Discretization-related issues in the Kardar-Parisi-Zhang equation: consistency, Galilean-invariance violation, and fluctuation-dissipation relation.

In order to perform numerical simulations of the Kardar-Parisi-Zhang (KPZ) equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a Hopf-Cole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the Hopf-Cole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between real-space and pseudospectral discrete representations. In addition we discuss the relevance of the Galilean-invariance violation in these consistent discretization schemes and the alleged conflict of standard discretization with the fluctuation-dissipation theorem, peculiar of one dimension.

Coupling-reentrant phase transition, complex hysteretic behavior, and efficiency optimization in coupled phase oscillators submitted to colored flashing potentials.

A recent mean-field analysis of a model consisting of N nonlinear phase oscillators-under the joint influence of global periodic coupling with strength K0 and of local multiplicative and additive noises-has shown a nonequilibrium phase transition towards a broken-symmetry phase exhibiting noise-induced transport, or "ratchet" behavior. In a previous paper we focused on the relationship between the character of the (mean velocity vs load force F) hysteresis loop, the number of "homogeneous" mean-field solutions, and the shape of the stationary mean-field probability distribution function (PDF). Here we assume that the multiplicative noises of the model are Ornstein-Uhlenbeck with common strength Q and self-correlation time tau. By resorting to an effective Markovian approximation, we study the tau dependence of the phase boundary, and that of the line signaling the transition from the "interaction-driven regime" to the "noise-driven regime." We also study-for selected representative points of the K0 vs Q phase diagram-the tau dependence of the transport properties induced by coupling and colored multiplicative noise (including the efficiency epsilon of the mechanical rectification process) and that of the above-mentioned PDF.

Transition from anomalous to normal hysteresis in a system of coupled Brownian motors: a mean-field approach.

We address a recently introduced model describing a system of periodically coupled nonlinear phase oscillators submitted to multiplicative white noises, wherein a ratchetlike transport mechanism arises through a symmetry-breaking noise-induced nonequilibrium phase transition. Numerical simulations of this system reveal amazing novel features such as negative zero-bias conductance and anomalous hysteresis, explained by performing a strong-coupling analysis in the thermodynamic limit. Using an explicit mean-field approximation, we explore the whole ordered phase finding a transition from anomalous to normal hysteresis inside this phase, estimating its locus, and identifying (within this scheme) a mechanism whereby it takes place.