Limit velocity for a driven particle in a random medium with mass aggregation

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Abstract

We study a one-dimensional infinite system of particles driven by a constant positive force F which acts only on the leftmost particle which is regarded as the tracer particle (t.p.). All other particles are field neutral, do not interact among themselves, and independently of each other with probability 0<p≤1 are either perfectly inelastic and “stick” to the t.p. after the first collision, or with probability 1−p are perfectly elastic, mechanically identical and have the same mass m. At initial time all particles are at rest, and the initial measure is such that the interparticle distances ξi's are i.i.d. r.v.'s. with absolutely continuous density. We show that for any value of the field F>0, the velocity of the t.p. converges to a limit value, which we compute.

Résumé

On étudie un système de particules infini unidimensionnel soumis à une force constante positive F qui agit seulement sur la particule la plus à gauche, vue comme la particule traceuse (p.t.).

Toutes les autres particules sont en champ neutre, et n'interagissent pas entre elles. Indépendamment unes des autres, elles sont soit, avec probabilité 0<p<1, parfaitement inélastiques et “se collent” à la p.t. après première collision, soit, avec probabilité 1−p, parfaitement élastiques, mécaniquement identiques avec même masse m. A l'origine toutes les particules sont au repos, et la mesure initiale est telle que les distances interparticules ξi sont des variables aléatoires i.i.d. de densité absolument continue. On montre que pour toute valeur du champ F>0, la vitesse de la p.t. converge vers une valeur limite que nous calculons.

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