We characterize the time evolution of a $d$-dimensional probability distribution by the value of its final entropy. If it is near the maximally possible value we call the evolution mixing, if it is near zero we say it is purifying. The evolution is determined by the simplest nonlinear equation and contains a $d\times d$ matrix as input. Since we are not interested in a particular evolution but in the general features of evolutions of this type, we take the matrix elements as uniformly distributed random numbers between zero and some specified upper bound. Computer simulations show how the final entropies are distributed over this field of random numbers. The result is that the distribution crowds at the maximum entropy, if the upper bound is unity. If we restrict the dynamical matrices to certain regions in matrix space, to diagonal or triangular matrices, for instance, then the entropy distribution is maximal near zero, and the dynamics typically becomes purifying.

Segregation in vertically vibrated binary granular mixtures with the same size is studied experimentally. The partially segregated state occurring in this system is observed carefully. We find that the characteristic of the partially segregated state is that the lighter particles tend to rise and form a pure layer on the top of the system while the heavier particles and some of the lighter ones stay at the bottom and form a mixed layer. The ratio of the thickness of the pure top layer to that of the whole system can be taken as an order parameter, which describes the degree of the segregation quantitatively and is useful in the investigation of the system. By use of it, we find that the segregation state is only dependent on the density ratio of the two kinds of particles. The dependent of the segregation on the vibration frequency is also studied by use of this order parameter, and finally, two typical phase diagrams are given.

Many physical and engineering systems exhibit cascades of periodic attractors arranged in period increment and period adding sequences as a parameter is varied. Such systems have been found to yield piecewise smooth maps, and in some cases the obtained map is discontinuous. By investigating the normal form of such maps, we have detected a type of codimension-three bifurcation which serves as the organizing center of periodic and aperiodic dynamics in the parameter space. The results will help in understanding the occurrence and structure of such cascades observed in many nonsmooth systems in science and engineering.

We present an experimental study of cessations and reversals of the large-scale circulation (LSC) in turbulent thermal convection in a cylindrical cell of aspect ratio $\left(\Gamma \right)$ $1∕2$. It is found that cessations and reversals of the LSC occur in $\Gamma =1∕2$ geometry an order-of-magnitude more frequently than they do in $\Gamma =1$ cells, and that after a cessation the LSC is most likely to restart in the opposite direction, i.e., reversals of the LSC are the most probable cessation events. This contrasts sharply to the finding in $\Gamma =1$ geometry and implies that cessations in the two geometries are governed by different dynamics. It is found that the occurrence of reversals is a Poisson process and that a stronger rebound of the flow strength after a reversal or cessation leads to a longer period of stability of the LSC. Several properties of reversals and cessations in this system are found to be statistically similar to those of geomagnetic reversals. A direct measurement of the velocity field reveals that a cessation corresponds to a momentary decoherence of the LSC.

A discrete nonlinear system driven at one end by a periodic excitation of frequency above the upper band edge (the discreteness induced cutoff) is shown to be a means to (1) generate propagating breather excitations in a long chain and (2) reveal the bistable property of a short chain. After detailed numerical verifications, the bistability prediction is demonstrated experimentally on an electrical transmission line made of 18 inductance-capacitance $\left(LC\right)$ cells. The numerical simulations of the $LC$-line model allow us also to verify the breather generation prediction with a striking accuracy.

Random sequential adsorption (RSA), on a two-dimensional continuum substrate, of different types of zero area objects that disallow domain formation and hence lead to jamming, is examined by simulation. In all the cases, in the asymptotic time regime, the approach of the number density $\rho \left(t\right)$ at instant $t$ to jamming density $\rho \left(\infty \right)$ is found to exhibit power law $\rho \left(\infty \right)-\rho \left(t\right)\sim t^{-p}$ as that for RSA of finite area objects. These results suggest the possibility of the power law being universal for all jamming systems in RSA on a continuum substrate. A generalized analytical treatment is also proposed.

In this work we derive an analytic expression for the Kolmogorov-Sinai entropy of dilute wet granular matter, valid for any spatial dimension. The grains are modeled as hard spheres and the influence of the wetting liquid is described according to the capillary model, in which dissipation is due to the hysteretic cohesion force of capillary bridges. The Kolmogorov-Sinai entropy is expanded in a series with respect to density. We find a rapid increase of the leading term when liquid is added. This demonstrates the sensitivity of the granular dynamics to humidity, and shows that the liquid significantly increases the chaoticity of the granular gas.

