Jorge A. Revelli

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.

Recent developments on the Kardar-Parisi-Zhang surface-growth equation

The stochastic nonlinear partial differential equation known as the Kardar–Parisi–Zhang (KPZ) equation is a highly successful phenomenological mesoscopic model of surface and interface growth processes. Its suitability for analytical work, its explicit symmetries and its prediction of an exact dynamic scaling relation for a one-dimensional substratum led people to adopt it as a ‘standard’ model in the field during the last quarter of a century. At the same time, several conjectures deserving closer scrutiny were established as dogmas throughout the community. Among these, we find the beliefs that ‘genuine’ non-equilibrium processes are non-variational in essence, and that the exactness of the dynamic scaling relation owes its existence to a Galilean symmetry. Additionally, the equivalence among planar and radial interface profiles has been generally assumed in the literature throughout the years. Here—among other topics—we introduce a variational formulation of the KPZ equation, remark on the importance of consistency in discretization and challenge the mainstream view on the necessity for scaling of both Galilean symmetry and the one-dimensional fluctuation–dissipation theorem. We also derive the KPZ equation on a growing domain as a first approximation to radial growth, and outline the differences with respect to the classical case that arises in this new situation.

KPZ equation: Galilean-invariance violation, consistency, and fluctuation-dissipation issues in real-space discretization

Strong constraints are drawn for the choice of real-space discretization schemes, using the known fact that the KPZ equation results from a diffusion equation (with multiplicative noise) through a Hopf-Cole transformation. Whereas the nearest-neighbor discretization passes the consistency tests, known examples in the literature do not. We emphasize the importance of the Lyapunov functional as natural starting point for real-space discretization and, in the light of these findings, challenge the mainstream opinion on the relevance of Galilean invariance.

Diffusion in fluctuating media: first passage time problem

Abstract We study the actual and important problem of Mean First Passage Time (MFPT) for diffusion in fluctuating media. We exploit van Kampens technique of composite stochastic processes , obtaining analytical expressions for the MFPT for a general system, and focus on the two state case where the transitions between the states are modelled introducing both Markovian and non-Markovian processes. The comparison between the analytical and simulations results show an excellent agreement.

Resonant phenomena in extended chaotic systems subject to external noise: The Lorenz’96 model case

We have investigated the effects of noise on an extended chaotic system. The chosen model is the Lorenz’96, a type of “toy” model used for climate studies. Through the analysis of the system’s time evolution and its time and space correlations, we have obtained numerical evidence for two distinct stochastic resonance-like behaviors. Such behaviors are seen when both the usual and a generalized signal-to-noise ratio functions are depicted as a function of the external noise intensity, or of the system size. The underlying mechanisms seem to be associated with a noise-induced chaos reduction. The possible relevance of these and other findings for an optimal climate prediction are discussed.

Bulk mediated surface diffusion: finite bulk case

Within the framework of a Master Equation scheme, we address the dynamics of adsorbed molecules (a fundamental issue in surface physics) and study the diffusion of particles in a finite cubic lattice whose boundaries are at the z = 1 and the z = L planes where L = 2; 3; 4; ..., while the x and y directions are unbounded. As we are interested in the effective diffusion process at the interface z = 1, we calculate analytically the conditional probability for finding the particle on the z = 1 plane as well as the surface dispersion as a function of time and compare these results with Monte Carlo simulations finding an excellent agreement. These results show that: there exists an optimal number of layers that maximizes 〈r2(t)〉 on the interface; for a small number the layers the long-time effective diffusivity on the interface is normal, crossing over abruptly towards a subdiffusive behavior as the number of layers increases.

Non Local Effects in the Sznajd Model: Stochastic resonance aspects

The Sznajd model is basically an Ising spin model, widely used in sociophysics studies as a simple mechanism for predicting decision-making in a closed community through interactions among the nearest neighbors. In the present work we aim to deepen our understanding of this model by analyzing not only local or first neighbor interactions but also long range ones. Besides, we consider the system as being subjected to two signals, a stochastically social internal one, mimicking “social temperature”, and an external periodic signal playing the role of the effects of fashion or propaganda. Under these conditions, we show the occurrence of a double stochastic resonance phenomenon when depicting signal-to-noise ratio as a function of both, the social temperature and the non local interaction parameters.

Optimal intermittent search strategies: smelling the prey

We study the kinetics of the search of a single fixed target by a searcher/walker that performs an intermittent random walk, characterized by different states of motion. In addition, we assume that the walker has the ability to detect the scent left by the prey/target in its surroundings. Our results, in agreement with intuition, indicate that the preys survival probability could be strongly reduced (increased) if the predator is attracted (or repelled) by the trace left by the prey. We have also found that, for a positive trace (the predator is guided towards the prey), increasing the inhomogeneitys size reduces the preys survival probability, while the optimal value of ? (the parameter that regulates intermittency) ceases to exist. The agreement between theory and numerical simulations is excellent.

Bulk-mediated surface diffusion: return probability in an infinite system

We analyse the dynamics of adsorbed molecules within the bulk-mediated surface diffusion framework. We consider that the particles desorption mechanism is characterized by a non-Markovian process, while the particles motion in the bulk is governed by Markovian dynamics, and also include the effect of a Markovian absorption probability on the surface. We study this system for the diffusion of particles in a semi-infinite lattice, analysing the return probability to the reference absorptive plane as well as the mean return time to such a surface. Comparisons with numerical simulations show an excellent agreement.