Long data sets are one of the prime requirements of time series analysis techniques to unravel the dynamics of an underlying system. However, acquiring long data sets is often not possible. In this paper, we address the question of whether it is still possible to understand the complete dynamics of a system if only short (but many) time series are observed. The key idea is to generate a single long time series from these short segments using the concept of recurrences in phase space. This long time series is constructed so as to exhibit a dynamics similar to that of a long time series obtained from the corresponding underlying system.

We derive a family of differential identities for parametric correlation functions in disordered systems by casting them as first- or second-order Ward identities of an associated matrix model. We show that this approach allows for a systematic classification of such identities, and provides a template for deriving higher-order results. We also reestablish and generalize some identities of this type which had been derived previously using a different method.

Experimentally, it turns out that radiation forces from a cw laser on a liquid-liquid interface are able to produce giant deformations (up to about $100\mu \mathrm{m}$), if the system is close to the critical point where the surface tension becomes small. We present a model for such a fingerlike deformation, implying that the system is described as an optical fiber. One reason for introducing such a model is that the refractive index difference in modern experiments, such as those of the Bordeaux group, is small, of the same order as in practical fibers in optics. It is natural therefore to adopt the hybrid ${\mathrm{HE}}_{11}$ mode, known from fiber theory as the fundamental mode for the liquid system. We show how the balance between hydrodynamical and radiation forces leads to a stable equilibrium point for the liquid column. Also, we calculate the narrowing of the column radius as the depth increases. Comparison with experimental results of the Bordeaux group yields quite satisfactory agreement as regards the column width.

A beer bottle or soda can on a table, when slightly tipped and released, falls to an upright position and then rocks up to a somewhat opposite tilt. Superficially this rocking motion involves a collision when the flat circular base of the container slaps the table before rocking up to the opposite tilt. A keen eye notices that the after-slap rising tilt is not generally just diametrically opposite the initial tilt but is veered to one side or the other. Cushman and Duistermaat [Regular Chaotic Dyn. 11, 31 (2006)] recently noticed such veering when a flat disk with rolling boundary conditions is dropped nearly flat. Here, we generalize these rolling disk results to arbitrary axi-symmetric bodies and to frictionless sliding. More specifically, we study motions that almost but do not quite involve a face-down collision of the round container’s bottom with the tabletop. These motions involve a sudden rapid motion of the contact point around the circular base. Surprisingly, similar to the rolling disk, the net angle of motion of this contact point is nearly independent of initial conditions. This angle of turn depends simply on the geometry and mass distribution but not on the moment of inertia about the symmetry axis. We derive simple asymptotic formulas for this “angle of turn” of the contact point and check the result with numerics and with simple experiments. For tall containers (height much bigger than radius) the angle of turn is just over $\pi$ and the sudden rolling motion superficially appears as a nearly symmetric collision leading to leaning on an almost diametrically opposite point on the bottom rim.

Telomeres are DNA repeats protecting chromosomal ends which shorten with each cell division, eventually leading to cessation of cell growth. We present a population mixture model that predicts an exponential decrease in telomere length with time. We analytically solve the dynamics of the telomere length distribution. The model provides an excellent fit to available telomere data and accounts for the previously unexplained observation of telomere elongation following stress and bone marrow transplantation, thereby providing insight into the nature of the telomere clock.

Experiments suggest that localization via self-trapping plays a dominant role in the behavior of a low-mass particle, e.g., an electron, positron, or positronium atom, in both liquids and supercritical fluids. In the latter case, the behavior is dominated by the liquid-vapor critical point. However, because of its large isothermal compressibility, the critical point is difficult to probe both experimentally and theoretically. Here we present the results of path-integral computations of the characteristics of a generic self-trapped particle at the critical point of a Lennard-Jones fluid for a positive particle-atom scattering length. We carefully investigate the dependence of the properties of both the self-trapped quantum particle and the proximal fluid on the range of the direct particle-atom interaction. To the extent that the generic particle mimics the behavior of ortho-positronium, qualitative information is provided on the pick-off decay rate. In general, compared with self-trapping at higher temperature, we find that the localized quantum defect has a much larger range of influence on the host fluid. In particular, it appears that long-range density oscillations are induced in the fluid surrounding the defect. The results also suggest that, even at the critical point, there is a minimum interaction range below which self-trapping does not occur.

