We studied an anomaly in fractal dimensions measured from the attractors of dynamical systems driven by stochastically switched inputs. We calculated the dimensions for different switching time lengths in two-dimensional linear dynamical systems, and found that changes in the dimensions due to the switching time length had a singular point when the system matrix had two different real eigenvalues. Using partial dimensions along each eigenvector, we explicitly derived a generalized dimension $D_q$ and a multifractal spectrum $f\left(\alpha \right)$ to explain this anomalous property. The results from numerical calculations agreed well with those from analytical equations. We found that this anomaly is caused by linear independence, inhomogeneity of eigenvalues, and overlapping conditions. The mechanism for the anomaly could be identified for various inhomogeneous systems including nonlinear ones, and this reminded us of anomalies in some physical values observed in critical phenomena.

We investigate the emergence of singular solutions in a nonlocal model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the nonlocal effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of nonlocality on the system’s stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.

The simple model of a low Reynolds number swimmer made from three spheres that are connected by two arms is considered in its general form and analyzed. The swimming velocity, force-velocity response, power consumption, and efficiency of the swimmer are calculated both for general deformations and also for specific model prescriptions. The role of noise and coherence in the stroke cycle is also discussed.

The effects of elasticity on filament thinning and breakup are investigated in microchannel cross flow. When a viscous solution is stretched by an external immiscible fluid, a low $100\mathrm{ppm}$ polymer concentration strongly affects the breakup process, compared to the Newtonian case. Qualitatively, polymeric filaments show much slower evolution, and their morphology features multiple connected drops. Measurements of filament thickness show two main temporal regimes: flow- and capillary-driven. At early times both polymeric and Newtonian fluids are flow-driven, and filament thinning is exponential. At later times, Newtonian filament thinning crosses over to a capillary-driven regime, in which the decay is algebraic. By contrast, the polymeric fluid first crosses over to a second type of flow-driven behavior, in which viscoelastic stresses inside the filament become important and the decay is again exponential. Finally, the polymeric filament becomes capillary-driven at late times with algebraic decay. We show that the exponential flow thinning behavior allows a measurement of the extensional viscosities of both Newtonian and polymeric fluids.

The turbulent near-wake of a crossbar is investigated numerically with the lattice Boltzmann method (LBM). The crossbar is made up of two perpendicular square bars arranged in a biplane configuration and is included in the computational domain. The Reynolds number based on a bar diameter is about 1600. The numerical results are first tested against results of both particle image velocimetry (PIV) and laser Doppler velocimetry (LDV). The LBM data compare well with the PIV and LDV data. In particular, the LBM reproduces the generation of vortical structures at the crossbar as observed in the PIV data. The numerical results reveal the presence of intermittent lateral motions along the span of the two bars, yielding fingerlike structures. It is argued that these motions contribute to the formation of streamwise vortical structures just behind the crossbar. These streamwise structures interlace with lateral structures also generated at the crossbar. The region over which this activity takes place is about four diameters. Within this region, the turbulent kinetic energy at the crossbar centerline increases and reaches a maximum at a distance of about three diameters. As the downstream distance increases, the individual wakes merge to form a single wake with features, for $x/D\ge 20$, similar to those observed in grid-generated turbulence.

The spatial dependence of proton acceleration at the rear surface of a target that is irradiated by high-contrast and ultraintense laser pulses is investigated. Lateral movement of the proton acceleration position at the rear surface is observed; this is tested by a two-pinhole measurement which results in the observation of protons with a narrow energy band. This drifting is only observed when relativistic-intensity laser pulses irradiate targets with a small preplasma at oblique incidence, as is confirmed by two-dimensional particle-in-cell simulations. This scenario of proton acceleration by the fast-moving sheath field leads to energy selection of the accelerated protons as a function of observing position.

We pinpoint the limitations in traditional electroviscous analysis for narrow fluidic confinements. We show that because of neglecting the convective transport of ions originated out of the established streaming field itself, the traditional approach may result in physically inconsistent flow rate predictions. We show that the larger the value of an ionic Peclet number and narrower the confinement, the more conspicuous are these erroneous predictions, within threshold limits of the nondimensional $\zeta$ potential. We come up with an improved mathematical model to overcome such discrepancies.

At equilibrium, a fluid element, within a larger heat bath, receives random impulses from the bath. Those impulses, which induce stochastic transitions in the system (the fluid element), respect the principle of detailed balance, because the bath is also at equilibrium. Under continuous shear, the fluid element adopts a nonequilibrium steady state. Because the surrounding bath of fluid under shear is also in a nonequilibrium steady state, the system receives stochastic impulses with a nonequilibrium distribution. Those impulses no longer respect detailed balance, but are nevertheless constrained by rules. The rules in question, which are applicable to a wide subclass of driven steady states, were recently derived [R. M. L. Evans, Phys. Rev. Lett. 92, 150601 (2004); J. Phys A 38, 293 (2005)] using information-theoretic arguments. In the present paper, we provide a more fundamental derivation, based on the uncontroversial, non-Bayesian interpretation of probabilities as simple ratios of countable quantities. We apply the results to some simple models of interacting particles, to investigate the nature of forces that are mediated by a nonequilibrium noise source such as a fluid under shear.

