Carlos E. Budde

Superdiffusion in decoupled continuous time random walks

Abstract Continuous time random walk models with decoupled waiting time density are studied. When the spatial one-jump probability density belongs to the Levy distribution type and the total time transition is exponential a generalized superdiffusive regime is established. This is verified by showing that the square width of the probability distribution (appropriately defined) grows as t2/γ with 0

Intermittent search strategies revisited: effect of the jump length and biased motion

We study the kinetics of a search of a single fixed target by a large number of searchers performing an intermittent biased random walk in a homogeneous medium. Our searchers carry out their walks in one of two states between which they switch randomly. One of these states (search phase) is a nearest-neighbor walk characterized by the probability of stepping in a given direction (i.e. the walks in this state are not necessarily isotropic). The other (relocation phase) is characterized by the length of the jumps (i.e. when in this state a walker does not perform a nearest-neighbor walk). Within such a framework, we propose a model to describe the searchers’ dynamics, generalizing results of our previous work. We have obtained, and numerically evaluated, analytic results for the mean number of distinct sites visited up to a maximum evolution time. We have studied the dependence of this quantity on both the transition probability between the states and the parameters that characterize each state. In addition to our theoretical approach, we have implemented Monte Carlo simulations, finding excellent agreement between the theoretical–numerical and simulations results.

The continuous-time resolvent matrix for non-markovian chains

Abstract A unified theory of discrete and continuous-time resolvent matrix method is described in terms of the waiting-time density. We obtain a differential evolution equation for the marginal probability distribution, after summing over the internal states of a Markovian chain. The example of the continuous-time Lorentz-gas model is presented as an application of this theory.

Stochastic escape processes from a nonsymmetric potential normal form

The lifetime from the normal form X=aX2+b+ square root ( epsilon xi )(t) is analytically studied in terms of the escape time of leaving the unstable point X=0. A perturbation theory, in the small noise parameter square root epsilon , is introduced to analyse the escape of the stochastic paths. We show that the first passage time density satisfies a scale transformation. The anomalous fluctuation of the phase-space variable X(t) (when there is saturation in the potential of the normal form) is analytically calculated using an instanton-like approximation. An emphasis is placed on thermal explosions in order to exemplify a system undergoing hysteresis in a first-order non-equilibrium phase transition. We carried out Monte Carlo simulations showing excellent agreement with our theoretical predictions.

Marginal distribution of a Markovian chain with internal states: the resolvent matrix

Abstract In order to obtain the marginal probability distribution, with respect to the internal states of a markovian chain, we prove that a resolvent matrix can be constructed. Non-markovian recurrence relations for its evolution are also established. We obtain the Green function of this equation and apply this theory to the one-dimensional Lorentz-gas model.

Compositional Construction of Importance Functions in Fully Automated Importance Splitting.

Importance splitting is a technique to accelerate discrete event simulation when the value to estimate depends on the occurrence of rare events. It requires a guiding importance function typically defined in an ad hoc fashion by an expert in the field, who could choose an inadequate function. In this article we present a compositional and automatic technique to derive the importance function from the model description, and analyze different composition heuristics. This technique is linear in the number of modules, in contrast to the exponential nature of our previous proposal. This approach was compared to crude simulation and to importance splitting using typical ad hoc importance functions. A prototypical tool was developed and tested on several models, showing the feasibility and efficiency of the technique.

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.

Bulk-mediated surface diffusion: non-Markovian desorption dynamics

Here we analyse the dynamics of adsorbed molecules within the bulk-mediated surface diffusion framework, when the particles desorption mechanism is characterized by a non-Markovian process, while the particles adsorption as well as its motion in the bulk is governed by Markovian dynamics. We study the diffusion of particles in both semi-infinite and finite cubic lattices, analysing the conditional probability to find the system on the reference absorptive plane as well as the surface dispersion as functions of time. The results are compared with known Markovian cases showing the differences that can be exploited to distinguish between Markovian and non-Markovian desorption mechanisms in experimental situations.

STOCHASTIC ESCAPE PROCESSES FROM A NON-SYMMETRIC POTENTIAL NORMAL FORM II : THE MARGINAL CASE

The first-passage time distribution to reach the attractor of the stochastic differential equation is analytically obtained by using a previously reported scheme: the stochastic path perturbation approach. A second-order perturbation theory, in the small noise parameter , is introduced to analyse the random escape, of the stochastic paths, from the marginal unstable state X = 0. The anomalous fluctuation of the phase-space variable X(t) is analytically calculated by using the instanton-like approximation. We have carried out Monte Carlo simulations showing good agreement with our theoretical predictions.

Exact and asymptotic properties of multistate random walks

A method is presented which allows one to obtain explicit analytical expressions (both exact and asymptotic) for many of the physically interesting quantities related to a multistate random walk (MRW). The exact results include the Laplace-Fourier-transformed probability distribution (continuous time) and generating function (discrete time), and closed evolution equations for the propagators related to each “internal” state of the walker. Analytical expressions for the scattering dynamical structure function and the frequency-dependent diffusion coefficient are given as illustrations. Asymptotic approximations to the single-state propagators are derived, allowing a detailed analysis of the longtime behavior and the calculation of asymptotic properties by single-state random walk standard methods. As an example, analytical expressions for the drift and diffusion coefficients are given.