A scaling theory of replica symmetry breaking (RSB) in the Sherrington-Kirkpatrick (SK) model is presented in the framework of critical phenomena for the scaling regime of large RSB orders $\kappa$, small temperatures $T$, and small (homogeneous) magnetic fields $H$. We employ the pseudodynamical picture [R. Oppermann, M. J. Schmidt, and D. Sherrington, Phys. Rev. Lett. 98, 127201 (2007)], where two critical points $\mathcal{C}\mathcal{P}1$ and $\mathcal{C}\mathcal{P}2$ are associated with the order function’s pseudodynamical limits ${\lim }_{a\to \infty }q\left(a\right)=1$ and ${\lim }_{a\to 0}q\left(a\right)=0$ at $(T=0$, $H=0$, $1∕\kappa =0)$. $\mathcal{C}\mathcal{P}1$- and $\mathcal{C}\mathcal{P}2$-dominated contributions to the free energy functional $F\left[q\left(a\right)\right]$ require an unconventional scaling hypothesis. We determine the scaling contributions in accordance with detailed numerical self-consistent solutions for up to 200 orders of RSB. Power laws, scaling functions, and crossover lines are obtained. $\mathcal{C}\mathcal{P}1$-dominated behavior is found for the nonequilibrium susceptibility, which decays like ${\chi }_1={\kappa }^{-5∕3}{f}_1(T∕{\kappa }^{-5∕3})$, for the entropy, which obeys $S(T=0)\sim {\chi }_1^2$, and for the subclass of diverging parameters $a_i={\kappa }^{5∕3}{f}_{a_i}(T∕{\kappa }^{-5∕3})$ [describing Parisi box sizes $m_i\left(T\right)\equiv a_i\left(T\right)T$], with $f_1\left(\zeta \right)\sim \zeta$ and $f_{a_i}\left(\zeta \right)\sim 1∕\zeta$ for $\zeta \to \infty$, while $f\left(0\right)$ is finite. $\mathcal{C}\mathcal{P}2$-dominated behavior, controlled by the magnetic field $H$ while temperature is irrelevant, is retrieved in the plateau height (or width) of the order function $q\left(a\right)$ according to $q_{pl}\left(H\right)={\kappa }^{-1}{f}_{pl}(H^{2∕3}∕{\kappa }^{-1})$ with $f_{pl}{\phantom{\mid }\left(\zeta \right)\mid }_{\zeta \to \infty }\sim \zeta$ and $f_{pl}\left(0\right)$ finite. Divergent characteristic RSB orders ${\kappa }_{\mathcal{C}\mathcal{P}1}\left(T\right)\sim T^{-3∕5}$ and ${\kappa }_{\mathcal{C}\mathcal{P}2}\left(H\right)\sim H^{-2∕3}$, respectively, describe the crossover from mean field SK- to RSB-critical behavior with rational-valued exponents extracted with high precision from our RSB data. The order function $q\left(a\right)$ is obtained as a fixed-point function $q^{*}\left(a^{*}\right)$ of RSB flow, in agreement with integrated fixed-point energy and susceptibility distributions.

We discuss how various models of scale-free complex networks approach their limiting properties when the size $N$ of the network grows. We focus mainly on equilibrated networks and their finite-size degree distributions. Our results show that the position of the cutoff in the degree distribution, $k_{\text{cutoff}}$, scales with $N$ in a different way than predicted for $N\to \infty$; that is, subleading corrections to the scaling $k_{\text{cutoff}}\sim N^{\alpha }$ are strong even for networks of order $N\sim {10}^9$ nodes. We observe also a logarithmic correction to the scaling for degenerated graphs with the degree distribution $\pi \left(k\right)\sim k^{-3}$. On the other hand, the distribution of the maximal degree $k_{\max }$ may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We argue that $k_{\max }\sim N^{{\alpha }^{\prime }}$ with an exponent ${\alpha }^{\prime }=\min [\alpha ,1∕(\gamma -1)]$, where $\gamma$ is the exponent in the power law $\pi \left(k\right)\sim k^{-\gamma }$. We also present some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.

