Static and Dynamic Correlations in Many-Particle Lyapunov Vectors

We introduce static and dynamic correlation functions for the spatial densities of Lyapunov vector fluctuations. They enable us to show, for the first time, the existence of hydrodynamic Lyapunov modes in chaotic many-particle systems with soft core interactions, which indicates universality of this phenomenon. Our investigations for Lennard-Jones fluids yield, in addition to the Lyapunov exponent - wave vector dispersion, the collective dynamic excitations of a given Lyapunov vector. In the limit of purely translational modes the static and dynamic structure factor are recovered.Comment: 5 pages, 5 figure

Preisach models of hysteresis driven by Markovian input processes

We study the response of Preisach models of hysteresis to stochastically fluctuating external fields. We perform numerical simulations which indicate that analytical expressions derived previously for the autocorrelation functions and power spectral densities of the Preisach model with uncorrelated input, hold asymptotically also if the external field shows exponentially decaying correlations. As a consequence, the mechanisms causing long-term memory and 1/f-noise in Preisach models with uncorrelated inputs still apply in the presence of fast decaying input correlations. We collect additional evidence for the importance of the effective Preisach density previously introduced even for Preisach models with correlated inputs. Additionally, we present some new results for the output of the Preisach model with uncorrelated input using analytical methods. It is found, for instance, that in order to produce the same long-time tails in the output, the elementary hysteresis loops of large width need to have a higher weight for the generic Preisach model than for the symmetric Preisach model. Further, we find autocorrelation functions and power spectral densities to be monotonically decreasing independently of the choice of input and Preisach density

Lyapunov instabilities of Lennard-Jones fluids

Recent work on many particle system reveals the existence of regular collective perturbations corresponding to the smallest positive Lyapunov exponents (LEs), called hydrodynamic Lyapunov modes. Until now, however, these modes are only found for hard core systems. Here we report new results on Lyapunov spectra and Lyapunov vectors (LVs) for Lennard-Jones fluids. By considering the Fourier transform of the coordinate fluctuation density $u^{(\alpha)}(x,t)$, it is found that the LVs with $\lambda \approx 0$ are highly dominated by a few components with low wave-numbers. These numerical results provide strong evidence that hydrodynamic Lyapunov modes do exist in soft-potential systems, although the collective Lyapunov modes are more vague than in hard-core systems. In studying the density and temperature dependence of these modes, it is found that, when the value of Lyapunov exponent $\lambda^{(\alpha)}$ is plotted as function of the dominant wave number $k_{max}$ of the corresponding LV, all data from simulations with different densities and temperatures collapse onto a single curve. This shows that the dispersion relation $\lambda^{(\alpha)}$ vs. $k_{max}$ for hydrodynamical Lyapunov modes appears to be universal irrespective of the particle density and temperature of the system. Despite the wave-like character of the LVs, no step-like structure exists in the Lyapunov spectrum of the systems studied here, in contrast to the hard-core case. Further numerical simulations show that the finite-time LEs fluctuate strongly. We have also investigated localization features of LVs and propose a new length scale to characterize the Hamiltonian spatio-temporal chaotic states.Comment: 12 pages, 20 figure

Comparison of covariant and orthogonal Lyapunov vectors

Two sets of vectors, covariant and orthogonal Lyapunov vectors (CLVs/OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover the previous classification of two universality classes works for CLVs as well, i.e. the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs by CLVs. For general dissipative systems with nonhyperbolic dynamics the long wave length structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the micro- reversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs and hyperbolicity under changes of coordinate are discussed in appendices

Infinite invariant density in a semi-Markov process with continuous state variables

We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the inter-event time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized L\'evy walk model and the energy of a particle in the trap model. In our model, the inter-event-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long inter-event times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these non-stationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.Comment: 16 pages, 7 figure

Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities

Anisotropic diffusion processes emerge in various fields such as transport in biological tissue and diffusion in liquid crystals. In such systems, the motion is described by a diffusion tensor. For a proper characterization of processes with more than one diffusion coefficient an average description by the mean squared displacement is often not sufficient. Hence, in this paper, we use the distribution of diffusivities to study diffusion in a homogeneous anisotropic environment. We derive analytical expressions of the distribution and relate its properties to an anisotropy measure based on the mean diffusivity and the asymptotic decay of the distribution. Both quantities are easy to determine from experimental data and reveal the existence of more than one diffusion coefficient, which allows the distinction between isotropic and anisotropic processes. We further discuss the influence on the analysis of projected trajectories, which are typically accessible in experiments. For the experimentally relevant cases of two- and three-dimensional anisotropic diffusion we derive specific expressions, determine the diffusion tensor, characterize the anisotropy, and demonstrate the applicability for simulated trajectories.Comment: v2: 14 pages, 4 figures, added section about curvature, added references, several clarifications and enhancements; v1: 13 pages, 4 figure

Dimensional collapse and fractal attractors of a system with fluctuating delay times

A frequently encountered situation in the study of delay systems is that the length of the delay time changes with time, which is of relevance in many fields such as optics, mechanical machining, biology or physiology. A characteristic feature of such systems is that the dimension of the system dynamics collapses due to the fluctuations of delay times. In consequence, the support of the long-trajectory attractors of this kind of systems is found being fractal in contrast to the fuzzy attractors in most random systems

Normal and anomalous random walks of 2-d solitons

Solitons, which describe the propagation of concentrated beams of light through nonlinear media, can exhibit a variety of behaviors as a result of the intrinsic dissipation, diffraction, and the nonlinear effects. One of these phenomena, modeled by the complex Ginzburg-Landau equation, are chaotic explosions, transient enlargements of the soliton that may induce random transversal displacements, which in the long run lead to a random walk of the soliton center. As we show in this work, the transition from non-moving to moving solitons is not a simple bifurcation but includes a sequence of normal and anomalous random walks. We analyze their statistics with the distribution of generalized diffusivities, a novel approach that has been used successfully for characterizing anomalous diffusion.Comment: 10 pages, 5 figure

Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions

We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a `simple' and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions it becomes characterized by normal diffusion.Comment: 6 pages, 6 figure

How to compare diffusion processes assessed by single-particle tracking and pulsed field gradient nuclear magnetic resonance

Heterogeneous diffusion processes occur in many different fields such as transport in living cells or diffusion in porous media. A characterization of the transport parameters of such processes can be achieved by ensemble-based methods, such as pulsed field gradient nuclear magnetic resonance (PFG NMR), or by trajectory-based methods obtained from single-particle tracking (SPT) experiments. In this paper, we study the general relationship between both methods and its application to heterogeneous systems. We derive analytical expressions for the distribution of diffusivities from SPT and further relate it to NMR spin-echo diffusion attenuation functions. To exemplify the applicability of this approach, we employ a well-established two-region exchange model, which has widely been used in the context of PFG NMR studies of multiphase systems subjected to interphase molecular exchange processes. This type of systems, which can also describe a layered liquid with layer-dependent self-diffusion coefficients, has also recently gained attention in SPT experiments. We reformulate the results of the two-region exchange model in terms of SPT-observables and compare its predictions to that obtained using the exact transformation which we derived.Comment: v2: 14 pages, 6 figures, several enhancements, added references; v1: 7 pages, 3 figure