Granger causality is a statistical concept devised to reconstruct and quantify predictive information flow between stochastic processes. Although the general concept can be formulated model-free it is often considered in the framework of linear stochastic processes. Here we show how local linear model descriptions can be employed to extend Granger causality into the realm of nonlinear systems. This novel treatment results in maps that resolve Granger causality in regions of state space. Through examples we provide a proof of concept and illustrate the utility of these maps. Moreover, by integration we convert the local Granger causality into a global measure that yields a consistent picture for a global Ornstein-Uhlenbeck process. Finally, we recover invariance transformations known from the theory of autoregressive processes.

We study the transport efficiency of excitations on complex quantum networks with loops. For this we consider sequentially growing networks with different topologies of the sequential subgraphs. This can lead either to a universal complete breakdown of transport for complete-graph-like sequential subgraphs or to optimal transport for ringlike sequential subgraphs. The transition to optimal transport can be triggered by systematically reducing the number of loops of complete-graph-like sequential subgraphs in a small-world procedure. These effects are explained on the basis of the spectral properties of the network's Hamiltonian. Our theoretical considerations are supported by numerical Monte Carlo simulations for complex quantum networks with a scale-free size distribution of sequential subgraphs and a small-world-type transition to optimal transport.

Polymer networks at the margins of mechanical stability are known to be highly sensitive to applied forces and fields and to exhibit an anomalously large resistance to deformation. In this paper, we study the effects of hydrodynamic interactions on the behavior of marginal networks using a hybrid molecular dynamics and multiparticle collision dynamics simulation technique. We examine how the filament and solvent properties affect the response of marginal networks to shear. We find that the stiffening of the network shows a stronger dependence on the shear frequency when hydrodynamic interactions are present than when they are not. The network shear modulus scales as $G^{\prime }\sim {\omega }^{{\alpha }_c}$, with a critical stiffening exponent ${\alpha }_c$ that can be controlled by varying the relative concentrations of the network and the solvent. Our results show that this arises due to the solvent aiding the relaxation of the network and suppressing the network nonaffinity, with the system deforming more affinely when hydrodynamic interactions are maximized.

A simple finite-element analysis with varying damping strength is used to model the athermal shear rheology of densely packed glassy systems at a continuum level. We focus on the influence of dissipation on bulk rheological properties. Our numerical studies, done over a wide range of damping coefficients, identify two well-separated rheological regimes along with a crossover region controlled by a critical damping. In the overdamped limit, inertial effects are negligible and the rheological response is well described by the commonly observed Herschel-Bulkley equation. In stark contrast, inertial vibrations in the underdamped regime prompt a significant drop in the mean-stress level, leading to a nonmonotonic constitutive relation. The observed negative slope in the flow curve, which is a signature of mechanical instability and thus permanent shear banding, arises from the sole influence of inertia, in qualitative agreement with the recent molecular dynamics study of Nicolas et al., Phys. Rev. Lett. 116, 058303 (2016).

We theoretically and experimentally investigate the flow field that emerges from a rodlike microrotor rotating about its center in a nonaxisymmetric manner. A simple theoretical model is proposed that uses a superposition of two rotlets as a fundamental solution to the Stokes equation. The predictions of this model are compared to measurements of the azimuthal and radial microfluidic velocity field components that are induced by a rotor composed of fused microscopic spheres. The rotor is driven magnetically and the fluid flow is measured with the help of a probe particle fixed by an optical tweezer. We find considerable deviations of the mere azimuthal flow pattern induced by a single rotating sphere as it has been reported by Di Leonardo et al. [Phys. Rev. Lett. 96, 134502 (2006)]. Notably, the presence of a radial velocity component that manifests itself by an oscillation of the probe particle with twice the rotor frequency is observed. These findings open up a way to discuss possible radial transport in microfluidic devices.

The effects of flow and particle properties on margination of particles in red blood cell (RBC) suspensions is investigated using direct numerical simulation (DNS) of cellar blood flow. We focus on margination of particles in the flow of moderately dense suspensions of RBCs. We hypothesize that margination rate in nondilute suspensions is mainly driven by the RBC-enhanced diffusion of marginating particles in the RBC-filled region. We derive a scaling law for margination length in a straight channel. Margination length increases cubically with channel height and is independent of shear rate. We verify this scaling law for margination length by DNS of flowing RBCs and marginating particles. We also show that rigidity and size both lead to particle margination with rigidity having a more significant effect compared to size within the range of parameters in this study.