We present an in-depth analysis of the geometrical percolation behavior in the continuum of random assemblies of hard oblate ellipsoids of revolution. Simulations were carried out by considering a broad range of aspect ratios, from spheres up to aspect-ratio-100 platelike objects, and with various limiting two-particle interaction distances, from 0.05 times the major axis up to 4.0 times the major axis. We confirm the widely reported trend of a consistent lowering of the hard particle critical volume fraction with increase of the aspect ratio. Moreover, by assimilating the limiting interaction distance to a shell of constant thickness surrounding the ellipsoids, we propose a simple relation based on the total excluded volume of these objects which allows us to estimate the critical concentration from a quantity that is quasi-invariant over a large spectrum of limiting interaction distances. Excluded volume and volume quantities are derived explicitly.

We propose a general approach to study spin models with two symmetric absorbing states. Starting from the microscopic dynamics on a square lattice, we derive a Langevin equation for the time evolution of the magnetization field, that successfully explains coarsening properties of a wide range of nonlinear voter models and systems with intermediate states. We find that the macroscopic behavior only depends on the first derivatives of the spin-flip probabilities. Moreover, an analysis of the mean-field term reveals the three types of transitions commonly observed in these systems—generalized voter, Ising and directed percolation. Monte Carlo simulations of the spin dynamics qualitatively agree with theoretical predictions.

We study the time evolution of different Laponite suspensions from a low-viscosity ergodic state to a viscoelastic nonergodic state over a wide range of volume fractions and salt contents. We find that the evolution of nonergodicity parameter (Debye-Waller factor) splits into two branches for all the samples, which correspond to two distinct dynamically arrested states. At moderately high salt concentrations, on the other hand, a third nonergodic state appears that is different from the above two nonergodic states. Measurement of the conductivity of Laponite solutions in pure water shows that the contribution of counterions in the ionic strength is considerable and their role should be taken into account in interpretations of aging dynamics and the phase diagram. Based on these data and available data in the literature, we propose a (nonequilibrium) phase diagram for Laponite suspensions.

We report the study of the effect of a static electric field on the huge optical nonlinearity of methyl-red doped nematic liquid crystals. Experimental data are well fitted using a theoretical model that takes into account the modulation of the surface charge density due to the impinging light beam. It is demonstrated that the optical nonlinearity can be varied by orders of magnitude with application of a low voltage below the threshold of the Fredericks transition. These results confirm the previously proposed model of surface induced nonlinear effects.

We introduce a generalization of the parallel, or Crow-Kimura, and Eigen models of molecular evolution to represent the exchange of genetic information between individuals in a population. We study the effect of different schemes of genetic recombination on the steady-state mean fitness and distribution of individuals in the population, through an analytic field theoretic mapping. We investigate both horizontal gene transfer from a population and recombination between pairs of individuals. Somewhat surprisingly, these nonlinear generalizations of quasispecies theory to modern biology are analytically solvable. For two-parent recombination, we find two selected phases, one of which is spectrally rigid. We present exact analytical formulas for the equilibrium mean fitness of the population, in terms of a maximum principle, which are generally applicable to any permutation invariant replication rate function. For smooth fitness landscapes, we show that when positive epistatic interactions are present, recombination or horizontal gene transfer introduces a mild load against selection. Conversely, if the fitness landscape exhibits negative epistasis, horizontal gene transfer or recombination introduces an advantage by enhancing selection towards the fittest genotypes. These results prove that the mutational deterministic hypothesis holds for quasispecies models. For the discontinuous single sharp peak fitness landscape, we show that horizontal gene transfer has no effect on the fitness, while recombination decreases the fitness, for both the parallel and the Eigen models. We present numerical and analytical results as well as phase diagrams for the different cases.

