We consider magnetization reversal due to thermal fluctuations in thin, submicron-scale rings. These mesoscopic ferromagnetic particles are of particular interest as potential information storage components in magnetoelectronic devices, because their lack of sharp ends result in a magnetization density that is significantly more stable against reversal than in thin needles and other geometries. Their two-dimensional nature and rotational symmetry allow us to incorporate long-range magnetostatic forces in a fully analytic treatment, which is not possible in most geometries. We uncover a type of 'phase transition' between different activation regimes as magnetic field is varied at fixed ring size. Previous studies of such transitions in classical activation behavior have found that they occur as length is varied, which cannot be realized easily or continuously for most systems. However, the different activation regimes in a single mesoscopic ferromagnet should be experimentally observable by changing the externally applied magnetic field, and by tuning this field the transition region itself can be studied in detail.

We study the Turing instability to stationary spatial patterns in reaction-transport systems with inertia. After a brief discussion of hyperbolic reaction-diffusion equations and reaction-Cattaneo equations we focus on reaction random walks, which are the most natural generalization of reaction-diffusion equations. We analyze the effect of inertia in the transport on spatial instabilities of the homogeneous steady state. Direction-dependent reaction walks, where the interaction between particles depends on the direction in which the particles are moving, allow us to take account of an energy requirement for reactive interactions, i.e., an activation energy, in the reaction-transport equation. We compare bifurcation conditions for direction-independent reaction walks and for direction-dependent reaction walks to assess the effect of inertia in the transport and the effect of activation energies in the kinetics on the Turing instability.

Reaction dynamics involving subdiffusive species is an interesting topic with only few known results, especially when the motion of different species is characterized by different anomalous diffusion exponents. Here we study the reaction dynamics of a (sub)diffusive particle surrounded by a sea of (sub)diffusive traps in one dimension. Under some reasonable assumptions we find rigorous results for the asymptotic survival probability of the particle in most cases, but have not succeeded in doing so for a particle that diffuses normally while the anomalous diffusion exponent of the traps is smaller than 2/3.

We analyze periodically driven bistable systems by two different approaches. The first approach is a linearization of the stochastic Langevin equation of our system by the response on small external force. The second one is based on the Gaussian approximation of the kinetic equations for the cumulants. We obtain with the first approach the signal power amplification and output signal-to-noise ratio for a model piece-wise linear bistable potential and compare with the results of linear response approximation. By using the second approach to a bistable quartic potential, we obtain the set of nonlinear differential equations for the first and the second cumulants.

In this paper we revisit the asymmetric binary channel from the double point of view of detection theory and information theory. We first evaluate the capacity of the asymmetric binary channel as a function of the probabilities of false alarm and of detection, thus allowing a noise distribution independent analysis. This sets the a priori probabilities of the hypotheses and couples the two points of view. We then study the simple realization of the asymmetric binary channel using a threshold device. We particularly revisit noise-enhanced processing for subthreshold signals using the aforementioned parametrization of the capacity, and we report a somewhat paradoxical effect: using the channel at its capacity precludes in general an optimal detection.

The motion of elastically coupled Brownian particles in ratchet-like potentials has attracted much recent interest due to its application to transport processes in many fields, including models of DNA polymers. We consider the influence of the type of interacting force on the transport of two particles in a one-dimensional flashing ratchet. Our aim is to examine whether the common assumption of elastic coupling captures the important features of ratchet transport when the inter-particle forces are more complex. We compare Lennard-Jones type interaction to the classical case of elastically coupled particles. Numerical simulations agree with analytical formulas for the limiting cases where the coupling is very weak or very strong. Parameter values where the Lennard-Jones force is not well approximated by a linearization of the force about the equilibrium distance are identified.

We explore the phase diagram of a recently introduced gradient driven anisotropic 3D sandpile model of vortex dynamics. Two distinct phases are observed: one is a self-organized critical state characterized by avalanches of vortex motion that obey finite-size scaling and that has a finite critical current density; the other one has vortices that cluster together and occupy only every other lattice site in the X-Y plane. The critical current density is zero in the clustered phase. Detailed results of a finite-size scaling analysis of the avalanches in the self-organized phase is discussed, including critical exponents that differ from the corresponding 2D model.

