We introduce a model of interacting Random Walk, whose hopping amplitudedepends on the number of walkers/particles on the link. The mesoscopiccounterpart of such a microscopic dynamics is a diffusing system whosediffusivity depends on the particle density. A non-equilibrium stationary fluxcan be induced by suitable boundary conditions, and we show indeed that it ismesoscopically described by a Fourier equation with a density dependentdiffusivity. A simple mean-field description predicts a critical diffusivity ifthe hopping amplitude vanishes for a certain walker density. Actually, weevidence that, even if the density equals this pseudo-critical value, thesystem does not present any criticality but only a dynamical slowing down. Thisproperty is confirmed by the fact that, in spite of interaction, the particledistribution at equilibrium is simply described in terms of a product ofPoissonians. For mesoscopic systems with a stationary flux, a very effect ofinteraction among particles consists in the amplification of fluctuations,which is especially relevant close to the pseudo-critical density. This agreeswith analogous results obtained for Ising models, clarifying that largerfluctuations are induced by the dynamical slowing down and not by a genuinecriticality. The consistency of this amplification effect with altered colourednoise in time series is also proved.
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