The diffusion equation fails to offer a satisfactory description of dynamics when the correlation between successive motions of dispersive particle is large. Further, the equation is associated with the pitfall of predicting the infinite propagation speed of the diffusive particles. Both limitations arise as the Brownian picture (on which the equation is based on) does not take into account the inertia of the diffusive particles. This could be overcome by introducing a delay in the diffusive flux. The resultant delay-diffusion equation may be converted to an ordinary differential equation by linearizing the flux with respect to the delay, a technique valid in general for small delays. In this article I show that the condition for a spatial bifurcation induced by diffusive delay in the delay differential model is significantly different from the condition derived in an earlier work modifying the reaction-telegraphic equation based on a microscopic approach. The latter necessitates criteria not realizable in common reaction-diffusion models and therefore effectively rules out such kinds of instability in these models. I show here that the instability condition derived with the original nontruncated version of the delayed reaction-diffusion equation does not impose any constraint on the model kinetics, but only requires the diffusive memory to be large enough. Numerical simulation with three well-known reaction-diffusion models corroborates with the predictions of the linear analysis and ensures the that spatiotemporal structure generated is stable in the long term.

The synchronization of self-propelled particles (SPPs) is a fascinating instance of emergent behavior in living and manmade systems, such as colonies of bacteria, flocks of birds, robot ensembles, and many others. The recent discovery of chimera states in coupled oscillators opens up new perspectives and indicates that other emergent behaviors may exist for SPPs. Indeed, for a minimal extension of the classical Vicsek model we show the existence of chimera states for SPPs, i.e., one group of particles synchronizes while others wander around chaotically. Compared to chimeras in coupled oscillators where the site position is fixed, SPPs give rise to new distinctive forms of chimeric behavior. We emphasize that the found behavior is directly implied by the structure of the deterministic equation of motion and is not caused by exogenous stochastic excitation. In the scaling limit of infinitely many particles, we show that the chimeric state persists. Our findings provide the starting point for the search or elicitation of chimeric states in real world SPP systems.

Mesoscopic pattern extraction (MPE) is the problem of finding a partition of the nodes of a complex network that maximizes some objective function. Many well-known network inference problems fall in this category, including, for instance, community detection, core-periphery identification, and imperfect graph coloring. In this paper, we show that the most popular algorithms designed to solve MPE problems can in fact be understood as special cases of the maximum likelihood formulation of the stochastic block model (SBM) or one of its direct generalizations. These equivalence relations show that the SBM is nearly universal with respect to MPE problems.

This work addresses a modification of the random geometric graph (RGG) model by considering a set of points uniformly and independently distributed on the surface of a $(d-1)$ sphere with radius $r$ in a $d$-dimensional Euclidean space, instead of on a unit hypercube ${[0,1]}^d$. Then, two vertices are connected by a link if their great circle distance is at most $s$. In the case of $d=3$, the topological properties of the random spherical graphs (RSGs) generated by this model are studied as a function of $s$. We obtain analytical expressions for the average degree, degree distribution, connectivity, average path length, diameter, and clustering coefficient for RSGs. By setting $r=\sqrt{\pi }/\left(2\pi \right)$, we also show the differences between the topological properties of RSGs and those of two-dimensional RGGs and random rectangular graphs. Surprisingly, in terms of the average clustering coefficient, RSGs look more similar to the analytical estimation for RGGs than RGGs themselves, when their boundary effects are considered. We support all our findings by computer simulations that corroborate the accuracy of the theoretical models proposed for RSGs.

The physical properties of collectively swimming bacteria have been thoroughly investigated both experimentally and theoretically using simulations. While models successfully predict some aspects of the dynamics observed in experiments, both models and experiments vary in their underlying assumptions and physical conditions. Hence, it is not clear which models are appropriate for which experimental setups. Here, we study, both experimentally and using two types of models (agent-based and continuous), the statistics of two strains of Serratia marcescens, wild-type and a nontumbling strain, swimming on a two-dimensional monolayer at varying concentrations. The experimental setup allows for a direct comparison with simulation results. Both models capture some aspects of the dynamics but fail at displaying others, especially at high densities. In particular, the effect of tumbling is much more significant than mere rotational (angular) diffusion.

