Biased Diffusion with Correlated Noise

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 d _LR, the long-ranged noise is always relevant. In contrast, for d<d _LR, 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.     

Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes

Motivated by recent studies of gas permeation through polymer networks, we consider a collection of ordinary random walks of fixed length ℓ, placed randomly on the bonds of a square lattice. These walks model polymers, each with ℓ segments. Using computer simulations, we find the critical concentration of occupied bonds (i.e., the critical occupation probability) for such a network to percolate the system. Though this threshold decreases monotonically with ℓ, the critical “mass” density, defined as the total number of segments divided by total number of bonds in the system, displays a more complex behavior. In particular, for fixed mass densities, the percolation characteristics of the network can change several times, as shorter polymers are linked to form longer ones.     

High Temperature Expansion for a Driven Bilayer System

Based directly on the microscopic lattice dynamics, a simple high temperature expansion can be devised for non-equilibrium steady states. We apply this technique to investigate the disordered phase and the phase diagram for a driven bilayer lattice gas at half filling. Our approximation captures the phases first observed in simulations, provides estimates for the transition lines, and allows us to compute signature observables of non-equilibrium dynamics, namely, particle and energy currents. Its focus on non-universal quantities offers a useful analytic complement to field-theoretic approaches.     

Field theory of critical behaviour in driven diffusive systems

We present a field theoretic renormalisation group study for the critical behaviour of a diffusive system with a single conserved density subjected to an external driving force. The anisotropies induced by the external field require the introduction of two critical parameters associated with transverse and longitudinal order. The transition to transverse order is governed by a fixed point which is infrared stable below five dimensions. With the help of Ward-Takahashi identities based on Galilei invariance, we derive scaling forms for density correlation functions, critical exponents to all orders in =5–d, and the equation of state, taking care of a dangerous irrelevant composite operator. The transition is continuous and of mean-field type, with anomalous long-wavelength and long-time correlations in the longitudinal direction only. For the transition to longitudinal order, no infrared stable fixed point is found. An analysis of the mean-field equations indicates that the transition is discontinuous.     

Field theory of long time behaviour in driven diffusive systems

We present a renormalisation group study for the long time behaviour of a diffusive system with a single conserved density which is subjected to an external driving force. In the asymptotic long wavelength limit the system approaches an infrared stable fixed point where detailed balance is satisfied. We obtain the exact scaling form of the density correlation function. In one dimension, the corresponding universal amplitude agrees excellently with a recent Monte Carlo simulation.     

Power spectra of the total occupancy in the totally asymmetric simple exclusion process

As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of nonequilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a 1-dimensional open lattice and its power spectrum. Using both Monte Carlo simulations and analytic methods, we explore its behavior in different characteristic regimes. In the maximal current phase and on the coexistence line (between high and low density phases), the power spectrum displays algebraic decay, with exponents -1.62 and -2.00, respectively. Deep within the high and low density phases, we find pronounced oscillations, which damp into power laws. This behavior can be understood in terms of driven biased diffusion with conserved noise in the bulk.     

On singularities in the disordered phase of a driven diffusive system

Using renormalization group techniques, we investigate the large distance behavior of a driven, interacting lattice gas in the disordered phase. Unlike the equilibrium Ising model, its behavior, ford>2, is controlled by aline of fixed points, each of which is interpreted as a dynamical system violating the fluctuation-dissipation theorem (FDT). As a consequence, correlation functions at large distances typically decay according to a power law instead of an exponential. Ford2, the renormalization group flows towards an FDT-satisfying fixed point, which corresponds to the high-temperature, strong-drive limit. In the steady state of such a model (a driven, free lattice gas), correlations are known to be exactly zero. Nevertheless, our correlations are still dominated by power laws, since the FDT-breaking operators aredangerously irrelevant (marginal ind=2). Thus, for anyd, the long wavelength properties cannot be obtained by taking either the non-interacting or theT limit, unlike for the equilibrium Ising model.