Retardation between sensation and action is an inherent biological trait. Here we study its effect in the Vicsek model, which is a paradigmatic swarm model. We find that (1) a discrete time delay in the orientational interactions diminishes the ability of strongly aligned swarms to follow a leader and, in return, increases their stability against random orientation fluctuations; (2) both longer delays and higher speeds favor ballistic over diffusive spreading of information (orientation) through the swarm; (3) for short delays, the mean change in the total orientation (the order parameter) scales linearly in a small orientational bias of the leaders and inversely in the delay time, while its variance first increases and then saturates with increasing delays; and (4) the linear response breaks down when orientation conservation is broken.

A hydrogel which spontaneously swells in an aqueous polymer solution was observed to produce a new hydrogel film coated on its swollen surface. Here, inspired by this phenomenon, we theoretically formulate the dynamics of isotropic-to-nematic ($I\text{−}N$) phase transition caused by swelling a hydrogel substrate (HS) in a dilute nanoplatelet suspension, and quantitatively characterize a self-growing nano-liquid-crystal (NLC) film coated on the swollen HS surface. We show that as the HS gets softer, the resulting NLC film can form earlier and achieve greater thickness (up to hundreds of micrometers). Our results and the existing experiments confirm that the growth dynamics of the NLC film or hydrogel film is exclusively regulated by the swelling behaviors of the HS instead of suspension configurations, e.g., $I\text{−}N$ phase transition or sol-gel transition, suggesting a universal signature for the solutes ranging from molecules to colloids. However, both the maximum thickness of the NLC film and the corresponding characteristic time rely highly on the inherent elasticity of the HS and nanoplatelet aspect ratio. We demonstrate that the swelling quasiequilibrium state rather than the equilibrium state of the HS is more qualified to formulate a condition which is practically significant in preestimating the moment when the maximum thickness of the NLC film appears. Our theoretical framework serves as a robust paradigm to extensively rationalize (bio)film coatings which self-integrate with diverse nanostructural configurations via swelling-induced phase transition.

We study geometrical clues of a rigidity transition due to the emergence of a system-spanning state of self-stress in underconstrained systems of individual polygons and spring networks constructed from such polygons. When a polygon with harmonic bond edges and an area spring constraint is subject to an expansive strain, we observe that convexity of the polygon is a necessary condition for such a self-stress. We prove that the cyclic configuration of the polygon is a sufficient condition for the self-stress. This correspondence of geometry and rigidity is akin to the straightening of a one dimensional chain of springs to rigidify it. We predict the onset of the rigidity transition and estimate the transition strain using purely geometrical methods. These findings help determine the rigidity of an area-preserving polygon just by knowing its geometry. Since two-dimensional spring networks can be considered as a network of polygons, we look for similar geometric features in underconstrained spring networks under isotropic expansive strain. We observe that all polygons attain convexity at the rigidity transition such that the fraction of convex, but not cyclic, polygons predicts the onset of the rigidity transition. Acyclic polygons in the network correlate with larger tensions, forming effective force chains.

KPP (Kolmogorov-Petrovsky-Piskunov) solutions of the reaction-diffusion equation have application in various physical phenomena occurring in biology, ecology, and reacting flows. In particular, these solutions are commonly used in turbulent combustion to scale turbulent burning velocities. Subject to certain conditions on reaction rate profile through the flame brush and turbulent diffusivity, this theory relates the turbulent burning velocity to the derivative of the reaction rate $\left(\tilde{\omega }\right)$ at the leading edge of the flame brush $\left(d\tilde{\omega }/d\tilde{c}{|}_{\tilde{c}=0}\right)$. Such waves are often referred to as “pulled fronts.” However, turbulent flames never actually satisfy the KPP conditions for a pulled front, as the turbulent flame brush, parametrized here by the thickness ${\delta }_t$, consists of an ensemble of laminar flamelets of thickness $\delta$, where $\epsilon =\delta /{\delta }_t\ll 1$ is very small, but nonzero, and $d\tilde{\omega }/d\tilde{c}$ tends to zero at the brush leading edge for high activation energy, combustion-type kinetics. This paper analyzes these effects on KPP wave solutions, parametrized by $\epsilon =\delta /{\delta }_t$ and Zeldovich number $\mathrm{Ze}$ focusing on whether turbulent flames retain their pulled front character and what the correction to the KPP wave speed is. Variational solutions of the reaction-diffusion equation show that the solution can be expanded in powers of $1/\left|ln\epsilon \right|$. Both numerical and asymptotic results are presented, showing that the wave still exhibits pulled front solutions but with significant corrections to the KPP result. The leading order correction is of the form ${\left|ln\epsilon \right|}^{-2}$ and independent of $\mathrm{Ze}$. Higher order corrections are function of both $\epsilon$ and $\mathrm{Ze}$. However, the dominant factor influencing the wave speed correction is due to the finite $\epsilon$, with $\mathrm{Ze}$ exhibiting a weaker effect.