We investigate orientational relaxation of a model discotic liquid crystal, consisting of disclike molecules, by molecular dynamics simulations along two isobars starting from the high temperature isotropic phase. The two isobars have been so chosen that (a) the phase sequence isotropic- $\left(I\text{−}\right)$ nematic- $\left(N\text{−}\right)$ columnar $\left(C\right)$ appears upon cooling along one of them and (b) the sequence isotropic- $\left(I\text{−}\right)$ columnar- $\left(C\right)$ along the other. While the orientational relaxation in the isotropic phase near the $I\text{−}N$ phase transition in system (a) shows a power law decay at short to intermediate times, such power law relaxation is not observed in the isotropic phase near the $I\text{−}C$ phase boundary in system (b). In order to understand this difference (the existence or the absence of the power law decay), we calculated the growth of the orientational pair distribution functions (OPDFs) near the $I\text{−}N$ phase boundary and also near the $I\text{−}C$ phase boundary. We find that the OPDF shows a marked growth in long range correlation as the $I\text{−}N$ phase boundary is approached in the $I\text{−}N\text{−}C$ system (a), but such a growth is absent in the $I\text{−}C$ system, which appears to be consistent with the result that $I\text{−}N$ phase transition in the former is weakly first order while the $I\text{−}C$ phase transition in the latter is not weak. As the system settles into the nematic phase, the decay of the single-particle second-rank orientational time correlation function follows a pattern that is similar to what is observed with calamitic liquid crystals and supercooled molecular liquids.

In this work, we analyze the defect and antidefect distribution in the nematic calamitic phase of a lyotropic liquid crystal [the ternary mixture formed by potassium laurate (KL), decanol (DeOH), and water]. We obtain defects with wedge disclinations of strength $\pm 1∕2$, and the scaling exponent determined by the defect-antidefect correlation was $0.29\pm 0.07$. This value is in good agreement with the theoretical value of $1∕4$ obtained by the Kibble mechanism. The constant of the scaling relation of the defect and antidefect distribution is also discussed. We compare our results with the values obtained by Digal et al. [Phys. Rev. Lett. 83, 5030 (1999)] who used a thermotropic liquid crystal.

The effect of polymer polydispersity on the depletion interaction between two plates immersed in a nonadsorbing polymer solution is studied by self-consistent-field theory. The depletion potential is calculated numerically for the Schulz molecular weight distribution. The results show that as the two plates approach the polymers with different chain lengths are excluded from the gap gradually for conformational entropy penalty, and the range of the depletion potential increases and the depth of the potential decreases with increasing polydispersity. For the case of two large spheres the Derjaguin approximation is used to study the effect of polydispersity. The result shows that the polydispersity has little effect on the contact potential. However, it induces more broad interaction than the monodisperse polymers.

We determine the mobility of dye-labeled polystyrene molecules in solution by fluorescence correlation spectroscopy (FCS) over a wide range of concentrations and molecular weights (ranging from $3.9\times {10}^3\text{to}1550\times {10}^3\mathrm{g}∕\mathrm{mol}$). In order to obtain absolute values of the diffusion coefficient, which can be compared to diffusion coefficients determined by other methods, the size of the focal volume has been determined by independent experiments and theoretical calculations. All data demonstrate that FCS is uniquely suited to explore polymer dynamics in solution. The mobility of the chains as expressed through the self-diffusion coefficient is significantly slowed down above the overlap concentration $c^{*}$. The dependence of $c^{*}$ on molecular weight is well described by the power law $c^{*}\propto M_w^{1-3\nu }$ ($\nu$: Flory exponent). A comparison with the data taken from the literature demonstrates that the overlap concentration presents a robust concept that holds for a wide range of molecular weights.

Simulation studies are presented that probe the possibility of using high-field $(>100\mathrm{kV}∕\mathrm{cm})$, short-duration $(\sim 50\mathrm{ns})$ electrical pulses for nonthermal and reversible cessation of biological electrical signaling pathways. This would have obvious applications in neurophysiology, clinical research, neuromuscular stimulation therapies, and even nonlethal bioweapons development. The concept is based on the creation of a sufficiently high density of pores on the nerve membrane by an electric pulse. This modulates membrane conductance and presents an effective “electrical short” to an incident voltage wave traveling across a nerve. Net blocking of action potential propagation can then result. A continuum approach based on the Smoluchowski equation is used to treat electroporation. This is self-consistently coupled with a distributed circuit representation of the nerve dynamics. Our results indicate that poration at a single neural segment would be sufficient to produce an observable, yet reversible, effect.

We study the phenomenon of nonlinear stochastic resonance (SR) in a complex noisy system formed by a finite number of interacting subunits driven by rectangular pulsed time periodic forces. We find that very large SR gains are obtained for subthreshold driving forces with frequencies much larger than the values observed in simpler one-dimensional systems. These effects are explained using simple considerations.

