Hydrodynamic limits for long-range asymmetric interacting particle systems

TL;DR

This paper studies the large-scale behavior of long-range asymmetric particle systems on bZ^n, deriving new integro-PDEs or Burgers equations depending on the decay rate of jump probabilities.

Contribution

It introduces a unified framework for analyzing hydrodynamic limits of long-range asymmetric systems, revealing different PDEs based on the decay parameter lpha.

Findings

For 0<lpha<1, a new integro-PDE describes the anomalous hydrodynamics.

For lpha1, the classical Burgers equation emerges in the hydrodynamic limit.

The results extend hydrodynamic limit theory to systems with long-range interactions.

Abstract

We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on ${\mathbb Z}^n$, where the jump rates are asymmetric and long-range of order $\|x\|^{-(n+\alpha)}$ for a particle displacement of order $\|x\|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<\alpha<1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $\alpha\geq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.