We investigate the effects of random perturbations on fully chaotic open systems. Perturbations can be applied to each trajectory independently (white noise) or simultaneously to all trajectories (random map). We compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time-dependent conditionally-map-invariant measure. For the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map. In random maps we show that the escape rate $\kappa$ and dimensions $D$ of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation. We discuss the accuracy (bias) and precision (variance) of finite-size estimators of $\kappa$ and $D$, and show that the improvement of the precision of the estimations with the number of trajectories $N$ is extremely slow ($\propto 1/\ln N$). We also argue that the finite-size $D$ estimators are typically biased. General theoretical results are combined with analytical calculations and numerical simulations in area-preserving baker maps.

The transition from a two-dimensional to three-dimensional flow in systems with spatial O(2) symmetry and spatiotemporal Z${}_2$ symmetry happens in many fluid systems, like wakes or periodically forced flows. In most of these systems, the dynamics after the first bifurcation is very complex and involves cascades of bifurcations in a very narrow parameter range. A numerical study of a flow in an enclosed cylindrical cavity driven by axial oscillations of the sidewall, which allows a detailed study of the secondary bifurcations and the corresponding mode interactions, is presented. The study focuses on a codimension-2 point that acts as the organizing center of the dynamics for moderate values of the forcing frequency. The unraveled dynamics is very rich, including slow-fast dynamics and hysteresis, and may help understand the bifurcation cascades in more complex systems.

A lattice Boltzmann model (LBM) is proposed based on the phase-field theory to simulate incompressible binary fluids with density and viscosity contrasts. Unlike many existing diffuse interface models which are limited to density matched binary fluids, the proposed model is capable of dealing with binary fluids with moderate density ratios. A new strategy for projecting the phase field to the viscosity field is proposed on the basis of the continuity of viscosity flux. The new LBM utilizes two lattice Boltzmann equations (LBEs): one for the interface tracking and the other for solving the hydrodynamic properties. The LBE for interface tracking can recover the Chan-Hilliard equation without any additional terms; while the LBE for hydrodynamic properties can recover the exact form of the divergence-free incompressible Navier-Stokes equations avoiding spurious interfacial forces. A series of 2D and 3D benchmark tests have been conducted for validation, which include a rigid-body rotation, stationary and moving droplets, a spinodal decomposition, a buoyancy-driven bubbly flow, a layered Poiseuille flow, and the Rayleigh-Taylor instability. It is shown that the proposed method can track the interface with high accuracy and stability and can significantly and systematically reduce the parasitic current across the interface. Comparisons with momentum-based models indicate that the newly proposed velocity-based model can better satisfy the incompressible condition in the flow fields, and eliminate or reduce the velocity fluctuations in the higher-pressure-gradient region and, therefore, achieve a better numerical stability. In addition, the test of a layered Poiseuille flow demonstrates that the proposed scheme for mixture viscosity performs significantly better than the traditional mixture viscosity methods.

We present a computational method to calculate arbitrary pair correlation functions of an orthorhombic system in the most efficient way. The algorithm is demonstrated by the calculation of the radial distribution function of shock compressed liquid hydrogen.

We consider fractional Stefan melting problems which involve a memory of the latent-heat accumulation. We show that the manner in which the memory of the latent-heat accumulation is recorded depends on the assumed nature of the transition between the liquid and the solid phases. When a sharp interface between the liquid and the solid phases is assumed, the memory of the accumulation of the latent heat is “lumped” in the history of the speed of the interface. In contrast, when a diffuse interface is assumed, the memory of the accumulation is “distributed” throughout the liquid phase. By use of an example problem, we demonstrate that the equivalence of the sharp- and diffuse-interface models can only occur when there is no memory in the system.

We study theoretically the smectic-$A$ to -$C$ phase transition in isotropic disordered environments. Surprisingly, we find that, as in the clean smectic-$A$ to -$C$ phase transition, smectic layer fluctuations do not affect the nature of the transition, in spite of the fact that they are much stronger in the presence of the disorder. As a result, we find that the universality class of the transition is that of the “random field $XY$ model” ($RFXY$).

A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor $\mathsf{D}$ as well as a positive definite scalar $\chi$ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of $\chi$ and $\mathsf{D}$ and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution $P(\chi ,\mathsf{D})$ and the two-point spatial correlation functions for $\chi$ and $\mathsf{D}$. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume $\Omega$ and dimensionality. As a corollary to our work, we identify the field $h_{\chi }$ conjugate to $\chi$ and show that this field may be tuned to produce a transition to a state where the ensemble average $\langle \chi \rangle$ and the correlation length of $\chi$ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.

