Untitled

Abstract

The diffusion of hard-core particles subject to a global bias is described by a nonlinear, anisotropic generalization of the diffusion equation with conserved, local noise. Using renormalization group techniques, we analyze the effect of an additional noise term, with spatially long-ranged correlations, on the long-time, long-wavelength behavior of this model. Above an upper critical dimension dLR, the long-ranged noise is always relevant. In contrast, for d < dLR, we find a "weak noise" regime dominated by short-range noise. As the range of the noise correlations increases, an intricate sequence of stability exchanges between different fixed points of the renormalization group occurs. Both smooth and discontinuous crossovers between the associated universality classes are observed, reflected in the scaling exponents. We discuss the necessary techniques in some detail since they are applicable to a much wider range of problems.

Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem

“…Alternatively, one may utilize the mapping to the Burgers equation and hence to driven diffusive systems, for which a well-defined (2 − d) expansion exists. Adding long-range correlated noise, this actually leads to the identical stability condition ρ < 1/4 for the shortrange fixed point in d = 1 [30]. In the long-range regime, the case ρ = 1, corresponding to the Burgers equation with non-conserved noise, is accessible through an ε expansion below the upper critical dimension d uc = 4 of this model [2].…”

“…First, from the RG analysis exploiting the Cole-Hopf transformation we have learned that the strong-coupling phase above d c (ρ) is not accessible by perturbation theory even to infinite order. On the other hand, the rough phase at d = 1 is accessible by standard perturbation theory using a mapping of the KPZ equation to a driven diffusion model [30]. Second, for ρ = 0 an explicit two-loop calculation [8] shows that the fixed-point value of the coupling constant g approaches infinity as the lower critical dimension approaches 2 from below.…”

Driven diffusive system with non-local perturbations

“…As the short-range limit is approached, this deviation is small and is treated on equal footing with ǫ. The problem of driven diffusive system with a long-range spatially correlated noise [15] is more intricate due to the existence of two fixed points in the short-range case and only one fixed point in the corresponding long-range version. In our case, the discontinuity related problem is naturally taken care of by the local term that is generated.…”

“…In driven diffusive systems, the noise is usually considered to have a Gaussian distribution with a short-range correlation in space and time. Though a long-range spatial correlation in the noise in the high temperature version of the KLS model [14] exhibits interesting crossover behavior [15] with the increase of the range of the noise correlation, the effect of a long-range temporal correlation near the critical point need not be so. We have already pointed out that several of the known results depend crucially on the Galilean invariance [9] ensuing from the delta-correlation of the noise in time.…”