Interacting random walkers and non-equilibrium fluctuations

We introduce a model of interacting Random Walk, whose hopping amplitude depends on the number of walkers/particles on the link. The mesoscopic counterpart of such a microscopic dynamics is a di_using system whose di_usivity depends on the particle density. A non-equilibrium stationary ux can be induced by suitable boundary conditions, and we show indeed that it is mesoscopically described by a Fourier equation with a density dependent di_usivity. A simple mean-_eld description predicts a critical di_usivity if the hopping amplitude vanishes for a certain walker density. Actually, we evidence that, even if the density equals this pseudo-critical value, the system does not present any criticality but only a dynamical slowing down. This property is con_rmed by the fact that, in spite of interaction, the particle distribution at equilibrium is simply described in terms of a product of Poissonians. For mesoscopic systems with a stationary ux, a very e_ect of interaction among particles consists in the ampli_cation of uctuations, which is especially relevant close to the pseudo-critical density. This agrees with analogous results obtained for Ising models, clarifying that larger uctuations are induced by the dynamical slowing down and not by a genuine criticality. The consistency of this ampli_cation e_ect with altered coloured noise in time series is also proved.