Diffusion in Fluctuating Media: The Resonant Activation Problem

We present a one-dimensional model for diffusion in a fluctuating lattice; that is a lattice which can be in two or more states. Transitions between the lattice states are induced by a combination of two processes: one periodic deterministic and the other stochastic. We study the dynamics of a system of particles moving in that medium, and characterize the problem from different points of view: mean first passage time (MFPT), probability of return to a given site ($P_{s_0}$), and the total length displacement or number of visited lattice sites ($\Lambda$). We observe a double {\it resonant activation}-like phenomenon when we plot the MFPT and $P_{s_0}$ as functions of the intensity of the transition rate stochastic component.Comment: RevTex, 15 pgs, 8 figures, submitted to Eur.Phys.J.

Noise effects in extended chaotic system: study on the Lorenz'96 model

We investigate the effects of a time-correlated noise on an extended chaotic system. The chosen model is the Lorenz'96, a kind of toy model used for climate studies. The system is subjected to both temporal and spatiotemporal perturbations. Through the analysis of the system's time evolution and its time correlations, we have obtained numerical evidence for two stochastic resonance-like behaviors. Such behavior is seen when a generalized signal-to-noise ratio function are depicted as a function of the external noise intensity or as function of the system size. The underlying mechanism seems to be associated to a noise-induced chaos reduction. The possible relevance of those findings for an optimal climate prediction are discussed, using an analysis of the noise effects on the evolution of finite perturbations and errors.Comment: To appear in Statistical Mechanics Research Focus, Special volume (Nova Science Pub., NY, in press) (LaTex, 16 pgs, 14 figures

Bulk Mediated Surface Diffusion: Non Markovian Desorption with Finite First Moment

Here we address a fundamental issue in surface physics: the dynamics of adsorbed molecules. We study this problem when the particle's desorption is characterized by a non Markovian process, while the particle's adsorption and its motion in the bulk are governed by a Markovian dynamics. We study the diffusion of particles in a semi-infinite cubic lattice, and focus on the effective diffusion process at the interface $z = 1$. We calculate analytically the conditional probability to find the particle on the $z=1$ plane as well as the surface dispersion as functions of time. The comparison of these results with Monte Carlo simulations show an excellent agreement.Comment: 16 pages, 7 figs. European Physical Journal B (in press

Bulk Mediated Surface Diffusion: The Infinite System Case

An analytical soluble model based on a Continuous Time Random Walk (CTRW) scheme for the adsorption-desorption processes at interfaces, called bulk-mediated surface diffusion, is presented. The time evolution of the effective probability distribution width on the surface is calculated and analyzed within an anomalous diffusion framework. The asymptotic behavior for large times shows a sub-diffusive regime for the effective surface diffusion but, depending on the observed range of time, other regimes may be obtained. Montecarlo simulations show excellent agreement with analytical results. As an important byproduct of the indicated approach, we present the evaluation of the time for the first visit to the surface.Comment: 15 pages, 7 figure

Bulk Mediated Surface Diffusion: Finite System Case

We address the dynamics of adsorbed molecules (a fundamental issue in surface physics) within the framework of a Master Equation scheme, 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 system 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.Comment: 19 pages, 8 figure

Invited review: KPZ. Recent developments via a variational formulation

Recently, a variational approach has been introduced for the paradigmatic Kardar--Parisi--Zhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansion that the KPZ nonequilibrium potential (NEP) admits. Such expansion becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradient-flow counterpart to the KPZ equation. A dynamic renormalization group analysis at one-loop order of this new mesoscopic model yields the KPZ scaling relation alpha+z=2, as a consequence of the exact cancelation of the different contributions to vertex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a path-integral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation.Comment: 16 pages, 2 figure

Variational Formulation for the KPZ and Related Kinetic Equations

We present a variational formulation for the Kardar-Parisi-Zhang (KPZ) equation that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations. For the KPZ case, with the knowledge of such a potential we prove some global shift invariance properties previously conjectured by other authors. We also show a few results about the form of the stationary probability distribution function for arbitrary dimensions. The procedure used for KPZ was extended in order to derive more general forms of such a functional leading to other nonlinear kinetic equations, as well as cases with density dependent surface tension.Comment: RevTex, 8pgs, double colum

The Use of Rank Histograms and MVL Diagrams to Characterize Ensemble Evolution in Weather Forecasting

13 páginas, 9 figuras.-- El pdf del artículo es la versión pre-print.Rank Histograms are suitable tools to assess the quality of ensembles within an ensemble prediction system or framework. By counting the rank of a given variable in the ensemble, we are basically making a sample analysis, which does not allow us to distinguish if the origin of its variability is external noise or comes from chaotic sources. The recently introduced Mean to Variance Logarithmic (MVL) Diagram accounts for the spatial variability, being very sensitive to the spatial localization produced by infinitesimal perturbations of spatiotemporal chaotic systems. By using as a benchmark a simple model subject to noise, we show the distinct information given by Rank Histograms and MVL Diagrams. Hence, the main effects of the external noise can be visualized in a graphic. From the MVL diagram we clearly observe a reduction of the amplitude growth rate and of the spatial localization (chaos suppression), while from the Rank Histogram we observe changes in the reliability of the ensemble. We conclude that in a complex framework including spatiotemporal chaos and noise, both provide a more complete forecasting picture.We acknowledge financial support from MEC, Spain, through Grant No. CGL2007- 64387/CLI, and also thank the AECID, Spain, for support through projects A/013666/07 and A/018685/08. JAR thanks the MEC, Spain, for the award of a Juan de la Cierva fellowship. HSW thanks to the European Commission for the award of a Marie Curie Chair during part of the development of this work.Peer reviewe

Discretization-related issues in the KPZ equation: Consistency, Galilean-invariance violation, and fluctuation--dissipation relation

In order to perform numerical simulations of the 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 \emph{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 \emph{space} and the Hopf--Cole transformation is \emph{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 pseudo-spectral 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 1D.Comment: RevTex, 23pgs, 2 figures, submitted to Phys. Rev.