Hydrodynamic limits for long-range asymmetric interacting particle systems

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 $\mathrm{\parallel }x{\mathrm{\parallel }}^{-(n+\alpha )}$ for a particle displacement of order $\mathrm{\parallel }x\mathrm{\parallel }$. 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 $\alpha \ge 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.