Euler hydrodynamics of one-dimensional attractive particle systems

Abstract

We consider attractive irreducible conservative particle systems on $\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1--30], from which we relax the assumption that the process has explicit invariant measures.Comment: Published at http://dx.doi.org/10.1214/009117906000000115 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

q-Zero Range has Random Walking Shocks

“…This has in fact been proved for several cases of the rates g, see e.g. Rezakhanlou [27] or Bahadoran, Guiol, Ravishankar and Saada [2] for details. It has also been shown in [8] that H is convex, resp.…”

Hydrodynamics in a condensation regime: The disordered asymmetric zero-range process

“…The measure (6) is always supported on X if β ∈ (0, c) ∪ {0} (11) When β = c > 0, conventions (9)-(10) yield a measure supported on configurations with infinitely many particles at all sites x ∈ Z that achieve the infimum in (7), and finitely many particles at other sites. In particular, this measure is supported on X when the infimum in (7) is not achieved.…”

Constructive Euler Hydrodynamics for One-Dimensional Attractive Particle Systems

“…In this paper, we review successive stages ( [6,7,8,38,9]) of a constructive approach to hydrodynamic limits given by equations of the type (1), which ultimately led us in [9] to a very general hydrodynamic limit result for attractive particle systems in one dimension in ergodic random environment.…”

“…Our method is based on an interplay of macroscopic properties for the conservation law and analogous microscopic properties for the particle system. The next stage, achieved in [7], was to derive hydrodynamics (which was still a weak law) for attractive processes without explicit invariant measures. In the same setting, we then obtained almost sure hydrodynamics in [8], relying on a graphical representation of the dynamics.…”

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We review a (constructive) approach first introduced in [6] and further developed in [7,8,38,9] for hydrodynamic limits of asymmetric attractive particle systems, in a weak or in a strong (that is, almost sure) sense, in an homogeneous or in a quenched disordered setting.

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