We study the percolation problem on the Apollonian network model. The Apollonian networks display many interesting properties commonly observed in real network systems, such as small-world behavior, scale-free distribution, and a hierarchical structure. By taking advantage of the deterministic hierarchical construction of these networks, we use the real-space renormalization-group technique to write exact iterative equations that relate percolation network properties at different scales. More precisely, our results indicate that the percolation probability and average mass of the percolating cluster approach the thermodynamic limit logarithmically. We suggest that such ultraslow convergence might be a property of hierarchical networks. Since real complex systems are certainly finite and very commonly hierarchical, we believe that taking into account finite-size effects in real-network systems is of fundamental importance.

It is shown that the information entropy based on waiting-time distributions (WTDs) offers a natural and robust measure of nonexponentiality of the distributions in the form of the Schrödinger-Brillouin negentropy, or equivalently the Kullback-Leibler divergence, and has a straightforward interpretation in terms of transition state theory. Other measures of nonexponentiality of WTDs, based on comparison of the standard deviation and the median with the mean waiting time, are also discussed. The theoretical analysis is illustrated with results from protein folding studies.

We examine spatial chaos in a one-dimensional attractive Bose-Einstein condensate interacting with a Gaussian-like laser barrier and perturbed by a weak optical lattice. For a low laser barrier, chaotic regions of the parameters are demonstrated and the chaotic and regular states are illustrated numerically. In the high-barrier case, bounded perturbed solutions that describe a set of discrete chaotic states are constructed for discrete barrier heights and magic numbers of condensed atoms. Chaotic density profiles are exhibited numerically for the lowest quantum number, and analytically bounded but numerically unbounded Gaussian-like configurations are confirmed. It is shown that the chaotic wave packets can be controlled experimentally by adjusting the laser barrier potential.

We experimentally report the cyclic scenario of birth and annihilation of periodic orbits in a photonic integrated circuit as the feedback phase of the electric field varies. The latter is also shown to result in minimal alterations in the statistical properties of the chaotic attractor, with simultaneously transiting the Hurst exponent $H$, erratically, below and above the critical value of $H=0.5$ that indicates regular Brownian motion. Consequently there is an indication of the most effective operating regions with minimized predictability, which hinders eavesdropping and the progress of forecasting the development of the chaotic light carrier.

We use speckle-visibility spectroscopy to measure the time dependence of bubble rearrangement events that are driven by coarsening in an aqueous foam. This technique gives the time trace for the average scattering site speed within a prescribed volume of the sample. Results are analyzed in terms of distributions of event times, event speeds, and event displacements. The distribution of rest times between successive events is also measured; comparison with diffusing-wave spectroscopy results shows that the spatial structure of a typical event consists of a core of only a few bubbles which undergo topology change plus a surrounding shell of bubbles which shift by an amount that decays to one wavelength at four to five bubbles away. No correlations are found between the durations, speeds, and rest times between successive events.

Recently, Pantina and Furst [Phys. Rev. Lett. 94, 138301 (2005)] experimentally demonstrated the existence of tangential forces between bonded colloidal particles and the capability of these bonds to supporting bending moments. We introduce a model to be used in computer simulations that describes these tangential interactions. We show how the model parameters can be determined from experimental data. Simulations using the model are in agreement to the measurement by Pantina and Furst. Application of the model to an aggregate with fractal structure leads to more realistic behavior than using classical approaches only.

The evolution model with parallel mutation-selection scheme is solved for the case when selection is accompanied by base substitutions, insertions, and deletions. The fitness is assumed to be either a single-peak function (i.e., having one finite discontinuity) or a smooth function of the Hamming distance from the reference sequence. The mean fitness is calculated exactly in large-genome limit. In the case of insertions and deletions the evolution characteristics depend on the choice of reference sequence.