A baseline mathematical Data Model for rapid detection of events affecting school attendance was established using measured absenteeism. We derived a process model in the form of a differential equation of baseline or nominal behavior of sufficient fidelity to enable predictive what-if excursions to be done. Also, we derived a change detector equation enabling aberrant absenteeism to be flagged on the first day of detection. Used together, both mathematical models provide a powerful tool set for analysis and examining what unusual absenteeism patterns might be like due to chem-bio attack. A convenient single page lookup table is provided to give predictive analysis capability without resorting to cumbersome calculations.

About a decade ago Brownian motors were introduced as a possible mechanism for motor protein mobility. Since then many theoretical and experimental papers have been published on the topic. While some experiments support Brownian motor mechanisms, others are more consistent with traditional power stroke models. Taking into account recent experimental data and molecular level simulations, we have developed a stochastic model which incorporates both power stroke and Brownian motor mechanisms. Depending on parameter values, this motor works as a power stroker, a Brownian motor or a hybrid of the two. Using this model we investigate the motility of single-head myosins, two-head myosins and a group of myosins (muscle). The results are compared with some experimental data.

Canalization is a form of network robustness found in genetic regulatory networks that results in a reduction of the variation of phenotypic expression relative to the complexity of the genome. Recently, it was discovered that canalization can evolve in a complex network through a self-organization of node (gene) behavior resulting from a competition of a network's nodes that selects for a diversity of behavior [K.E. Bassler, C. Lee, and Y. Lee, Phys. Rev. Lett. 93, 038101 (2004)]. Previously, this "self-organized" mechanism for the evolution of canalization has been studied only in deterministic model systems. This paper considers the effects of stochastic noise in the signals exchanged between nodes on the self-organized evolution of canalization. We find that small levels of stochastic noise increase the amount of canalization produced. At higher levels of noise, the amount of canalization produced levels off and reaches a maximum value, before it reduces at large levels of noise.

The theory of particle transport with branching in a medium randomly varying in time is developed in this paper. We consider an evolution equation for the probability distribution of the number of particles in a multiplying system whose properties jump randomly between two discrete states. A forward type master equation is derived for the probability distributions, and from these, the first two factorial moments are calculated, including the variance. This model can be considered the unification of stochastic methods that were used either for the particle fluctuations in a constant multiplying medium via the master equation technique, or in a fluctuating medium via the Langevin technique. The results obtain show a much richer variety of behaviour than any of the above two methods separately can reconstruct.

We considered diffusion driven processes on power-law small-world networks: random walk on randomly folded polymers and surface growth related to synchronization problems. We found a rich phase diagram, with different transient and recurrent phases. The calculations were done in two limiting cases: the annealed case, when the rearrangement of the random links is fast (the configuration of the polymer changes fast) and the quenched case, when the link rearrangement is slow (the polymer configuration is static) compared to the motion of the random walker. In the quenched case, the random links introduced in small-world networks often lead to mean-filed coupling (i.e., the random links can be treated in an annealed fashion) but in some systems mean-field predictions break down, such as for diffusion in one dimension. This break-down can be understood treating the random links perturbatively where the mean field prediction appears as the lowest order term of a naive perturbation expansion. Our results were obtained using self-consisten perturbation theory. Numerical results will also be shown as a confirmation of the theory.

Motion in Gaussian random space-time fields ("dynamical disorder") is proposed as a model for certain dynamical systems where fluctuations play an important role. Analytical and numerical methods adapted from the study of passive scalar turbulence are applied to two examples: phase diffusion in noisy nonlinear oscillators and demonstrating the existence of 1/f phase noise in the mean field of Kuramoto's coupled oscillators model.

We study the stochastic properties of gene regulation taking into account the non-Markovian character of gene transcription and translation. We show that time delay in protein production or degradation may change the behavior of the system from stationary to oscillatory even when a deterministic counterpart of the stochastic system exhibits no oscillations. Assuming significant decorrelation on the time scale of gene transcription, we deduce a truncated master equation of the reactive system and derive an analytical expression for the autocorrelation function of the protein concentration. For weak feedback the theory agrees well with with numerical simulations based on the modified direct Gillespie method.

