Hydrodynamic limits for long-range asymmetric interacting particle systems

Abstract

We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on Z n , where the jump rates are asymmetric and long-range of order x −(n+α) 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 < α < 1, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when α ≥ 1, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.2010 Mathematics Subject Classification. 60K35.

Hydrodynamic limits: The emergence of fractional boundary conditions

“…Note that the infinite variance case corresponds to γ ∈ (0, 2) since, in this range, ∑ z z 2 p(z) = ∞. When p(⋅) is asymmetric, one can obtain an integro-PDE [49]. All these equations can be supplemented with several types of boundary conditions by superposing the dynamics described above with another one, for example, by 1.…”

Non-equilibrium stationary properties of the boundary driven zero-range process with long jumps

“…More recently, it has appeared several studies of interacting particle systems whose hydrodynamic limit is given by a fractional diffusion equation or a fractional conservation law [27,5,10,11,12,13,18,22,23,24,37,38]. In the models considered in those articles, the fractional nature is induced by the presence of non-local interactions in the microscopic dynamics (the reader can think for example of a system of independent random walks with a transition probability which has infinite variance).…”

Hydrodynamic behavior of long-range symmetric exclusion with a slow barrier: diffusive regime

“…By taking the anomalous time scale n γ , the hydrodynamic equation was given by the fractional heat equation. In [13] it was considered a general class of misanthrope interacting particle systems on Z d , which included both the exclusion process and the zero-range process. In this general class of models the dynamics conserved the number of particles.…”