Dynamics of electron heating in the radiation-pressure-driven acceleration through self-induced transparency (SIT) is investigated with the help of particle-in-cell simulations. The SIT is achieved through laser filamentation which is seeded by the transverse density modulations due to the Rayleigh-Taylor-like instability. We observe stronger SIT induced electron heating for the longer duration laser pulses leading to deterioration of accelerated ion beam quality (mainly energy spread). Such heating can be controlled to obtain a quasimonoenergetic beam by cascaded foils targets where a second foil behind the main accelerating foil acts as a laser reflector to suppress the SIT.

We analyze the FitzHugh-Nagumo equations subject to time-delayed self-feedback in the activator variable. Parameters are chosen such that the steady state is stable independent of the feedback gain and delay $\tau$. We demonstrate that stable large-amplitude $\tau$-periodic oscillations can, however, coexist with a stable steady state even for small delays, which is mathematically counterintuitive. In order to explore how these solutions appear in the bifurcation diagram, we propose three different strategies. We first analyze the emergence of periodic solutions from Hopf bifurcation points for $\tau$ small and show that a subcritical Hopf bifurcation allows the coexistence of stable $\tau$-periodic and stable steady-state solutions. Second, we construct a $\tau$-periodic solution by using singular perturbation techniques appropriate for slow-fast systems. The theory assumes $\tau =O\left(1\right)$ and its validity as $\tau \to 0$ is investigated numerically by integrating the original equations. Third, we develop an asymptotic theory where the delay is scaled with respect to the fast timescale of the activator variable. The theory is applied to the FitzHugh-Nagumo equations with threshold nonlinearity, and we show that the branch of $\tau$-periodic solutions emerges from a limit point of limit cycles.

By means of a perturbative scheme, we determine analytically the capillary energy of a spheroidal colloid floating on a deformed fluid interface in terms of the local curvature tensor of the background deformation. We validate our results, that hold for small ellipticity of the particle and small deformations of the surface, by an exact numerical calculation. As an application of our perturbative approach, we determine the asymptotic interaction, for large separations $d$, between two different spheroidal particles. The dominant contribution is quadrupolar and proportional to $d^{-4}$. It coincides with the known superposition approximation and is zero if one of the two particles is spherical. The next to leading approximation, proportional to $d^{-8}$, is always attractive and independent of the orientation of the two colloids. It is the dominant contribution to the interaction between a spheroidal and a spherical colloid.

We show that the flexoelectric model of chiral and achiral modulated nematics predicts the compression modulus that is by orders of magnitude lower than the measured values. The discrepancy is much larger in the chiral modulated nematic phase, in which the measured value of the compression modulus is of the same order of magnitude as in achiral modulated nematics, even though the heliconical pitch is by an order of magnitude larger. The relaxation of a one-constant approximation in the biaxial elastic model used for chiral modulated nematics does not solve the problem. Therefore, we propose a structural model of the modulated nematic phase, which is consistent with the current experimental evidence and can also explain large compression modulus: the structure consists of short-range smectic clusters with a fourfold symmetry and periodicity of two molecular distances. In chiral systems, chiral interactions lead to a helicoidal structure of such clusters.

We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder—an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)—and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment ${\tilde{m}}_2$ of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling ${\tilde{m}}_2\sim t^{\gamma }$ (for some constant $\gamma$) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, ${\tilde{m}}_2$ and the inverse participation ratio $P^{-1}$ satisfy scaling relations ${\tilde{m}}_2\sim t^{\gamma }$ and $P^{-1}\sim t^{-\eta }$, and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with $\eta <0.1$) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly $\gamma \approx 1.7$ and $\eta \approx 1$) for all three types of disorder and for both types of initial conditions.

We investigate the energy transport in one-dimensional disordered granular solids by extensive numerical simulations. In particular, we consider the case of a polydisperse granular chain composed of spherical beads of the same material and with radii taken from a random distribution. We start by examining the linear case, in which it is known that the energy transport strongly depends on the type of initial conditions. Thus, we consider two sets of initial conditions: an initial displacement and an initial momentum excitation of a single bead. After establishing the regime of sufficiently strong disorder, we focus our study on the role of nonlinearity for both sets of initial conditions. By increasing the initial excitation amplitudes we are able to identify three distinct dynamical regimes with different energy transport properties: a near linear, a weakly nonlinear, and a highly nonlinear regime. Although energy spreading is found to be increasing for higher nonlinearities, in the weakly nonlinear regime no clear asymptotic behavior of the spreading is found. In this regime, we additionally find that energy, initially trapped in a localized region, can be eventually detrapped and this has a direct influence on the fluctuations of the energy spreading. We also demonstrate that in the highly nonlinear regime, the differences in energy transport between the two sets of initial conditions vanish. Actually, in this regime the energy is almost ballistically transported through shocklike excitations.

