of two-dimensional molecular dynamics simulations of point-like particles. Cell motion is governed by the gradient of the concentration of a chemical substance that they produce (chemotaxis). The typical time of degradation of the chemical substance introduces a characteristic length in the system. We show that point-like model cells form network resembling structures tuned by this characteristic length, before collapsing altogether. Successively, we improve the non-realistic point-like model cells by introducing an isotropic repulsive force between them and a velocity dependent force mimicking the observed peculiarity of endothelial cells of preserving the direction of their motion (persistence). This more realistic model does not show a clear network formation. We ascribe this partial fault in reproducing the experiments to the static geometry of our model cells that, in reality, change their shapes by elongating toward neighboring cells.

The competition-diffusion limit of a stochastic growth model

A competition-diffusion system, where populations of healthy and ill cells compete and move on a neutral matrix, is analyzed. A coupled system of nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The space dependent solutions show a behavior markedly different from the associated ODE system. For a large class of initial conditions, the asymptotic behavior of the system can be described through the analysis of associated travelling waves.

Quantum Statistical Mechanics: Overview

An overview of the birth of quantum statistical mechanics is given;the path from the early second quantization to more recent developments is traced,with some reference to the mathematical tools and the present challenges.