Irreversible phase transitions in contact processes with Lévy exchanges and long-range interactions.

Abstract

The contact process (CP) is generalized allowing the exchange of particles via L\'evy flights, where the flying length (l) is a random variable with a probability distribution given by P(l)\ensuremath{\propto}${\mathit{l}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$, where d is the spacial dimension and \ensuremath{\sigma} is the dimension of the random walk. The contact process with L\'evy flights (CPLF) exhibits irreversible phase transitions between an active state and a vacuum state. It is show that within the superdiffusive regime of the walkers (i.e., \ensuremath{\sigma}1), the L\'evy mechanism effectively build up additional long-range correlations, therefore the critical exponents of the CPLF model depart from those of the standard CP and they are tunable functions of \ensuremath{\sigma}. Comparison of the critical exponents characteristic of branching annihilating L\'evy walkers [E. Albano Europhys. Lett. 34, 97 (1996)] and those of the CPLF gives strong evidences on a universality class which comprises second order irreversible phase transitions in systems involving L\'evy exchanges and/or flights. It is suggested that the CPLF is equivalent to the standard CP with long-range interactions generated by a potential decaying with distance r as a power law of the form V(r)\ensuremath{\propto}${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathit{d}\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\sigma}}}$. \textcopyright{} 1996 The American Physical Society.