The Ruppeiner geometry has been shown to provide novel ways for constructing the phase boundaries and the Widom line of certain fluids. This paper examines the applicability of these geometric constructions to more general fluids. We develop a general equation-of-state expansion for fluids near a critical point that mainly assumes analyticity with respect to the number density. Based on this general parametrization of fluids, we prove the equivalence of the Ruppeiner geometric construction and the standard Maxwell construction of phase boundaries near the critical point. In contrast, we find that the usual prescription based on the Ruppeiner geometry for the Widom line does not produce the expected Widom line for arbitrary cases of our general fluid equation of state. This usual prescription relies on the Ruppeiner metric induced on a particular hypersurface of the thermodynamic manifold. We show that by choosing a different hypersurface, which we call the Ruppeiner-$N$ surface, and using its associated induced metric, the Ruppeiner construction generates the entire Widom line of the van der Waals fluid exactly, even away from the critical point. Interestingly, this alternative hypersurface yields another benefit. It improves the classification scheme originally proposed by Diósi et al. for partitioning the van der Waals state space into its different phases using geodesics of a thermodynamic metric. We argue that, whereas the original Diósi boundaries did not correspond to any clear thermodynamic lines, the corresponding boundaries based on the Ruppeiner-$N$ metric become sensitive to the presence of the van der Waals Widom line and provide the correct classification of all van der Waals states. These results suggest that the Ruppeiner-$N$ surface may be the more appropriate hypersurface to use when studying phase diagrams with thermodynamic geometry.

Extreme events are by nature rare and difficult to predict, yet are often much more important than frequent, typical events. An interesting counterpoint to the prediction of such events is their retrodiction—given a process in an outlier state, how did the events leading up to this endpoint unfold? In particular, was there only a single, massive event, or was the history a composite of multiple, smaller but still significant events? To investigate this problem we take heavy-tailed stochastic processes (specifically, the symmetric, $\alpha$-stable Lévy processes) as prototypical random walks. A natural and useful characteristic scale arises from the analysis of processes conditioned to arrive in a particular final state (Lévy bridges). For final displacements longer than this scale, the scenario of a single, long jump is most likely, even though it corresponds to a rare, extreme event. On the other hand, for small final displacements, histories involving extreme events tend to be suppressed. To further illustrate the utility of this analysis, we show how it provides an intuitive framework for understanding three problems related to boundary crossings of heavy-tailed processes. These examples illustrate how intuition fails to carry over from diffusive processes, even very close to the Gaussian limit. One example yields a computationally and conceptually useful representation of Lévy bridges that illustrates how conditioning impacts the extreme-event content of a random walk. The other examples involve the conditioned boundary-crossing problem and the ordinary first-escape problem; we discuss the observability of the latter example in experiments with laser-cooled atoms.

The structure of a network is an unlabeled graph, yet graphs in most models of complex networks are labeled by meaningless random integers. Is the associated labeling noise always negligible, or can it overpower the network-structural signal? To address this question, we introduce and consider the sparse unlabeled versions of popular network models and compare their entropy against the original labeled versions. We show that labeled and unlabeled Erdős-Rényi graphs are entropically equivalent, even though their degree distributions are very different. The labeled and unlabeled versions of the configuration model may have different prefactors in their leading entropy terms, although this remains conjectural. Our main results are upper and lower bounds for the entropy of labeled and unlabeled one-dimensional random geometric graphs. We show that their unlabeled entropy is negligible in comparison with the labeled entropy. This means that in sparse networks the entropy of meaningless labeling may dominate the entropy of the network structure. The main implication of this result is that the common practice of using exchangeable models to reason about real-world networks with distinguishable nodes may introduce uncontrolled aberrations into conclusions made about these networks, suggesting a need for a thorough reexamination of the statistical foundations and key results of network science.

Exploiting the mathematical curiosity of intransitive dice, we present a simple theoretical model for coevolution that captures scales ranging from the genome of the individual to the system-wide emergence of species diversity. We study a set of evolving agents that interact competitively in a closed system, in which both the dynamics of mutations and competitive advantage emerge directly from interpreting a genome as the sides of a die. The model demonstrates sympatric speciation where new species evolve from existing ones while in contact with the entire ecosystem. Allowing free mutations both in the genomes and the mutation rates, we find, in contrast to hierarchical models of fitness, the emergence of a metastable state of finite mutation rate and diversity.

We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations and computer codes based on combinatorial recursions are included.

Glasses exhibit spatially localized vibrations in the low-frequency regime. These localized modes emerge below the boson peak frequency ${\omega }_{\text{BP}}$, and their vibrational densities of state follow $g\left(\omega \right)\propto {\omega }^4$ ($\omega$ is frequency). Here, we attempt to address how the localized vibrations behave through the ideal glass transition. To do this, we employ a random pinning method, which enables us to study the thermodynamic glass transition. We find that the localized vibrations survive even in equilibrium glass states. Remarkably, the localized vibrations still maintain the properties of appearance below ${\omega }_{\text{BP}}$ and $g\left(\omega \right)\propto {\omega }^4$. Our results provide important insight into the material properties of ideal glasses.

