Hydrodynamic Limit of Brownian Particles Interacting with Short and Long Range Forces

We investigate the time evolution of a model system of interacting particles, moving in a $d$-dimensional torus. The microscopic dynamics are first order in time with velocities set equal to the negative gradient of a potential energy term $\Psi$ plus independent Brownian motions: $\Psi$ is the sum of pair potentials, $V(r)+\gamma^d J(\gamma r)$, the second term has the form of a Kac potential with inverse range $\gamma$. Using diffusive hydrodynamical scaling (spatial scale $\gamma^{-1}$, temporal scale $\gamma^{-2}$) we obtain, in the limit $\gamma\downarrow 0$, a diffusive type integro-differential equation describing the time evolution of the macroscopic density profile.Comment: 37 pages, in TeX (compile twice), to appear on J. Stat. Phys., e-mail addresses: [email protected], [email protected]

Dynamics of Infinite Classical Anharmonic Crystals

We consider an unbounded lattice and at each point of this lattice an anharmonic oscillator, that interacts with its first neighborhoods via a pair potential V and is subjected to a restoring force of potential U . We assume that U and V are even nonnegative polynomials of degree 2σ1 and 2σ2. We study the time evolution of this system, with a control of the growth in time of the local energy, and we give a nontrivial bound on the velocity of propagation of a perturbation. This is an extension to the case σ1 < 2σ2 − 1 of some already known results obtained for σ1 ≥ 2σ2 − 1

Local mean field models of uniform to nonuniform density (fluid-crystal) transitions

We investigate the existence of nontranslation invariant (periodic) density profiles, for systems interacting via translation invariant long-range potentials, as minimizers of local mean field free energy functionals. The existence of a second-order transition from a uniform to a nonuniform density at a specified temperature beta(-1)(0) is proven for a class of model systems

From particle systems to the BGK equation

In [Phys. Rev. 94 (1954), 511–525] the authors introduced a kinetic equation (the BGK equation), effective in physical situations where the Knudsen number is small compared to the scales where Boltzmann’s equation can be applied, but not enough for using hydrodynamic equa- tions. In this paper, we consider the stochastic particle system (inhomoge- neous Kac model) underlying Bird’s direct simulation Monte Carlo method (DSMC), with tuning of the scaled variables yielding kinetic and/or hy- drodynamic descriptions. Although the BGK equation cannot be obtained from pure scaling, it does follow from a simple modification of the dynam- ics. This is proposed as a mathematical interpretation of some arguments in [Phys. Rev. 94 (1954), 511–525], complementing previous results in [Arch. Ration. Mech. Anal. 240 (2021), 785–808] and [Kinet. Relat. Mod- els 16 (2023), 269–293]

Front fluctuations for the stochastic Cahn–Hilliard equation

We consider the Cahn–Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of small intensity, and we investigate the effect of the noise on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given with probability going to one as the noise intensity vanishes, the solution remains close to a front for long times, and we study the fluctuations of the front in this time scaling. They are given by a one dimensional continuous process, self similar of order one and non-Markovian, related to a fractional Brownian motion and for which a couple of representations are given

Motion by mean curvature by scaling a nonlocal equation: Convergence at all times in the two-dimensional case

The convergence to a motion by mean curvature by diffusively scaling a nonlocal evolution equation, describing the macroscopic behavior of a ferromagnetic spin system with Kac interaction and Glauber dynamics has recently been proved. The convergence is proven up to the times when the motion by curvature is regular. Here we show the convergence at all times in the two-dimensional case. Since, in this case, the only singularity is the shrinking to a point of a closed curve, we verify that the curve actually disappears past the singularity. © 1994 Kluwer Academic Publishers

ON THE VALIDITY OF AN EINSTEIN RELATION IN MODELS OF INTERFACE DYNAMICS

We consider models of interface dynamics derived from Ising systems with Kac interactions and we prove the validity of the ''Einstein relation'' theta = musigma, where theta is the proportionality coefficient in the motion by curvature, mu is the interface mobility, and sigma is the surface tension

Note del corso di Meccanica Razionale

Il testo è una presentazione degli argomenti trattati nel corso di Meccanica Razionale per gli studenti della Laurea Triennale in Matematica dell’Università Sapienza di Roma

DIRECTED CURRENT IN QUASI-ADIABATICALLY AC-DRIVEN NONLINEAR SYSTEMS

We rigorously prove the existence of directed transport for a certain class of ac-driven nonlinear one-dimensional systems, namely the generation of transport with a preferred direction in the absence of a net driving force