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PubMed2002

Chaotic bursting at the onset of unstable dimension variability.

Dynamical systems possessing symmetries have invariant manifolds. According to the transversal stability properties of this invariant manifold, nearby trajectories may spend long stretches of time in its vicinity before being repelled from it as a chaotic burst, after which the trajectories return to their original laminar behavior. The onset of chaotic bursting is determined by the loss of transversal stability of low-period periodic orbits embedded in the invariant manifold, in such a way that the shadowability of chaotic orbits is broken due to unstable dimension variability, characterized by finite-time Lyapunov exponents fluctuating about zero. We use a two-dimensional map with an invariant subspace to estimate shadowing distances and times from the statistical properties of the bursts in the transversal direction. A stochastic model (biased random walk with reflecting barrier) is used to relate the shadowability properties to the distribution of the finite-time Lyapunov exponents.

Journal Article↗
PubMed2002

Structure of best possible strategies for finding ground states.

Finding the ground state of a system with a complex energy landscape is important for many physical problems including protein folding, spin glasses, chemical clusters, and neural networks. Such problems are usually solved by heuristic search methods whose efficacy is judged by empirical performance on selected examples. We present a proof that for a wide range of objective functions threshold accepting is the best possible strategy within a large class of algorithms that simulate random walks on the landscape. In particular, it can perform better than simulated annealing, Tsallis and Glauber statistics.

Journal Article↗
PubMed2002

Cluster persistence in one-dimensional diffusion-limited cluster-cluster aggregation.

The persistence probability, P(C)(t), of a cluster to remain unaggregated is studied in cluster-cluster aggregation, when the diffusion coefficient of a cluster depends on its size s as D(s) approximately s(gamma). In the mean field the problem maps to the survival of three annihilating random walkers with time-dependent noise correlations. For gamma> or =0 the motion of persistent clusters becomes asymptotically irrelevant and the mean-field theory provides a correct description. For gamma<0 the spatial fluctuations remain relevant and the persistence probability is overestimated by the random walk theory. The decay of persistence determines the small size tail of the cluster size distribution. For 0<gamma<2 the distribution is flat and, surprisingly, independent of gamma.

Journal Article↗
PubMed2002

Generalization of the Wang-Landau method for off-lattice simulations.

We present a rigorous derivation for off-lattice implementations of the so-called "random-walk" algorithm recently introduced by Wang and Landau [Phys. Rev. Lett. 86, 2050 (2001)]. Originally developed for discrete systems, the algorithm samples configurations according to their inverse density of states using Monte Carlo moves; the estimate for the density of states is refined at each simulation step and is ultimately used to calculate thermodynamic properties. We present an implementation for atomic systems based on a rigorous separation of kinetic and configurational contributions to the density of states. By constructing a "uniform" ensemble for configurational degrees of freedom-in which all potential energies, volumes, and numbers of particles are equally probable-we establish a framework for the correct implementation of simulation acceptance criteria and calculation of thermodynamic averages in the continuum case. To demonstrate the generality of our approach, we perform sample calculations for the Lennard-Jones fluid using two implementation variants and in both cases find good agreement with established literature values for the vapor-liquid coexistence locus.

Journal Article↗
PubMed2002

Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition.

We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

Biophysical Phenomena↗
PubMed2002

Reaction-diffusion wave fronts: multigeneration biological species under climate change.

A generalization of reaction-diffusion models to multigeneration biological species is presented. It is based on more complex random walks than those in previous approaches. The new model is developed analytically up to infinite order. Our predictions for the speed agree to experimental data for several butterfly species better than existing models. The predicted dependence for the speed on the number of generations per year allows us to explain the change in speed observed for a specific invasion.

Journal Article↗
PubMed2002

Stability of shortest paths in complex networks with random edge weights.

