C. Boldrighini

One-Dimensional Hard Rod Caricature of Hydrodynamics

We give here a rigorous deduction of the “hydrodynamic” equation which holds in the hydrodynamic limit, for a model system of one-dimensional identical hard rods interacting through elastic collisions. The equation should be considered as the analog of the Euler equation of real hydrodynamics. Owing to the degeneracy of the model, it is written in terms of a functiong(q, v, t) expressing the density of particles with velocityv at the pointq at timet. For this equation we prove an existence and uniqueness theorem in some natural class of functions. Our main result is the proof that if {∈, ∈ >0} is a class of initial states which are homogeneous on a scale much less than ε−1, and if the corresponding particle densities tend, asε→0, in the proper scale, to the initial hydrodynamic densitygo(q,v), then, under some general assumptions on the states ∈− and ong0, the particle densities of the evolved states at timeε−1t, tend asε→0 to the unique solution of the hydrodynamic equation with initial conditiong0. The proof is completed by exhibiting a large class of initial families {∈, ∈ >0} which possess the required properties.

A five-dimensional truncation of the plane incompressible Navier-Stokes equations

A five-modes truncation of the Navier-Stokes equations for a two dimensional incompressible fluid on a torus is considered. A computer analysis shows that for a certain range of the Reynolds number the system exhibits a stochastic behaviour, approached through an involved sequence of bifurcations.

Computer simulation of shock waves in the completely asymmetric simple exclusion process

AbstractWe study the evolution of the completely asymmetric simple exclusion process in one dimension, with particles moving only to the right, for initial configurations corresponding to average densityρ− (ρ+) left (right) of the origin,ρ−⩽ρ+. The microscopic shock position is identified by introducing a “second-class” particle. Results indicate that the shock profile is stable, and that the distribution as seen from the shock positionN(t) tends, as time increases, to a limiting distribution, which is locally close to an equilibrium distribution far from the shock. Moreover

Nonequilibrium fluctuations in particle systems modelling reaction-diffusion equations

N(t)_{\frown \,}^\smile V \cdot t

Drift and diffusion for a mechanical system

, withV=1−ρ−−ρ+, as predicted, and the dispersion ofN(t), σ2(t), behaves linearly, for not too small values ofρ+−ρ−, i.e.,

Convergence to a stationary state and diffusion for a charged particle in a standing medium

\sigma ^2 (t)_{\frown \,}^\smile S \cdot t