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.

System size stochastic resonance: general nonequilibrium potential framework.

We study the phenomenon of system size stochastic resonance within the nonequilibrium potential framework. We analyze three different cases of spatially extended systems, exploiting the knowledge of their nonequilibrium potential, showing that through the analysis of that potential we can obtain a clear physical interpretation of this phenomenon in wide classes of extended systems. Depending on the characteristics of the system, the phenomenon is associated with a breaking of the symmetry of the nonequilibrium potential or a deepening of the potential minima yielding an effective scaling of the noise intensity with the system size.

Stochastic resonance between dissipative structures in a bistable noise-sustained dynamics.

We study an extended system that without noise shows a monostable dynamics, but when submitted to an adequate multiplicative noise, an effective bistable dynamics arises. The stochastic resonance between the attractors of the noise-sustained dynamics is investigated theoretically in terms of a two-state approximation. The knowledge of the exact nonequilibrium potential allows us to obtain the output signal-to-noise ratio. Its maximum is predicted in the symmetric case for which both attractors have the same nonequilibrium potential value.

Limit cycle induced by multiplicative noise in a system of coupled Brownian motors.

We study a model consisting of N nonlinear oscillators with global periodic coupling, and local multiplicative and additive noises. The model was shown to undergo a nonequilibrium phase transition towards a broken-symmetry phase exhibiting noise-induced "ratchet" behavior. A previous study [H. S. Wio, S. Mangioni, and R. Deza, Physica D 168-169, 184 (2002)] focused on the relationship between the character of the hysteresis loop, the number of "homogeneous" mean-field solutions, and the shape of the stationary mean-field probability distribution function. Here, we show-as suggested by the absence of stable solutions when the load force is beyond a critical value-the existence of a limit cycle induced by both multiplicative noise and global periodic coupling.

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.

Experimental evidence of stochastic resonance without tuning due to non-Gaussian noises.

In order to test theoretical predictions, we have studied the phenomenon of stochastic resonance in an electronic experimental system driven by white non-Gaussian noise. In agreement with the theoretical predictions our main findings are an enhancement of the sensibility of the system together with a remarkable widening of the response (robustness). This implies that even a single resonant unit can reach a marked reduction in the need for noise tuning.

Pattern dynamics in bidimensional oscillatory media with bistable inhomogeneities.

By means of numerical simulations, we study pattern dynamics in selected examples of inhomogeneous active media described by a reaction diffusion model of the activator-inhibitor type. We consider inhomogeneities corresponding to a variation in space of the (nonlinear) reaction characteristics of the system or the diffusion constants. Three different bidimensional systems are analyzed: an oscillatory medium in a square reactor with a circular central bistable domain, and cases of a bistable stripe immersed in an oscillatory medium in a trapezoidal reactor and in a rectangular reactor with inhomogeneous diffusion. The different types of complex behavior that arise in these systems are analyzed.

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.

Stochastic resonance in extended bistable systems: the role of potential symmetry.

We study the role of potential symmetry in a three-field reaction-diffusion system presenting bistability by means of a two-state theory for stochastic resonance in general asymmetric systems. By analyzing the influence of different parameters in the optimization of the signal-to-noise ratio, we observe that this magnitude always increases with the symmetry of the system's potential, indicating that it is this feature which governs the optimization of the system's response to periodic signals.

Exact expression for the diffusion propagator in a family of time-dependent anharmonic potentials.

We have obtained the exact expression of the diffusion propagator in the time-dependent anharmonic potential V(x,t)=1 / 2a(t)x(2)+b ln x. The underlying Euclidean metric of the problem allows us to obtain analytical solutions for a whole family of the elastic parameter a(t), exploiting the relation between the path integral representation of the short time propagator and the modified Bessel functions. We have also analyzed the conditions for the appearance of a nonzero flow of particles through the infinite barrier located at the origin (b<0).

Reaction kinetics of diffusing particles injected into a reactive substrate.

We analyze the kinetics of trapping (A+B-->B) and annihilation (A+B-->0) processes on a one-dimensional substrate with homogeneous distribution of immobile B particles while the A particles are supplied by a localized source. For the imperfect reaction case, we analyze both problems by means of a stochastic model and compare the results with numerical simulations. In addition, we present the exact analytical results of the stochastic model for the case of perfect trapping.

An exact analytical solution of a three-component model for competitive coexistence.

A three-component competition system is modeled as a reaction-diffusion process. An exact analytical solution has been found that indicates that in certain situations the classical results on extinction and coexistence of Lotka-Volterra-type equations are no longer valid. Cases with one or both predators diffuse are analyzed, and the stability question is discussed.