We claim that looking at probability distributions of finite time largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincaré recurrences in the-quite-delicate case of dynamical systems with weak chaotic properties.

By exploiting the error functions of explicit symplectic integrators for solving separable Hamiltonians, I show that it is possible to develop explicit time-reversible symplectic integrators for solving nonseparable Hamiltonians of the product form. The algorithms are unusual in that they are of fractional orders.

For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix $R_{mn}$, whose elements converge to two constants. This allows for an effective extrapolation of the equation of state for these models. Due to a nearby (nonphysical) singularity on the negative real $z$ axis, standard methods (e.g., Padé approximants based on the cluster integrals expansion) fail to capture the behavior of these models near the ordering transition, and, in particular, do not detect the critical point. A recent work [E. Eisenberg and A. Baram, Proc. Natl. Acad. Sci. U.S.A. 104, 5755 (2007)] has shown that the critical exponents $\sigma$ and ${\sigma }^{\prime }$, characterizing the singularity of the density as a function of the activity, can be exactly calculated if the decay of the $R$ matrix elements to their asymptotic constant follows a $1/n^2$ law. Here we employ renormalization group (RG) arguments to extend this result and analyze cases for which the asymptotic approach of the $R$ matrix elements toward their limiting value is of a more general form. The relevant asymptotic correction terms (in RG sense) are identified, and we then present a corrected exact formula for the critical exponents. We identify the limits of usage of the formula and demonstrate one physical model, which is beyond its range of validity. The formula is validated numerically and then applied to analyze a number of concrete physical models.

Modeling of recorded time series may be used as a method of analysis for heart rate variability studies. In particular, the extraction of the first two Kramers-Moyal coefficients has been used in this context. Recently, the method was applied to a wide range of signal analysis: from financial data to physiological and biological time series. Modeling of the signal is important for the prediction and interpretation of the dynamics underlying the process. The method requires the determination of the Markov time. Obtaining the drift and diffusion term of the Kramers-Moyal expansion is crucial for the modeling of the original time series with the Langevin equation. Both Tabar et al. [Comput. Sci. Eng. 8, 54 (2006)] and T. Kuusela [Phys. Rev. E 69, 031916 (2004)] suggested that these terms may be used to distinguish healthy subjects from those with heart failure. The research groups applied a somewhat different methodology and obtained substantially different ranges of the Markov time. We show that the two studies may be considered consistent with each other as Kuusela analyzed 24 h recordings while Tabar et al. analyzed daytime and nighttime recordings, separately. However, both groups suggested using the Langevin equation for modeling of time series which requires the fluctuation force to be a Gaussian. We analyzed heart rate variability recordings for ten young male (age $26-4+3\text{y}$) healthy subjects. 24 h recordings were analyzed and 6-h-long daytime and nighttime fragments were selected. Similar properties of the data were observed in all recordings but all the nighttime data and seven of the ten 24 h series exhibited higher-order, non-negligible Kramers-Moyal coefficients. In such a case, the reconstruction of the time series using the Langevin equation is impossible. The non-negligible higher-order coefficients are due to autocorrelation in the data. This effect may be interpreted as a result of a physiological phenomenon (especially occurring for nighttime data): respiratory sinus arrhythmia (RSA). We detrended the nighttime recordings for the healthy subjects and obtained an asymmetry in the dependence of the diffusion term on the rescaled heart rate. This asymmetry seems to be an effect of different time scales during the inspiration and the expiration phase of breathing. The asymmetry was significantly decreased in the diffusion term found for detrended nighttime recordings obtained from five hypertrophic cardiomyopathy (HCM) patients. We conclude that the effect of RSA is decreased in the heart rate variability of HCM patients—a result which may contribute to a better medical diagnosis by supplying a new quantitative measure of RSA.

We use Monte Carlo simulations of the Lennard-Jones model to study the nucleation of a crystal phase at a flat surface. Our motivation is the observation that crystal phases almost always nucleate at a surface. We find that a surface phase transition (prefreezing) can control nucleation of the bulk crystal. This finding should be general and so surface phase behavior should be considered whenever nucleation of bulk phases at surfaces is considered. Also, nucleation of the bulk crystal transforms smoothly into the nucleation of a surface crystal layer as the bulk transition is crossed.