We study shortest paths and spanning trees of complex networks with random edge weights. Edges which do not belong to the spanning tree are inactive in a transport process within the network. The introduction of quenched disorder modifies the spanning tree such that some edges are activated and the network diameter is increased. With analytic random-walk mappings and numerical analysis, we find that the spanning tree is unstable to the introduction of disorder and displays a phase-transitionlike behavior at zero disorder strength epsilon =0. In the infinite network-size limit (N--> infinity ), we obtain a continuous transition with the density of activated edges Phi growing like Phi approximately epsilon (1) and with the diameter-expansion coefficient Upsilon growing like Upsilon approximately epsilon (2) in the regular network, and first-order transitions with discontinuous jumps in Phi and Upsilon at epsilon=0 for the small-world (SW) network and the Barabási-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when N>>N(c), where the crossover size scales as N(c) approximately epsilon (-2) for the regular network, N(c) approximately exp(alpha epsilon (-2)) for the SW network, and N(c) approximately exp(alpha|ln epsilon | epsilon (-2)) for the SF network. In a transient regime with N<<N(c), there is an infinite-order transition with Phi approximately Upsilon approximately exp[-alpha/(epsilon (2)ln N)] for the SW network and approximately exp[-alpha/(epsilon (2)ln N/ln ln N)] for the SF network. It shows that the transport pattern is practically most stable in the SF network.

Journal Article↗
PubMed2003

Anomalous diffusion and Hall effect on comb lattices.

In this paper we study the effects of a magnetic field on the discrete time random walk of a classical charged particle moving on a comb lattice. We develop an analytical technique to study the Lorentz force effects on the asymptotic diffusion laws. This approach also allows the description of the combined action of an electric and a magnetic field (Hall effect). The generalization to other comblike branched structures is discussed.

Journal Article↗
PubMed2003

Hopping in a supercooled Lennard-Jones liquid: metabasins, waiting time distribution, and diffusion.

We investigate the jump motion among potential energy minima of a Lennard-Jones model glass former by extensive computer simulation. From the time series of minima energies, it becomes clear that the energy landscape is organized in superstructures called metabasins. We show that diffusion can be pictured as a random walk among metabasins, and that the whole temperature dependence resides in the distribution of waiting times. The waiting time distribution exhibits algebraic decays: tau(-1/2) for very short times and tau(-alpha) for longer times, where alpha approximately 2 near T(c). We demonstrate that solely the waiting times in the very stable basins account for the temperature dependence of the diffusion constant.

Journal Article↗
PubMed2003

Anomaly of the height-height correlation functions in self-flattening surface growth.

By Monte Carlo simulations and scaling theories, we consider the height-height correlation function G(r,t;L) of the one-dimensional equilibrium self-flattening surface growths, where the deposition (evaporation) attempt only at the globally highest (lowest) site is suppressed. G(r,t:L) is shown to satisfy the anomalous scaling behavior G(r,t;L)=L(2alpha)g(1)(r/L(delta),t/L(z)) or G(r,t;L)=t(2beta)g(2)(r/t(1/z(')),L/t(1/z)). Here alpha, beta, and z are the roughness, growth, and dynamic exponents, respectively, for the surface width, with alpha=1/3 and z=alpha/beta=3/2. Anomalous exponents z(') and delta are found to satisfy z(')=9/4 and delta=z/z('). We also show that anomalous behavior of G(r,t;L) can be understood from a scaling theory based on the competition between local random-walk-like behavior and the global-length-scale suppression.

Journal Article↗
PubMed2003

Non-Markovian stochastic Liouville equation and its Markovian representation.

The non-Markovian variant of the stochastic Liouville equation (SLE) is studied within the continuous time random walk approach (CTRWA). The CTRWA-based non-Markovian SLE is shown to be equivalently represented by the corresponding conventional Markovian SLE. This Markovian representation provides a rigorous method for deriving the non-Markovian SLE and allows for a physically clear interpretation of the specific features of this SLE. It also enables one to develop convenient non-Markovian models useful for applications, some of which are discussed in detail. Special attention is given to the discussion of anomalous long-tailed CTRW processes and non-Markovian SLE. The obtained results are applied to the analysis of the effect of rate fluctuations on chemical reaction kinetics. It is shown, in particular, that the anomalous fluctuations not only influence the reaction rate but also change the reaction kinetics itself.