We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which types of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem—e.g., dynamical, rigidity, and satisfiability-unsatisfiability (SAT-UNSAT) transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance, we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so-called rigidity transition where the thermodynamically relevant clusters become frozen. These nonequilibrium solutions belong to the most probable unfrozen clusters.

In a previous paper [E. Altshuler et al., Phys. Rev. Lett. 91, 014501 (2003)], the mechanism of “revolving rivers” for sandpile formation is reported: As a steady stream of dry sand is poured onto a horizontal surface, a pile forms which has a river of sand on one side flowing from the apex of the pile to the edge of the base. For small piles the river is steady, or continuous. For larger piles, it becomes intermittent. In this paper we establish experimentally the “dynamical phase diagram” of the continuous and intermittent regimes, and give further details of the piles’ “topography,” improving the previous kinematic model to describe it and shedding further light on the mechanisms of river formation. Based on experiments in Hele-Shaw cells, we also propose that a simple dimensionality reduction argument can explain the transition between the continuous and intermittent dynamics.

A systematic investigation by dielectric spectroscopy of 18 different water-rich mixtures with very different hydrophilic substances shows universal features for the water dynamics. The temperature dependence of the relaxation times exhibits a crossover from non-Arrhenius to Arrhenius behavior at the $T_g$ range of the mixtures. Furthermore, the temperature dependence of the relaxation times presents a universal behavior both above and below the crossover temperature. We also show that these features suggest that the observed crossover is associated with the emergence of confinement effects.

A simple flashing ratchet model in two dimensions is proposed to simulate the hand-over-hand motion of two head molecular motors such as kinesin. Extensive Langevin simulations of the model are performed. Good qualitative agreement with the expected behavior is observed. We discuss different regimes of motion and efficiency depending on model parameters.

We consider a linear reversible isomerization reaction $A\rightleftharpoons B$ under subdiffusion described by continuous time random walks (CTRW). The reactants’ transformations take place independently on the motion and are described by constant rates. We show that the form of the ensuing system of mesoscopic reaction-subdiffusion equations is unusual: the equation for time derivative of say $A\left(x,t\right)$ contains the terms depending not only on $\Delta A$, but also on $\Delta B$. This mirrors the fact that in subdiffusion the flux of particles at time $t$ is defined by the distributions of the particles’ concentrations at all previous times. Since the particles which jump as $A$ at time $t$ could previously be both $A$ or $B$, this flux depends on both $A$ and $B$ concentrations.

Patterns in dissipative systems with weakly broken symmetry are studied based upon the simplest canonical equation (generalized Nikolaevskiy model). A generic cubic dispersion equation governing stability of steady spatially periodic patterns is derived and analyzed. A domain of stable states in the space of the problem parameters (stability balloon) is obtained. It is shown that the domain is characterized by unusual scaling properties, so that its different parts obey different scalings. The results obtained may be applied to describe instabilities of advancing fronts and interfaces, pattern formation in reaction-diffusion systems, nonlinear evolution of seismic waves, and other phenomena.

We rigorously derive a dramatically simplified kinetic model for fluids with internal degrees of freedom. With proper discretization in velocity space, the model leads to a lattice Boltzmann model for polyatomic gases. The macroscopic recovery of correct hydrodynamics is theoretically shown and numerically validated.

A detailed study was performed of the fracture behavior of toasted rusk rolls, a cellular solid food, at different water activities and morphologies. We find that the energies of the emitted sound pulses follow Gutenberg-Richter power laws with characteristic exponents $b\sim 1.5$. The scaling exponents varied only within a range of 0.2 when the method of fracture, humidity, or morphology was changed. However, differences in $b$ were observed, indicating nonuniversal behavior, that seems to be related to morphology and water activity. Also, power law scaling behavior was observed for the waiting time distributions with an exponent $a\sim 1.9$.

The stability of synchronized states (frequency locked states) in networks of phase oscillators is investigated for several network topologies. It is shown that for some topologies there is more than one stable synchronized state according to the sign of coupling strength between oscillators. It is also shown that in some cases the synchronized state corresponds to zero order parameter.

We study the effects of kinetic helicity fluctuations in a turbulence with large-scale shear using two different approaches: the spectral $\tau$ approximation and the second-order correlation approximation (or first-order smoothing approximation). These two approaches demonstrate that homogeneous kinetic helicity fluctuations alone with zero mean value in a sheared homogeneous turbulence cannot cause a large-scale dynamo. A mean-field dynamo is possible when the kinetic helicity fluctuations are inhomogeneous, which causes a nonzero mean $\alpha$ effect in a sheared turbulence. On the other hand, the shear-current effect can generate a large-scale magnetic field even in a homogeneous nonhelical turbulence with large-scale shear. This effect was investigated previously for large hydrodynamic and magnetic Reynolds numbers. In this study we examine the threshold required for the shear-current dynamo versus Reynolds number. We demonstrate that there is no need for a developed inertial range in order to maintain the shear-current dynamo (e.g., the threshold in the Reynolds number is of the order of 1).

Governing equations for two-dimensional inviscid free-surface flows with constant vorticity over arbitrary nonuniform bottom profile are presented in exact and compact form using conformal variables. An efficient and very accurate numerical method for this problem is developed.