Dynamic field pumping principle has been developed utilizing the interactions of both the director and velocity fields and a temperature gradient $\mathbf{\nabla }T$. The orientational dynamics in the hybrid-oriented liquid-crystal (HOLC) microvolume confined between two infinitely long coaxial cylinders under the influence of the radially directed $\mathbf{\nabla }T$ has been investigated. We have carried out a numerical study of a system of hydrodynamic equations including director reorientation, fluid flow, and temperature redistribution across the HOLC cavity between two cylinders under the influence of $\mathbf{\nabla }T$, when the liquid-crystal cavity is heated both from outer (inner) to inner (outer) bounding cylinders. Calculations show that under the influence of $\mathbf{\nabla }T$ the initially quiescent HOLC drop settles down to a stationary flow regime, with the horizontal $u_{\text{st}}\left(r\right)$ component of velocity. The effects of $\mathbf{\nabla }T$, of the character of the preferred anchoring of the average molecular direction to the restricted cylinders, and of the size of the HOLC cavity on magnitude and direction of hydrodynamic flow—for a number of hydrodynamic regimes—has been investigated.

The flexoelectro-optic effect describes the rotation of the optic axis of a short-pitch chiral nematic liquid crystal under the application of an electric field. We investigate the effect in the uniform standing helix, or “Grandjean” configuration. An in-plane electric field is applied. The director profile is determined numerically using a static one-dimensional continuum model with strong surface anchoring. The Berreman method is used to solve for plane-wave solutions to Maxwell’s equations, and predict the optical properties of the resulting structure in general cases. By using a chiral nematic with short pitch between crossed polarizers an optical switch may be generated. With no applied field the configuration is nontransmissive at normal incidence, but becomes transmissive with an applied field. For this case, numerical results using the Berreman method are supplemented with an analytic theory and found to be in good agreement. The transmitted intensity as a function of tilt, the contrast ratio, and the tilt required for full intensity modulation are presented. The angular dependence of the transmission is calculated and the isocontrast curves are plotted. For typical material and cell parameters a switching speed of 0.017 ms and contrast ratio of 1500:1 at normal incidence are predicted, at a switch-on tilt of 41.5 degrees. Experimental verification of the analytic and numerical models is provided.

X-ray lasers may allow structural studies on single particles and biomolecules without crystalline periodicity in the samples. We examine here the effect of sample dynamics as a source of structural heterogeneity on the resolution of the reconstructed image of a small protein molecule. Structures from molecular-dynamics simulations of lysozyme were sampled and aligned. These structures were then used to calculate diffraction patterns corresponding to different dynamic states. The patterns were incoherently summed and the resulting data set was phased using the oversampling method. Reconstructed images of hydrated and dehydrated lysozyme gave resolutions of $3.7\text{Å}$ and $7.6\text{Å}$, respectively. These are significantly worse than the root-mean-square deviation of the hydrated ($2.7\text{Å}$ for all atoms and $1.45\text{Å}$ for $\text{C-}\alpha$ positions) or dehydrated ($3.7\text{Å}$ for all atoms and $2.5\text{Å}$ for $\text{C-}\alpha$ positions) structures. The noise introduced by structural dynamics and incoherent addition of dissimilar structures restricts the maximum resolution to be expected from direct image reconstruction of dynamic systems. A way of potentially reducing this effect is by grouping dynamic structures into distinct structural substates and solving them separately.

In this work we describe the interaction of the responses of neuronal networks to pairs of electrical stimuli. For this we use networks of dissociated cortical neurons cultured on planar microelectrode arrays. We compare the response to pairs of stimuli with the response to a stimulus in isolation. To evaluate the influence of both stimuli we introduce a normalization of the root-mean square of the response. Furthermore we consider the response to a pair of stimuli as the linear superposition of the two constituents. The two methods combined show that the neuronal network strongly suppresses the second stimulus. At the same time we find a possible window of integration in which two stimuli from separate locations in the network can interact to form a new response.