The vibrational properties of model amorphous materials are studied by combining complete analysis of the vibration modes, dynamical structure factor, and energy diffusivity with exact diagonalization of the dynamical matrix and the kernel polynomial method, which allows a study of very large system sizes. Different materials are studied that differ only by the bending rigidity of the interactions in a Stillinger-Weber modelization used to describe amorphous silicon. The local bending rigidity can thus be used as a control parameter, to tune the sound velocity together with local bonds directionality. It is shown that for all the systems studied, the upper limit of the Boson peak corresponds to the Ioffe-Regel criterion for transverse waves, as well as to a minimum of the diffusivity. The Boson peak is followed by a diffusivity's increase supported by longitudinal phonons. The Ioffe-Regel criterion for transverse waves corresponds to a common characteristic mean-free path of 5–7 Å (which is slightly bigger for longitudinal phonons), while the fine structure of the vibrational density of states is shown to be sensitive to the local bending rigidity.

The motion of contracting and expanding bubbles in an incompressible chaotic flow is analyzed in terms of the finite-time Lyapunov exponents. The viscous forces acting on the bubble surface depend not only on the relative acceleration but also on the time dependence of the bubble volume, which is modeled by the Rayleigh-Plesset equation. The effect of bubble coalescence on the coherent structures that develop in the flow is studied using a simplified bubble merger model. Contraction and expansion of the bubbles is favored in the vicinity of the coherent structures. Time evolution of coalescence bubbles follows a Lévy distribution with an exponent that depends on the initial distance between bubbles. Mixing patterns were found to depend heavily on merging and on the time-dependent volume of the bubbles.

Surface growth by the conserved-noise restricted solid-on-solid model is investigated on diluted lattices, i.e., on percolation networks that are embedded in two spatial dimensions. The growth exponent $\beta$ and the roughness exponent $\alpha$ are defined, respectively, by the mean-square surface width via $W^2\left(t\right)\sim t^{2\beta }$ and the mean-square saturated width via $W_{\mathrm{sat}}^2\left(L\right)\sim L^{2\alpha }$, where $L$ is the system size. These are measured on both an infinite network and a backbone network and the results are compared with power-counting predictions obtained using the fractional Langevin equation. While the Monte Carlo results on deterministic fractal substrates show excellent agreement with the predictions [D. H. Kim and J. M. Kim, Phys. Rev. E 84, 011105 (2011)], the results on critical percolation networks deviate by 8%–12% from these predictions.

An analysis is given of numerical simulation data to size $L=128$ on the archetype square lattice Ising spin glasses (ISGs) with bimodal ($\pm J$) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent $\eta \equiv 0$, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature $T^{*}\left(L\right)$ there is a regime of effectively continuous energy levels; below $T^{*}\left(L\right)$ there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. $T^{*}\left(L\right)$ tends to zero at very large $L$, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable $t=T$ and with a scaling variable ${\tau }_b=T^2/(1+T^2)$ suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility $\chi ({\tau }_b,L)$ and second moment correlation length $\xi ({\tau }_b,L)$ in the thermodynamic limit regime are extrapolated to the ${\tau }_b=0$ critical limit. The Gaussian critical exponent estimates from the simulations, $\eta =0$ and $\nu =3.55\left(5\right)$, are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are $\eta =0.20\left(2\right)$ and $\nu =4.8\left(3\right)$, distinctly different from the Gaussian critical exponents.

We theoretically study how the measurements of the flexoelectric coefficients in nematic liquid crystals are affected by the inclusion of the director-density coupling energy. It is shown that this investigation is quite relevant for interpreting the data of experiments.

A propagating “beam” triggering a local phase transition in a nematic elastomer sets it into a crawling motion, which may morph due to buckling. We consider the motion of the various configurations of slender rods and thin stripes with both uniform and splayed nematic order in cross-section and detect the dependence of the gait and speed on flexural rigidity and substrate friction.

In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as “apples with apples.” Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.