Invariant submanifold for series arrays of Josephson junctions

We study the nonlinear dynamics of series arrays of Josephson junctions in the large-N limit, where N is the number of junctions in the array. The junctions are assumed to be identical, overdamped, driven by a constant bias current and globally coupled through a common load. Previous simulations of such arrays revealed that their dynamics are remarkably simple, hinting at the presence of some hidden symmetry or other structure. These observations were later explained by the discovery of (N - 3) constants of motion, each choice of which confines the resulting flow in phase space to a low-dimensional invariant manifold. Here we show that the dimensionality can be reduced further by restricting attention to a special family of states recently identified by Ott and Antonsen. In geometric terms, the Ott-Antonsen ansatz corresponds to an invariant submanifold of dimension one less than that found earlier. We derive and analyze the flow on this submanifold for two special cases: an array with purely resistive loading and another with resistive-inductive-capacitive loading. Our results recover (and in some instances improve) earlier findings based on linearization arguments.Comment: 10 pages, 6 figure

Synchronization of globally coupled two-state stochastic oscillators with a state dependent refractory period

We present a model of identical coupled two-state stochastic units each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state. Increasing coupling strength leads to a saddle node bifurcation, beyond which the quiescent state coexists with a stable limit cycle of nonlinear coherent oscillations. We explicitly determine the critical coupling constant for this transition

Signal processing with Levy information

Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a "message", then under the transformed measure the original Levy process takes on the character of an "information process". In this paper we develop a theory of such Levy information processes. The underlying Levy process, which we call the fiducial process, represents the "noise type". Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian, and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.Comment: 27 pages. Version to appear in: Proc. R. Soc. London

Opposite Thermodynamic Arrows of Time

A model in which two weakly coupled systems maintain opposite running thermodynamic arrows of time is exhibited. Each experiences its own retarded electromagnetic interaction and can be seen by the other. The possibility of opposite-arrow systems at stellar distances is explored and a relation to dark matter suggested.Comment: To appear in Phys. Rev. Let

Transfer Entropy as a Log-likelihood Ratio

Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic chi-squared distribution is established for the transfer entropy estimator. The result generalises the equivalence in the Gaussian case of transfer entropy and Granger causality, a statistical notion of causal influence based on prediction via vector autoregression, and establishes a fundamental connection between directed information transfer and causality in the Wiener-Granger sense

Self consistent determination of plasmonic resonances in ternary nanocomposites

We have developed a self consistent technique to predict the behavior of plasmon resonances in multi-component systems as a function of wavelength. This approach, based on the tight lower bounds of the Bergman-Milton formulation, is able to predict experimental optical data, including the positions, shifts and shapes of plasmonic peaks in ternary nanocomposites without using any ftting parameters. Our approach is based on viewing the mixing of 3 components as the mixing of 2 binary mixtures, each in the same host. We obtained excellent predictions of the experimental optical behavior for mixtures of Ag:Cu:SiO2 and alloys of Au-Cu:SiO2 and Ag-Au:H2 O, suggesting that the essential physics of plasmonic behavior is captured by this approach.Comment: 7 pages and 4 figure

Comment on “Surface Plasmons and Nonlocality: A Simple Model”

In the Comment [1], Schaich calculated the mode dispersion of surface plasmons supported by a planar metal-dielectric-metal (MIM) structure, and concluded that our model [2] fails to mimic the effect of nonlocality at high frequencies. Here, we shall clarify the difference between our calculations and that in Schaich’s Comment, and highlight the validity of our model for a general class of plasmonic structures.Published versio

Noise-induced dynamics in bistable systems with delay

Noise-induced dynamics of a prototypical bistable system with delayed feedback is studied theoretically and numerically. For small noise and magnitude of the feedback, the problem is reduced to the analysis of the two-state model with transition rates depending on the earlier state of the system. In this two-state approximation, we found analytical formulae for the autocorrelation function, the power spectrum, and the linear response to a periodic perturbation. They show very good agreement with direct numerical simulations of the original Langevin equation. The power spectrum has a pronounced peak at the frequency corresponding to the inverse delay time, whose amplitude has a maximum at a certain noise level, thus demonstrating coherence resonance. The linear response to the external periodic force also has maxima at the frequencies corresponding to the inverse delay time and its harmonics.Comment: 4 pages, 4 figures, submitted to Physical Review Letter

Transformation-optics description of nonlocal effects in plasmonic nanostructures

We develop an insightful transformation-optics approach to investigate the impact that nonlocality has on the optical properties of plasmonic nanostructures. The light-harvesting performance of a dimer of touching nanowires is studied by using the hydrodynamical Drude model, which reveals nonlocal resonances not predicted by previous local calculations. Our method clarifies the interplay between radiative and nonlocal effects in this nanoparticle configuration, which enables us to elucidate the optimum size that maximizes its absorption and field enhancement capabilitiesThis work was supported by the ESF plasmonbionanosense program, the Leverhulme Trust, and the Engineering and Physical Sciences Research Council (EPSRC