We investigated the effects of noise on periodic firing in the Hodgkin-Huxley nonlinear system. With mean input current $\mu$ as a bifurcation parameter, a bifurcation to repetitive spiking occurs at a critical value ${\mu }_c\approx 6.44$. The firing behavior was studied as a function of the mean and variance of the input current, firstly with initial resting conditions. Noise of a small amplitude can turn off the spiking for values of $\mu$ close to ${\mu }_c$, and the number of spikes undergoes a minimum as a function of the noise level. The robustness of these phenomena was confirmed by simulations with random initial conditions and with random time of commencement of the noise. Furthermore, their generality was indicated by their occurrence when additive noise was replaced by conductance-based noise. For long periods of observation, many frequent transitions may occur from spiking to nonspiking activity when the noise is sufficiently strong. Explanations of the above phenomena are sought in terms of the underlying bifurcation structure and the probabilities that noise shifts the process from the basin of attraction of a stable limit cycle to that of a stable rest state. The waiting times for such transitions depend strongly on the values of $\mu$ and $\sigma$ and on the forms of the basins of attraction. The observed effects of noise are expected to occur in diverse fields in systems with the same underlying dynamical structure.

The accuracy of earnings predictions is hampered by the several predominantly unpredictable effects due to the complex evolution of economy. Finding out which are the dominant market features embracing uncertainty is therefore the key to get beyond present state-of-art earnings forecasts. The analysis of annual revenues and earnings data (1954–2008) from the 500 largest-revenue U.S. companies suggests a linear relation between company expected mean profit and revenue. Annual profit fluctuations are then obtained as difference between actual annual profits and expected mean values. It is found that the temporal evolution of profit fluctuations for a single company displays a slowly decaying autocorrelation, yielding Hurst exponents in the range $H=0.75\pm 0.17$. The study of profits cross correlations between companies suggests a way to distinguish typical earnings years from anomalous ones by looking at minimal information structures contained within the space defined by the associated covariance metric.

We develop a renormalization group method to investigate synchronization clusters in a one-dimensional chain of nearest-neighbor coupled phase oscillators. The method is best suited for chains with strong disorder in the intrinsic frequencies and coupling strengths. The results are compared with numerical simulations of the chain dynamics and good agreement in several characteristics is found. We apply the renormalization group and simulations to Lorentzian distributions of intrinsic frequencies and couplings and investigate the statistics of the resultant cluster sizes and frequencies, as well as the dependence of the characteristic cluster length upon parameters of these Lorentzian distributions.

The Log-Poisson phenomenological description of the turbulent energy cascade is evoked to discuss high-order statistics of velocity derivatives and the mapping between their probability distribution functions at different Reynolds numbers. The striking confirmation of theoretical predictions suggests that numerical solutions of the flow obtained at low/moderate Reynolds numbers can play an important quantitative role in the analysis of experimental high Reynolds number phenomena, where small scales fluctuations are in general inaccessible from direct numerical simulations.

We discuss the propagation of nonlinear electromagnetic short waves in a magnetically saturated ferromagnetic thick film. The sample is magnetized to saturation by a field perpendicular to both the film plane and the propagation direction. A $\left(2+1\right)$ dimensional asymptotic model equation generalizing the sine-Gordon one is derived. Line soliton solutions are exhibited; their stability condition is derived. When unstable, line solitons decay into stable two-dimensional lumps, which are studied both numerically and analytically.

The Brownian dynamics of a single microorganism coupled by chemotaxis to a diffusing concentration field that is secreted by the microorganism itself is studied by computer simulations in spatial dimensions $d=1,2,3$. Both cases of a chemoattractant and a chemorepellent are discussed. For a chemoattractant, we find a transient dynamical arrest until the microorganism diffuses for long times. For a chemorepellent, there is a transient ballistic motion in all dimensions and a long-time diffusion. These results are interpreted with the help of a theoretical analysis.