The penetration of a rigid body with a randomly rough, self-affine surface in a half space filled with a linearly viscous elastomer is studied numerically using the method of boundary elements. Using Radok's principle of functional equations, it is shown analytically that this problem is closely related to the recently investigated problem of contact of self-affine surfaces with an elastic half space. We show that the penetration velocity occurs to be a power function of the applied force and time, the corresponding exponents depending only on the Hurst exponent. For comparison, the same problem is solved using the method of reduction of dimensionality. Both three-dimensional numerical results and the method of reduction of dimensionality support the analytical predictions provided by general scaling arguments.

The distribution of escape times is usually studied in open billiards theory. In this work, we will concentrate on another important question: The distribution of outgoing rays by exit directions, which we refer to as billiard's indicatrix. It can be obtained analytically and consists of two parts: the symmetric diffuse part and the asymmetric directed part. The criterion for the separation of the indicatrix into these two parts is established. The asymmetry of the directed part of the indicatrix and the influence of the billiard's borders on it is investigated. We also propose a method of the creation of a matte surface model using open billiards with a fully diffuse indicatrix.

We present results for objects settling in a cornstarch suspension. Two surprising phenomena can be found in concentrated suspensions. First, the settling object does not attain a terminal velocity but exhibits oscillations around a terminal velocity when traveling through the bulk of the liquid. Second, close to the bottom, the object comes to a full stop but then reaccelerates before coming to another stop. This cycle can be repeated up to 6 or 7 times before the object reaches the bottom to come to a final stop. For the bulk, we show that shear-thickening models are insufficient to account for the observed oscillations and that the history of the suspension needs to be taken into account. A hysteretic model, that goes beyond the traditional viscoelastic ones, describes the experiments quite well but still misses some details. The behavior at the bottom can be modeled with a minimal jamming model.

We analyze a class of estimators of the generalized diffusion coefficient for fractional Brownian motion $B_t$ of known Hurst index $H$, based on weighted functionals of the single-time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true ensemble-average generalized diffusion coefficient to any necessary precision from single-trajectory data, but at the expense of a progressively higher experimental resolution. Convergence is fastest around $H\simeq 0.30$, a value in the subdiffusive regime.

The Lévy walk model is a stochastic framework of enhanced diffusion with many applications in physics and biology. Here we investigate the time-averaged mean squared displacement $\overline{{\delta }^2}$ often used to analyze single particle tracking experiments. The ballistic phase of the motion is nonergodic and we obtain analytical expressions for the fluctuations of $\overline{{\delta }^2}$. For enhanced subballistic diffusion we observe numerically apparent ergodicity breaking on long time scales. As observed by Akimoto [Phys. Rev. Lett. 108, 164101 (2012)], deviations of temporal averages $\overline{{\delta }^2}$ from the ensemble average $\langle x^2\rangle$ depend on the initial preparation of the system, and here we quantify this discrepancy from normal diffusive behavior. Time-averaged response to a bias is considered and the resultant generalized Einstein relations are discussed.

We report the measurements of active and passive viscosities of a bent-core nematic liquid crystal. The active viscosity is measured using a rheometer and the passive viscosities are measured by measuring the self-diffusion coefficient of a microsphere in the aligned sample. The effective active viscosity is larger and the effect of presmectic fluctuations are observed much higher than the $N$–Sm-$C$ transition temperature than commonly seen in calamitic liquid crystals. The temperature dependence of passive viscosities is stronger than that of active viscosities. The self-diffusion of the microsphere is reduced and appears to be non-Gaussian as the $N$–Sm-$C$ transition is approached. The study of Brownian fluctuations is useful to understand the smectic fluctuations in bent-core nematic liquid crystals.

We study with numerical simulations the transverse fluctuations in quasi-one-dimensional systems of particles in a thermal bath, near the zigzag transition. We show that close to the zigzag threshold, the transverse fluctuations exhibit an anormal diffusion, characterized by a mean square displacement that increases as the square root of time. In contrast with the longitudinal fluctuations, this behavior of the transverse fluctuations cannot be explained by the single-file ordering. We provide an analytical modelization, and in the thermodynamic limit we demonstrate the existence of this subdiffusive regime near the zigzag transition, showing that it results from overdamped collective modes of the system. These calculations are extended to finite systems, in excellent agreement with the simulations data. We also exhibit some effects of the thermal fluctuations on the zigzag transition, and analyze them in the light of stochastic bifurcation theory.

