E. Presutti

Liquid–Vapor Phase Transitions for Systems with Finite-Range Interactions

We consider particles in ℝd, d≥2, interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean-field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid–gas phase transition when the interaction range is finite but long compared to the interparticle spacing for a range of temperature.

A particle model for spinodal decomposition

We study a one-dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a “small” drift toward regions of higher concentration. We prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift. There is a critical valueac ofa. Fora<ac, the density values are unrestricted, while fora⩾ac, they should all be to the right or to the left of a given interval ℐ(a). The diffusion coefficient of the limiting equation can be continued analytically to ℐ(a), and, in the interior of ℐ(a), it has negative values which should correspond to particle aggregation phenomena. We also show that the dynamics can be obtained as a limit of a Kawasaki evolution associated to a Kac potential. The coefficienta plays the role of the inverse temperatureβ. The critical value ofa coincides with the critical inverse temperature in the van der Waals limit and ℐ(a) with the spinodal region. It is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic, the system evolves according to an integrodifferential equation. The instanton solutions of this equation, as studied by Dal Passo and De Mottoni, are then related to the phase transition region in the thermodynamic phase diagram; analogies with the Cahn-Hilliard equations are also discussed.

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.

Rigorous Proof of a Liquid-Vapor Phase Transition in a Continuum Particle System

We consider particles in

A stochastic particle system modeling the Carleman equation

{R}^{d},d\ensuremath{\ge}2

Macroscopic Stochastic Fluctuations in a One-Dimensional Mechanical System

, interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid-gas phase transition when the interaction range is finite but long compared to the interparticle spacing.

Energy levels of a nonlocal functional

Two species of Brownian particles on the unit circle are considered; both have diffusion coefficient σ>0 but different velocities (drift), 1 for one species and −1 for the other. During the evolution the particles randomly change their velocity: if two particles have the same velocity and are at distance ⩽ε (ε being a positive parameter), they both may simultaneously flip their velocity according to a Poisson process of a given intensity. The analogue of the Boltzmann-Grad limit is studied when ε goes to zero and the total number of particles increases like ε−1. In such a limit propagation of chaos and convergence to a limiting kinetic equation are proven globally in time, under suitable assumptions on the initial state. If, furthermore, σ depends on ε and suitably vanishes when ε goes to zero, then the limiting kinetic equation (for the density of the two species of particles) is the Carleman equation.

Glauber evolution with Kac potentials: II. Fluctuations

This paper concerns the long-time behavior of a one-dimensional mechanical system, hard rods with equal masses and lengths interacting by elastic collisions. We have noticed that for much longer times than those at which the Euler equation is valid macroscopic observables develop stochastic behavior; this might be contrasted with the expected picture based on the description of the long-time behavior of the system in terms of a Navier-Stokes correction to the Euler equation. We propose a scheme for defining an operator reminiscent of the wave operator and of the M611er morphism in scattering theory, which could be considered in one case as defining the Navier-Stokes correction, while still being meaningful when the Navier-Stokes description fails. The model we consider is a system of hard core particles on the line moving with constant velocities except for elastic collisions. This is a system for which the hydrodynamic limit makes sense, but the Euler equation is not of the usual form, as the model has as many local conserved quantities as velocities in the system. Nevertheless, one carries out the hydrodynamic limiting procedure in the same way as for more physically realistic models: one introduces a scaling parameter e and imposes the condition that the state at time zero varies on a spatial scale e-1. Rescaling the time by the same factor, one obtains the Euter equation in the limit e~0. ~3~,~4) The degenerate nature of this model is offset by its mathematical tractability.

Microscopic models of hydrodynamic behavior

We study the critical points of a nonlocal free energy functional. The functional has two minimizers (ground states) m(±) with zero energy. We prove that there is a first excited state identified as the instanton mL, and that above the energy of the instanton there is a gap. We also characterize parts of the basin of attraction of m(±) and mL under a dynamics associated to the free energy functional. The result completes the analysis of tunneling from m(−) to m(+).

Self-diffusion for particles with stochastic collisions in one dimension

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.