Dynamical generation of long-range interactions: Random Levy flights in the kinetic Ising and spherical models

We study nonequilibrium steady states produced by a competition between anomalous diffusion (random Levy-flight exchanges) and local dynamics satisfying detailed balance for the equilibrium state of either the Ising chain or the spherical model. The phase transitions occurring in these systems are investigated via simulations and an exact solution. The results show that the steady-state properties are dominated by effective long-range interactions decaying with distance as ${\mathit{r}}^{\mathrm{-}\mathit{d}\mathrm{-}\mathrm{\sigma }}$, where d is the spatial dimension while σ is the dimension of the Levy flight.