Author(s):
Gonçalves, Patrícia 
Date: 2011
Persistent ID: http://hdl.handle.net/1822/16877
Origin: RepositóriUM - Universidade do Minho
Subject(s): Hyperbolic conservation law; Asymmetric zero-range and exclusion
Document details
A hyperbolic conservation law and Particle Systems
Description
In these notes we consider two particle systems: the totally asymmetric simple exclusion process and the totally asymmetric zero-range process. We introduce the notion of hydrodynamic limit and describe the partial differential equation that governs the evolution of the conserved quantity - the density of particles $\rho(t,\cdot)$. This equation is a hyperbolic conservation law of type $\partial_{t}\rho(t,u)+\nabla F(\rho(t,u))=0$, where the flux $F$ is a concave function. Taking these systems evolving on the Euler time scale $tN$, a Central Limit Theorem for the empirical measure holds and the temporal evolution of the limit density field is deterministic. By taking the system on a reference frame with constant velocity, the limit density field does not evolve in time. In order to have a non-trivial limit, time needs to be speeded up and for time scales smaller than $tN^{4/3}$ there is still no temporal evolution. As a consequence the current across a characteristic vanishes up to this longer time scale.
In these notes we consider two particle systems: the totally asymmetric simple exclusion process and the totally asymmetric zero-range process. We introduce the notion of hydrodynamic limit and describe the partial differential equation that governs the evolution of the conserved quantity - the density of particles $\rho(t,\cdot)$. This equation is a hyperbolic conservation law of type $\partial_{t}\rho(t,u)+\nabla F(\rho(t,u))=0$, where the flux $F$ is a concave function. Taking these systems evolving on the Euler time scale $tN$, a Central Limit Theorem for the empirical measure holds and the temporal evolution of the limit density field is deterministic. By taking the system on a reference frame with constant velocity, the limit density field does not evolve in time. In order to have a non-trivial limit, time needs to be speeded up and for time scales smaller than $tN^{4/3}$ there is still no temporal evolution. As a consequence the current across a characteristic vanishes up to this longer time scale.
Document Type
Article
Language English
Language English
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