Journal Article↗
PubMed2003

Linear and nonlinear response in the aging regime of the one-dimensional trap model.

We investigate the behavior of the response function in the one-dimensional trap model using scaling arguments that we confirm by numerical simulations. We study the average position of the random walk at time t(w)+t, given that a small bias h is applied at time t(w). Several scaling regimes are found, depending on the relative values of t, t(w), and h. Comparison with the diffusive motion in the absence of bias allows us to show that the fluctuation-dissipation relation is valid even in the aging regime, at least for times such that linear response is obeyed. However, for sufficiently long times, the response always becomes nonlinear in h.

Journal Article↗
PubMed2003

Measure synchronization in coupled phi4 Hamiltonian systems.

Transitions to measure synchronization both in the quasiperiodic and chaotic cases are investigated based on numerical computation of two coupled phi(4) equations. Some relevant quantities such as the bare energies, the interaction energy, and the phase difference of the two oscillators are computed to clarify the characteristics of the transitions and the measure-synchronous states. A bifurcation with discontinuous bare energy and continuous interaction energy, which takes the maximum value at the critical point, is found for the transition from the desynchronous quasiperiodic state to the measure-synchronous quasiperiodic state, and the related power law scalings are deduced. Stick-slip and random-walk-like behavior of the phase difference is found for the chaotic measure-synchronous state, and this explains the monotonous increase of the interaction energy with an increase of coupling.

Journal Article↗
PubMed2003

Avoiding boundary effects in Wang-Landau sampling.

A simple modification of the "Wang-Landau sampling" algorithm removes the systematic error that occurs at the boundary of the range of energy over which the random walk takes place in the original algorithm.

Journal Article↗
PubMed2003

Interplay of chemotaxis and chemokinesis mechanisms in bacterial dynamics.

Motivated by observations of the dynamics of Myxococcus xanthus, we present a self-interacting random walk model that describes the competition between chemokinesis and chemotaxis. Cells are constrained to move in one dimension, but release a chemical chemoattractant at a steady state. The bacteria sense the chemical that they produce. The probability of direction reversals is modeled as a function of both the absolute level of chemoattractant sensed directly under each cell as well as the gradient sensed across the length of the cell. If the chemical does not degrade or diffuse rapidly, the one-dimensional trajectory depends on the entire past history of the trajectory. We derive the corresponding Fokker-Planck equations, use an iterative mean-field approach that we solve numerically for short times, and perform extensive Monte Carlo simulations of the model. Cell positional distributions and the associated moments are computed in this feedback system. Average drift and mean squared displacements are found. Crossover behaviors among different diffusion regimes are found.

Bacterial Physiological Phenomena↗
PubMed2003

Shock formation in an exclusion process with creation and annihilation.

We investigate shock formation in an asymmetric exclusion process with creation and annihilation of particles in the bulk. We show how the continuum mean-field equations can be studied analytically and hence derive the phase diagrams of the model. In the large system-size limit direct simulations of the model show that the stationary state is correctly described by the mean-field equations, thus the predicted mean-field phase diagrams are expected to be exact. The emergence of shocks and the structure of the phase diagram are discussed. We also analyze the fluctuations of the shock position by using a phenomenological random walk picture of the shock dynamics. The stationary distribution of shock positions is calculated, by virtue of which the numerically determined finite-size scaling behavior of the shock width is explained.

Journal Article↗
PubMed2003

Transition from Knudsen to molecular diffusion in activity of absorbing irregular interfaces.

We investigate through molecular dynamics the transition from Knudsen to molecular diffusion transport towards two-dimensional absorbing interfaces with irregular geometry. Our results indicate that the length of the active zone decreases continuously with density from the Knudsen to the molecular diffusion regime. In the limit where molecular diffusion dominates, we find that this length approaches a constant value of the order of the system size, in agreement with theoretical predictions for Laplacian transport in irregular geometries. Finally, we show that all these features can be qualitatively described in terms of a simple random-walk model of the diffusion process.

Journal Article↗