Fluctuations in diversity and extinction sizes are discussed and compared for two different, individual-based models of biological coevolution. Both models display power-law distributions for various quantities of evolutionary interest, such as the lifetimes of individual species, the quiet periods between evolutionary upheavals larger than a given cutoff, and the sizes of extinction events. Time series of the diversity and measures of the size of extinctions give rise to flicker noise. Surprisingly, the power-law behaviors of the probability densities of quiet periods in the two models differ, while the distributions of the lifetimes of individual species are the same.

Single molecule techniques offer a unique tool studying the dynamical behaviour of individual molecules, and provide the possibility to construct distributions from individual events rather than from a signal stemming from an ensemble of molecules. Observing the activity of individual lipase molecules for extended periods of time (hours), we get long trajectories, made of "on-state" and "off-state" events. The waiting time probability density function (PDF) of the off-state and the state-correlaiton function fit stretched exponentials, independent of the substrate concentration in a certain range. The data analysis unravels correlations between off-state events. These findings imply that the fluctuating enzyme model, which involves a spectrum of enzymatic conformations that interconvert on the timescale of the catalytic activity, best describes the observed enzymatic activity. This contribution summarizes some of our recent published results.

A direct comparison between continuous and discrete forms of analysis of control and stability is investigated theoretically and numerically. We demonstrate that the continuous method provides a more energy-efficient means of controlling the switching of a periodically-driven class-B laser between its stable and unstable pulsing regimes. We provide insight into this result using the close correspondence that exists between the problems of energy-optimal control and the stability of a steady state.

We consider the following general problem of applied stochastic nonlinear dynamics. We observe a time series of signals y(t) = y(t₀+hn) corrupted by noise. The actual state and the nonlinear vector field of the dynamical system is not known. The question is how and with what accuracy can we determine x(t) and functional form of f(x). In this talk we discuss a novel approach to the solution of this problem based on the application of the path-integral approach to the full Bayesian inference. We demonstrate a reconstruction of a dynamical state of a system from corrupted by noise measurements. Next we reconstruct the corresponding nonlinear vector field. The emphasis are on the theoretical analysis. The results are compared with the results of earlier research.

Qualitative aspects of numerical methods for integration of systems of nonlinear ordinary stochastic differential equations (SDEs) with potential applicability to mechanical engineering are presented. In particular, we study the qualitative behavior of some linearly partial-implicit midpoint-type methods for numerical integration of infinite and finite systems of SDEs with cubic-type nonlinearity and Q-regular additive space-time noise. Construction and properties such as stability and convergence of such stochastic-numerical methods is strongly related to their uniform boundedness along Lyapunov-type functionals. Well-known convergence order bounds apart from further complexity issues forces us to focus our analysis on lower order Runge-Kutta methods rather than higher order Taylor methods. Nonstandard techniques such as partial-implicit difference methods for noisy ODEs/PDEs seem to be the most promising ones in view of adequate longterm integration of such nonlinear systems.

In the present article we extend the Random Matrix Theory to the theory of matrix-valued random fields. We derive the "equilibrium" probability distribution and "equilibrium" ensembles of matrix-valued random fields applying Maximum Entropy Principle. The inferred "equilibrium" probability density functionals are applied to Euclidean Quantum Field Theory and to the theory of two-dimensional Euclidean quantum gravity.

We present a numerical study of classical particles obeying a Langevin equation moving on a solid bcc(110) surface. The particles are subject to a two dimensional periodic and symmetric potential of rectangular symmetry and to an external dc field along one of the diagonals of the structure. One observes a bias current with a component orthogonal to the dc field. The drift velocity (magnitude and direction) and diffusion of the particle depend on the surface potential and external field parameters, the temperature, and the friction coefficient. We numerically explore these dependences. Because the potential perceived by a particle as well as its friction coefficient depend on the nature of the particle, so might the angle between the particle velocity and the dc field. This scenario may thus provide a useful particle sorting technique.