We studied the van der Waals interactions of two finite, solid, cylindrical rods at arbitrary angle and position with respect to each other. An analytic interpolative formula for the interaction potential energy is constructed based on various asymptotic cases. The potential can be readily used for numerical and analytic descriptions of multiwall carbon nanotubes, metallic nanorods, rod-shaped colloids, or any other similar objects with significant van der Waals interactions.

This article provides an explanation for the instability of toric focal conic domains and the stability of parabolic focal conic domains in the twist-bend nematic ($N_{\mathrm{TB}}$) phase, by reason of the presence (the first case) or absence (the second case) of defect densities attached to the conics, which are disclinations. This result is obtained after a reminder of the theory of attached defects (the extended Volterra process) and is followed by its usage in other cases (dislocations, disclinations, dispirations, double helices) in the $N_{\mathrm{TB}}$ phase. The notion of spin disclination is introduced.

Topological defects resulting from boundary constraints in confined liquid crystals have attracted extensive research interest. In this paper, we use numerical simulation to study the phase transition dynamics in the context of stochastic resonance in a bistable liquid crystal device containing defects. This device is made of nematic liquid crystals confined in a shallow square well and is described by the planar Lebwohl-Lasher model. The stochastic phase transition processes of the system in the presence of a weak oscillating potential is simulated using an overdamped Langevin dynamics. Our simulation results reveal that, depending on system size, the phase transition may follow two distinct pathways: In small systems the preexisting defect structures at the corners hold until the last stage and there is no newly formed defect point in the bulk during the phase transition and in large systems new defect points appear spontaneously in the bulk and eventually merge with the preexisting defects at the corners. For both transition pathways stochastic resonance can be observed, but the corresponding phase transition dynamics show a dramatic difference in their responses to the boundary anchoring strength. In small systems we observe a sticky-boundary effect for a certain range of anchoring strength in which the phase transition gets stuck and stochastic resonance becomes deactivated. Our work demonstrates the dynamical interplay among defects, noises, and boundary conditions in confined liquid crystals.

We study the meniscus on the outside of a small spherical particle with radius $R$ at a liquid-vapor interface. The liquid is confined in a cylindrical container with a finite radius $L$ and has a contact angle $\pi /2$ at the container surface. The center of the particle is placed at various heights along the central axis of the container. By varying $L$, we are able to systematically study the crossover of the meniscus from nanometer to macroscopic scales. The meniscus rise or depression on the particle is found to grow as $ln(2L/R)$ when $R\ll L\ll {\kappa }^{-1}$ with ${\kappa }^{-1}$ being the capillary length and saturate to a value predicted by the Derjaguin-James formula when $R\ll {\kappa }^{-1}\ll L$. The capillary force on the particle exhibits a linear dependence on the particle's displacement from its equilibrium position at the interface when the displacement is small. The associated spring constant is found to be $2\pi \gamma {ln}^{-1}(2L/R)$ for $L\ll {\kappa }^{-1}$ and saturate to $2\pi \gamma {ln}^{-1}(3.7{\kappa }^{-1}/R)$ for $L\gg {\kappa }^{-1}$. At nanometer scales, we perform molecular dynamics simulations of the described geometry and the results agree well with the predictions of the macroscopic theory of capillarity. At micrometer to macroscopic scales, comparison to experiments by Anachkov et al. [Soft Matter 12, 7632 (2016).] shows that the finite span of a liquid-vapor or liquid-liquid interface needs to be considered to interpret experimental data collected with $L\sim {\kappa }^{-1}$.

We experimentally study the transition from steady flow to unsteady flow in a quasi-two-dimensional granular heap when small amounts of water are added to monodisperse glass spheres. Particles flow uniformly down both sides of the heap for low water content, but unsteady flow occurs as the water content increases. The unsteady flow mode consists of a nondepositing downslope avalanche and an upslope propagating granular jump. The transition from steady to unsteady flow occurs when the slope exceeds a critical angle as a result of water-induced cohesion. Under unsteady flow conditions, the deposited heap consists of loosely packed and densely packed layers, the formation of which is closely related to the unsteady flow dynamics.