We use a topological approach to describe the frustration- and field-induced phase transitions exhibited by the infinite-range $XY$ model on the $A{B}_2$ chain, including noncollinear spin structures. For this purpose, we have computed the Morse number and the Euler characteristic, as well as other topological invariants, which are found to behave similarly as a function of the energy level in the context of Morse theory. In particular, we use a method based on an analogy with statistical mechanics to compute the Euler characteristic, which proves to be quite feasible. We also introduce topological energies which help us to clarify several properties of the transitions, both at zero and finite temperatures. In addition, we establish a nontrivial direct connection between the thermodynamics of the systems, which have been solved exactly under the saddle-point approach, and the topology of their configuration space. This connection allows us to identify the nondegeneracy condition under which the divergence of the density of Jacobian’s critical points $\left[j_l\left(E\right)\right]$ at the critical energy of a topology-induced phase transition, proposed by Kastner and Schnetz [Phys. Rev. Lett. 100, 160601 (2008)] as a necessary criterion, is suppressed. Finally, our findings and those available in the literature suggest that the cusplike singularity exhibited both by the Euler characteristic and the topological contribution for the entropy at the critical energy, put together with the divergence of $j_l\left(E\right)$, and emerge as necessary and sufficient conditions for the occurrence of the finite-temperature topology-induced phase transitions examined in this work. The general character of this proposal should be subject to a more rigorous scrutiny.

Dynamical scaling functions above $T_c$ for the characteristic frequencies and the dynamical correlation functions of the order parameter and the conserved density of model C are calculated in one loop order. By a proper exponentiation procedure these results can be extended in order to consider the changes in these functions using the fixed point values and exponents in two loop order. The dynamical amplitude ratio $R$ of the characteristic frequencies is generalized to the critical region. Surprisingly the decay of the shape functions at large scaled frequency does not behave as expected from applying scaling arguments. The exponent $\upsilon$ of the decay does not change when going from the critical to the hydrodynamic region although the shape functions change. The value of $\upsilon$ for the order parameter is in agreement with its value in the critical region, whereas for the conserved density it is equal to 2, the value in the hydrodynamic region.

We study the random walk problem on a class of deterministic scale-free networks displaying a degree sequence for hubs scaling as a power law with an exponent $\gamma =\text{log}3/\text{log}2$. We find exact results concerning different first-passage phenomena and, in particular, we calculate the probability of first return to the main hub. These results allow to derive the exact analytic expression for the mean time to first reach the main hub, whose leading behavior is given by $\tau \sim V^{1-1/\gamma }$, where $V$ denotes the size of the structure, and the mean is over a set of starting points distributed uniformly over all the other sites of the graph. Interestingly, the process turns out to be particularly efficient. We also discuss the thermodynamic limit of the structure and some local topological properties.

Water droplets can jump during vapor condensation on solid benzene near its melting point. This phenomenon, which can be viewed as a kind of micro scale steam engine, is studied experimentally and numerically. The latent heat of condensation transferred at the drop three phase contact line melts the substrate during a time proportional to $R$ (the drop radius). The wetting conditions change and a spontaneous jump of the drop results in random direction over length $\sim 1.5R$, a phenomenon that increases the coalescence events and accelerates the growth. Once properly rescaled by the jump length scale, the growth dynamics is, however, similar to that on a solid surface.

The aim of this paper is twofold: one is to give a detailed description of an alternative graph-based analysis method, which we call saddle connectivity graph, for analyzing the global topography and the dynamical properties of many-dimensional potential-energy landscapes and the other is to give examples of applications of this method in the analysis of the kinetics of realistic systems. A Dijkstra-type shortest path algorithm is proposed to extract dynamically dominant transition pathways by kinetically defining transition costs. The applicability of this approach is first confirmed by an illustrative example of a low-dimensional random potential. We then show that a coarse-graining procedure tailored for saddle connectivity graphs can be used to obtain the kinetic properties of 13- and 38-atom Lennard-Jones clusters. The coarse-graining method not only reduces the complexity of the graphs, but also, with iterative use, reveals a self-similar hierarchical structure in these clusters. We also propose that the self-similarity is common to many-atom Lennard-Jones clusters.