Ratchet effect and the phenomenon of stochastic resonance (SR) are investigated numerically in different random systems. For a simple flashing ratchet model, it is shown that both the power spectrum amplitude and the mean mobility of the system vary nonmonotonically in similar manners with the increase of the noise intensity. And for a Brownian motor with stochastic state-jumping, fixing the noise level in an anisotropic potential and merely increasing the thermal fluctuation in a flat potential not only causes the increasing of the mean mobility to a saturated value, but also induce the spectrum-measured SR. However, increasing the noise intensities in two potentials simultaneously induces mobility-measured SR, but no spectrum-measured SR can be observed.

We provide a complete solution of the problem of noise-induced escape in periodically driven systems. We show that both the exponent and the prefactor in the escape rate display scaling behavior with the field intensity. The corresponding scaling is related to synchronization of escape events by the modulating field. The onset of the synchronization with the increasing field and its loss as the field approaches a bifurcational value lead to a strongly nonmonotonic field dependence of the prefactor.

A model of relaxation in glassy materials, involving anomalous slow diffusion of defects, is reviewed. The movement of the defects causes a stretched exponential relaxation. If there is an attractive force between defects then as the temperature is lowered, or the pressure increased, the number of mobile defects will decrease. This loss of mobile defects produces a Vogel type law for the singular behavior of the relaxation time scale at a critical temperature.

The transition to chaos is a fundamental and widely studied problem in deterministic nonlinear dynamics. Well known routes to chaos, which include the period-doubling bifurcation route, the intermittency route, the quasiperiodic route, and the crisis route, describe transitions to low-dimensional chaotic attractors with one positive Lyapunov exponent. Transitions to high-dimensional chaotic attractors with multiple positive Lyapunov exponents have just started being addressed. In stochastic systems, transitions to chaotic-like behavior are less well characterized. Global analysis coupled with stochastic transport probability can explain emergent behavior in which stable and unstable manifolds may interact with noise to cause "stochastic chaos". Another stochastic route may induce chaotic signatures through a dimension changing bifurcation, whereby the topological dimension changes when the amplitude of the noise goes beyond a critical parameter. In this paper we present a theory of how the Lyapunov exponents may scale with the noise amplitude in general systems. A physical class of multiscale dynamical systems will be presented to show that noise may induce low dimensional chaos, or for other parameters, may induce chaos that bifurcates to an attractor contained in a high topological dimension. We present a numerical bifurcation analysis of the resulting system, illustrating the mechanism for the onset of high dimensional chaos. By computing the constrained invariant sets, we reveal the transition from low dimensional to high dimensional chaos. Applications include both deterministic and stochastic bifurcations.

We consider by means of the optimal fluctuation method the initial stage of the evolution of the noise-induced escape through various types of boundaries, especially concentrating on two types of the boundary - the wall and the boundary of the basin of attraction. We show in both cases that, if the damping is small enough, then the escape flux evolution possesses a remarkable property: it is it stairs-like i.e. intervals of a nearly constant flux alternate with intervals of a sharply increasing flux. This property is related to the successive increase of the number of turning points in the most probable escape path as time increases. Our results are relevant both for the absorbing and transparent boundaries. The major results of the theory are verified in computer simulations.

We discuss the close link between intermittent events ('quakes') and extremal noise fluctuations which has been advocated in recent numerical and theoretical work. From the idea that record-breaking noise fluctuations trigger the quakes, an approximate analytical description of non-equilibrium aging as a Poisson process with logarithmic time arguments can be derived. Theoretical predictions for measurable statistical properties of mesoscopic fluctuations are emphasized, and supporting numerical evidence is included from simulations of short-ranged Ising spin-glass models, of the ROM model of vortex dynamics in type II superconductors, and of the Tangled Nature model of biological evolution.

We consider two bistable systems, the double-well potential and the Schmitt-trigger, and examine whether the stochastic resonance occurring in these systems may produce output signals less noisy than the input. We apply cross-spectrum and cross-correlation based generalised measures to quantify noise content in the input and output, which enables us to use aperiodic or random sequences as input signals. We show that input-output signal improvement occurs even for these types of input.