Wrinkling patterns can be induced by the growth of a thin elastic film over a soft elastic substrate. While there is a good understanding of how this pattern is initiated on a flat geometry, wrinkling patterns over a curved surface are more complicated. Here, we consider this phenomenon within the framework of large deformation morphoelasticity by investigating surface wrinkling of a growing thin elastic film bonded to a large elastic cylinder. The system has two important dimensionless parameters: the ratio $\eta$ of the film thickness by the cylinder radius and the relative stiffness of the two layers $\xi$. Depending on the values of $\xi$ and $\eta$ we identify four different regimes for which we find the critical growth and wrinkling mode number. By combining asymptotic methods with numerical computations we determine the effect of the curvature on the bifurcation and establish that it always induces a delay at the bifurcation: Larger growth is needed on a curved surface to induce the same wrinkling instability. These results are crucial to understand pattern formation on surface with varying curvatures.

Permselective nanochannels can rectify the electric current transported through them similar to solid-state diodes. The rectification is due to symmetry breaking related to distribution of the nanochannels’ surface charge as well as the geometry. Thus far, most of the works related to the asymmetric current response have been primarily experimental. Here, we theoretically model a funnel-shaped nanochannel with a nonhomogeneous surface charge from which we derive a current-voltage relation (I-V). If the effects of the adjacent microchannels are ignored, the I-V is shown to behave like a unipolar diodes. When the effects of the adjacent microchannels are accounted for, the channel behaves like a diode only in a small voltage domain, while at larger voltages, the response is determined by the microchannels. The theoretical results are confirmed by numerical simulations.

When two bubbles encounter each other in a moderate ultrasound field they coalesce into a single bubble that shows purely spherical oscillations. For a sufficiently large acoustic field, this coalescence can lead to sustained nonspherical oscillations of the resulting bubble. We experimentally capture the time-resolved dynamics of the coalesced bubble, starting with the moment of film rupture. This allows the transient and steady-state regimes of the oscillating bubble to be studied. The amplitude dynamics for both of these regimes are successfully compared to numerical modeling, taking into account the coupling of volume and surface modes. Initial conditions for each surface mode are taken from the Legendre polynomial projection of the experimentally obtained bubble shape immediately after film rupture. We also observe that the symmetry axis of the zonal spherical harmonics is defined by the coalescence. The axis is identical to the rectilinear translational motion of the two approaching bubbles before impact. This high-amplitude coalescence technique provides a unique opportunity to study axisymmetric and sustained nonspherical bubble oscillations under controlled initial conditions.

Statistical theory of turbulence in viscid incompressible fluid, described by the Navier-Stokes equation driven by random force, is reformulated in terms of scale-dependent fields ${\mathbf{u}}_a\left(x\right)$, defined as wavelet-coefficients of the velocity field $\mathbf{u}$ taken at point $x$ with the resolution $a$. Applying quantum field theory approach of stochastic hydrodynamics to the generating functional of random fields ${\mathbf{u}}_a\left(x\right)$, we have shown the velocity field correlators $\langle {\mathbf{u}}_{a_1}\left(x_1\right)...{\mathbf{u}}_{a_n}\left(x_n\right)\rangle$ to be finite by construction for the random stirring force acting at prescribed large scale $L$. The study is performed in $d=3$ dimension. Since there are no divergences, regularization is not required, and the renormalization group invariance becomes merely a symmetry that relates velocity fluctuations of different scales in terms of the Kolmogorov-Richardson picture of turbulence development. The integration over the scale arguments is performed from the external scale $L$ down to the observation scale $A$, which lies in Kolmogorov range $l\ll A\ll L$. Our oversimplified model is full dissipative: interaction between scales is provided only locally by the gradient vertex $\left(\mathbf{u}\mathbf{\nabla }\right)\mathbf{u}$, neglecting any effects or parity violation that might be responsible for energy backscatter. The corrections to viscosity and the pair velocity correlator are calculated in one-loop approximation. This gives the dependence of turbulent viscosity on observation scale and describes the scale dependence of the velocity field correlations.

