Long-range interactions generated by random Lévy flights: Spin-flip and spin-exchange kinetic Ising model in two dimensions
Huang-Jian Xu, B. Bergersen, Zoltàn Ràcz
A d=2 dimensional kinetic Ising model that evolves by a combination of spin flips and spin exchanges is investigated. The spin flips satisfy detailed balance for the equilibrium state of the Ising model at temperature T while the spin exchanges are random L\'evy flights of dimension \ensuremath{\sigma}=1.5. Our Monte Carlo (MC) simulations show that the steady state of this system displays a second-order phase transition as T is lowered. Comparing the critical fluctuations of the magnetization to those of an Ising model in which the interaction decays with distance as ${\mathit{r}}^{\mathrm{\ensuremath{-}}3.5}$, we find that, within the resolution of the MC data, the critical exponents and the scaling functions of the two systems coincide. We argue that this coincidence indicates that a recent conjecture about the random L\'evy flights generating long-range interaction of the form ${\mathit{V}}_{\mathrm{eff}}$(r)\ensuremath{\sim}${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$ is valid not only in the spherical limit and in d=1 but also in d=2.
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Mar 1, 1993•10 citations