The dynamics of traveling waves in a nonlinear dissipative system are studied analytically and numerically. Spatiotemporal forcing and feedback forcing are applied to the traveling waves in a phase-separated system undergoing chemical reactions. The stability of the traveling waves and interesting, unexpected behavior, including the reversal of the propagation direction are analyzed in one dimension. The phase dynamical approach is applied to gain a theoretical understanding of the dynamics.

We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.

Spinodal decomposition can be driven by either diffusion or self-induced convection; the importance of convection relative to diffusion depends on the Péclet number, defined as the ratio between convective and diffusive mass fluxes. Diffusion is the dominating mechanism of phase segregation when the Péclet number is small—i.e., when viscosity and diffusivity are large—or when the domain characteristic size is small. For low-viscosity mixtures, convection is the dominating process and the segregation is very rapid as it takes a few seconds compared to the hours needed in the case of pure diffusion. In such cases, strong convective motion of the phase segregating domains is generated even in small-size systems and is almost independent of the temperature difference as long as it is below the transition value. We study experimentally the enhancement of heat transfer in a $1\text{−}\mathrm{mm}$-thick cell. A water-acetonitrile-toulene mixture is quenched into a two-phase region so as to induce convection-driven spinodal decomposition. The heat transfer rate is measured and compared to that obtained in the absence of convective motion. A substantial reduction in the cooling time obtains in the case of spinodal decomposition. The heat transfer enhancement induced by this self-induced, disordered but effectively convective effect may be exploited in the cooling or heating of small-scale systems whereby forced convection cannot be achieved because of the small sizes involved. A scaling analysis of the data based on the diffuse interface $H$ model for a symmetric mixture near the equilibrium point yields very encouraging agreement and insights.

Carrier wave shocking is studied using the pseudospectral spatial-domain (PSSD) technique. We describe the shock detection diagnostics necessary for this numerical study and verify them against theoretical shocking predictions for the dispersionless case. These predictions show a carrier envelope phase and pulse bandwidth sensitivity in the single-cycle regime. The flexible dispersion management offered by the PSSD enables us to independently control the linear and nonlinear dispersion. Customized dispersion profiles allow us to analyze the development of both carrier self-steepening and shocks. The results exhibit a marked asymmetry between normal and anomalous dispersion, both in the limits of the shocking regime and in the (near) shocked pulse wave forms. Combining these insights, we offer some suggestions on how carrier shocking (or at least extreme self-steepening) might be realized experimentally.

The simulation of liquid systems in a nonequilibrium steady state under planar elongational flow (PEF) for indefinite time is possible only with the use of the so-called Kraynik-Reinelt (KR) periodic boundary conditions (PBCs) on the simulation cell. These conditions admit a vast range of implementation parameters, which regulate how the unit lattice is deformed under elongation and periodically remapped onto itself. Clearly, nonequilibrium properties of homogeneous systems in a steady state have to be independent of the boundary conditions imposed on the unit cell. In order to confirm the independence of measurable properties of a system under PEF from the particular set of periodic boundary conditions, we compute the Lyapunov spectra, apply the conjugate pairing rule, and carefully analyze the so-called unpaired exponents for an atomic fluid of various sizes and state points. We further compute the elongational viscosity for various implementations of boundary conditions. All our results confirm the independence from KR PBCs for the dynamics of phase-space trajectories and for the transport coefficients.

The self-modulation of waves propagating in nonlinear magnetic metamaterials is investigated. Considering the propagation of a modulated amplitude magnetic field in such a medium, we show that the self-modulation of the carrier wave leads to a spontaneous energy localization via the generation of localized envelope structures (envelope solitons), whose form and properties are discussed. These results are also supported by numerical calculations.

Suprathreshold stochastic resonance (SSR) is a form of noise-enhanced signal transmission that occurs in a parallel array of independently noisy identical threshold nonlinearities, including model neurons. Unlike most forms of stochastic resonance, the output response to suprathreshold random input signals of arbitrary magnitude is improved by the presence of even small amounts of noise. In this paper, the information transmission performance of SSR in the limit of a large array size is considered. Using a relationship between Shannon’s mutual information and Fisher information, a sufficient condition for optimality, i.e., channel capacity, is derived. It is shown that capacity is achieved when the signal distribution is Jeffrey’s prior, as formed from the noise distribution, or when the noise distribution depends on the signal distribution via a cosine relationship. These results provide theoretical verification and justification for previous work in both computational neuroscience and electronics.