It was proposed recently by Loureiro and Boldyrev [Phys. Rev. Lett. 118, 245101 (2017)] and Mallet et al. [Mon. Not. R. Astron. Soc. 468, 4862 (2017)] that strongly anisotropic current sheets formed in the inertial range of magnetohydrodynamic turbulence become affected by the tearing instability at scales smaller than a critical scale ${\lambda }_{\mathrm{c}}$, and larger than the dissipation scale of turbulence. If true, this process can modify the nature of energy cascade at smaller scales, leading to a new, tearing-mediated regime of magnetohydrodynamic (MHD) turbulence. In this work, we present a numerical study of strongly anisotropic, two-dimensional turbulent eddies, and we demonstrate that the tearing instability can indeed compete with their nonlinear evolution. The results, therefore, provide direct numerical support for the picture that a new regime of MHD turbulence can exist below ${\lambda }_{\mathrm{c}}$.

In the present work we demonstrate how the diffusion constants $D_S$ for a tagged particle of species $s\left(=1,2\right)$ of a binary mixture depend on the mass ratio $\kappa =m_2/m_1$ of the two constituents. This dependence has also been reported earlier from simulations of a mixture. We use here a proper formulation of the self-consistent mode-coupling theory (MCT) of a two-component system. The effects of nonlocal and nonlinear coupling of slowly decaying hydrodynamic fluctuations on the long-time dynamics of a dense mixture is calculated here. The MCT memory functions are estimated here in the adiabatic approximation which assumes for the glassy state fast decay of momentum fluctuations compared to that of the number density. Plots $D_1$ and $D_2$ obtained from evaluation of the self-consistent MCT formulas, shows that for nearly equal-sized particles, the two self-diffusion coefficients are related as $D_1\sim D_2^a$ with a nonuniversal exponent $a$ and this relation holds over a wide range of $\kappa$ values. The simple MCT used here predicts an ideal ergodic-nonergodic transition from liquid to glassy state for critical packings $\{{\eta }_2^c,{\eta }_1^c\}$ of the two species. For high packing of one species ${\eta }_2$, at which the mixture is already in a nonergodic state, transition to an ergodic state occurs at a small ${\eta }_1^c$ and a reenetry into the nonergodic phase can occur again at an even larger value of ${\eta }_1^c$, while ${\eta }_2$ remains same as before.

We demonstrate that strongly repelling Brownian particles confined in two-dimensional microchannels with multiple layers show single-file diffusion. At long times the mean-square displacement (MSD) is proportional to $t^{1/2}$, as in the one-dimensional case. On an intermediate timescale the MSD is further reduced. It scales with a minimal exponent that decreases with the number of layers in the channel. In the limit of infinite width of the channel, the MSD time evolution shows a crossover to a logarithmic time dependence.

According to the Boltzmann distribution assumption of solitons, in thermal equilibrium, there is the most probable soliton whose average kinetic energy per site equals the thermal energy $k_{B}T$. Based on the momentum excitation method, the soliton can be numerically excited in the static Fermi-Pasta-Ulam (FPU) chains. By associating the excited soliton with the corresponding most probable soliton, the temperature dependence of the velocity of solitons in thermal equilibrium can be numerically evaluated. The results agree very well with the temperature dependence of the sound velocity of energy transfer. This confirms that solitons are promising candidates for energy carriers in FPU chains. Moreover, the validity of the Boltzmann distribution assumption of solitons in FPU chains is also confirmed. This work sheds light on how to numerically (even experimentally) investigate solitons in thermal equilibrium.

Clusters of ${\mathrm{IP}}_{\text{3}}$ receptor channels in the membranes of the endoplasmic reticulum of many nonexcitable cells release calcium ions in a cooperative manner giving rise to dynamical patterns such as ${\mathrm{Ca}}^{\text{2+}}$ puffs, waves, and oscillations that occur on multiple spatial and temporal scales. We introduce a minimal yet descriptive reaction-diffusion model of ${\mathrm{IP}}_{\text{3}}$ receptors for a saturating concentration of ${\mathrm{IP}}_{\text{3}}$ using a principled reduction of a detailed Markov chain description of individual channels. A dynamical systems analysis reveals the possibility of excitable, bistable, and oscillatory dynamics of this model that correspond to three types of observed patterns of calcium release: puffs, waves, and oscillations, respectively. We explain the emergence of these patterns via a bifurcation analysis of a coupled two-cluster model, compute the phase diagram, and quantify the speed of the waves and period of oscillations in terms of system parameters. We connect the termination of large-scale ${\mathrm{Ca}}^{\text{2+}}$ release events to ${\mathrm{IP}}_{\text{3}}$ unbinding or stochasticity.