We consider a driven tracer particle (TP) in a bath of hard-core particles undergoing continuous exchanges with a reservoir. We develop an analytical framework which allows us to go beyond the standard force-velocity relation used for this minimal model of active microrheology and quantitatively analyze, for any density of the bath particles, the fluctuations of the TP position and their correlations with the occupation number of the bath particles. We obtain an exact Einstein-type relation which links these fluctuations in the absence of a driving force and the bath particles density profiles in the linear driving regime. For the one-dimensional case we also provide an approximate but very accurate explicit expression for the variance of the TP position and show that it can be a nonmonotonous function of the bath particles density: counterintuitively, an increase of the density may increase the dispersion of the TP position. We show that this nontrivial behavior, which could in principle be observed in active microrheology experiments, is induced by subtle cross correlations quantified by our approach.

We investigate experimentally the drag force experienced by a rigid intruder immersed in a bidimensional dense and disordered granular flow as well as the displacement fields of grains around it. The approach to jamming is revealed through the drastic increase of force levels as well as the disappearance of the cavity that forms behind the intruder with increasing granular packing fraction. A simple picture based on the compressibility of the granular material is proposed to explain the dependence of the cavity area on the packing fraction, intruder size, and cell width.

The drift-diffusion equation is first solved analytically for the dissociation rate and lifetime of a biomolecular or colloidal dimer bonded by realistic intermolecular potentials, under shear flow. Then we show using rigidity percolation concepts that the lifetime of a generic cluster formed under shear is controlled by the typical lifetime of a single bond in its interior. The latter, however, is also affected by collective stress transmission from other bonds in the aggregate, which we account for by introducing a semiempirical, analytical stress transmission efficiency $0⩽\Gamma ⩽1$ calibrated on several simulation data sets. We show that aggregate breakup is a thermally activated process in which the activation energy is controlled by the interplay between intermolecular forces and the shear drift. The collective contribution to the overall shear drift term is dominant for large enough fractal aggregates, while surface erosion prevails for small and compact aggregates. The crossover between the two regimes occurs when $\Gamma N\simeq 2$, where both the number of particles in the cluster $N$ and the stress transmission efficiency $\Gamma$ depend on the aggregate structure through the fractal dimension $d_f$. The analytical framework for the aggregate breakup rate is in quantitative agreement with experiments and can be used in future studies in the population balance modeling of colloidal and protein aggregation.

Transmission spectra and rotation of the plane of polarization of light were measured on perfect samples of cholesteric liquid crystal. We found peculiarities of the spectral dependence of rotation of the plane of polarization related to Pendellösung oscillations that were predicted theoretically but never observed experimentally. The experimental data are well described by expressions based on an analytical solution of the Maxwell equations. An approach based on fundamental Kramers-Kronig relations is used to describe the optical properties of the photonic crystal. It is shown that the rotatory power of cholesterics consists of two parts: one of them is related to the diffraction-induced circular dichroism whereas the other one originates from the tails of the Mauguin rotation of the plane of light polarization.

The three-variable Goodwin model constitutes a prototypical oscillator based on a negative feedback loop. It was used as a minimal model for circadian oscillations. Other core models for circadian clocks are variants of the Goodwin model. The Goodwin oscillator also appears in many studies of coupled oscillator networks because of its relative simplicity compared to other biophysical models involving a large number of variables and parameters. Because the synchronization properties of Goodwin oscillators still remain difficult to explore mathematically, further simplifications of the Goodwin model have been sought. In this paper, we investigate the strong negative feedback limit of Goodwin equations by using asymptotic techniques. We find that Goodwin oscillations approach a sequence of decaying exponentials that can be described in terms of a single-variable leaky integrated-and-fire model.

We present a simple, intuitive, and effective approach for network clustering. It is based on basic concepts of linear algebra such as efficient calculation of spanning trees, and can be implemented in a few lines of code. We introduce the node separation measure spanning tree separation (STS) and the corresponding graph distance measure spanning tree vector similarity distance (STVSD). We demonstrate that the STS is a link salience measure able to identify the backbone of networks. The STVSD is used to reveal the hierarchical community structure of networks. We show that it, together with the clustering quality measure partition density, is on a par with the best graph or network clustering methods known, in terms of both quality and efficiency. In perspective, we note that our approach could also handle weighted and directed networks and could be used for identification of overlapping communities.