The patterns produced by dragging an atomic force microscope (AFM) tip over a polymer surface are studied using a mesoscopic model introduced by Gnecco and co-workers [E. Gnecco et al., New J. Phys. 17, 032001 (2015)]. We show that the problem can be reduced to solving a closed integrodifferential equation for a single degree of freedom, the position of the AFM tip. We find the steady-state solution to this equation and then carry out a linear stability analysis of it. The steady state is only stable if the dimensionless indentation rate $\alpha$ is less than a critical value ${\alpha }_c$ which depends on the dimensionless velocity of the rigid support $r$. Conversely, for $\alpha >{\alpha }_c$, periodic stick-slip motion sets in after a transient. Simulations show that the amplitude of these oscillations is proportional to ${(\alpha -{\alpha }_c)}^{1/2}$ for $\alpha$ just above ${\alpha }_c$. Our analysis also yields a closed equation that can be solved for the critical value ${\alpha }_c={\alpha }_c\left(r\right)$. If the steady-state motion is perturbed, as long as the deviation from the steady state is small, the deviation of the tip's position from the steady state can be written as a linear superposition of terms of the form $exp\left({\lambda }_{k}t\right)$, where the complex constants ${\lambda }_k$ are solutions to an integral equation. Finally, we demonstrate that the results obtained for the two-dimensional model of Gnecco et al. carry over in a straightforward way to the generalization of the model to three dimensions.

Wang-Landau simulations offer the possibility to integrate explicitly over a collective coordinate and stochastically over the remainder of configuration space. We propose to choose the so-called “slow mode,” which is responsible for large autocorrelation times and thus critical slowing down, for collective integration. We study this proposal for the Ising model and the linear-log-relaxation (LLR) method as simulation algorithm. We first demonstrate supercritical slowing down in a phase with spontaneously broken symmetry and for the heat-bath algorithms, for which autocorrelation times grow exponentially with system size. By contrast, using the magnetization as collective coordinate, we present evidence that supercritical slowing down is absent. We still observe a polynomial increase of the autocorrelation time with volume (critical slowing down), which is, however, reduced by orders of magnitude when compared to local update techniques.

While the support for the relevance of critical dynamics to brain function is increasing, there is much less agreement on the exact nature of the advocated critical point. Thus, a considerable number of theoretical efforts are currently concentrated on which mechanisms and what type(s) of transition can be exhibited by neuronal network models. In that direction, the present work describes the effect of incorporating a fraction of inhibitory neurons on the collective dynamics. As we show, this results in the appearance of a tricritical point for highly connected networks and a nonzero fraction of inhibitory neurons.

A characteristic feature of nondividing animal cells is the radial organization of microtubules (MTs), emanating from a single microtubule organizing center (MTOC). As generically these cells are not spherically symmetric, this raises the question of the influence of cell geometry on the orientational distribution of microtubules. We present a systematic study of this question in a simplified setting where MTs are nucleated from a single fixed MTOC in the center of an elliptical cell geometry. Within this context we introduce four models of increasing complexity, each one introducing additional mechanisms that govern the interaction of the MTs with the cell boundary. In order, we consider the cases: MTs that can bind to the boundary with a fixed mean residence time (M0), force-producing MTs that can slide on the boundary towards the cell poles (MS), MTs that interact with a generic polarity factor that is transported and deposited at the boundary, and which in turn stabilizes the MTs at the boundary (MP), and a final model in which both sliding and stabilization by polarity factors is taken into account (MSP). In the baseline model (M0), the exponential length distribution of MTs causes most of the interactions at the cell boundary to occur along the shorter transverse direction in the cell, leading to transverse biaxial order. MT sliding (MS) is able to reorient the main axis of this biaxial order along the longitudinal axis. The polarization mechanism introduced in MP and MSP overrules the geometric bias towards bipolar order observed in M0 and MS, and allows the establishment of unipolar order either along the short (MP) or the long cell axis (MSP). The behavior of the latter two models can be qualitatively reproduced by a very simple toy model with discrete MT orientations.

We study the one-dimensional active Ising model in which aligning particles undergo diffusion biased by the signs of their spins. The phase diagram obtained varying the density of particles, their hopping rate, and the temperature controlling the alignment shows a homogeneous disordered phase but no homogeneous ordered one, as well as two phases with localized dense structures. In the flocking phase, large ordered aggregates move ballistically and stochastically reverse their direction of motion. In what we termed the “aster” phase, dense immobile aggregates of opposite magnetization face each other, exchanging particles, without any net motion of the aggregates. Using a combination of numerical simulations and mean-field theory, we study the evolution of the shapes of the flocks, the statistics of their reversal times, and their coarsening dynamics. Solving exactly for the zero-temperature dynamics of an aster allows us to understand their coarsening, which shows extremal dynamics, while mean-field equations account for their shape.

Thin fluid or elastic films and membranes are found in nature and technology, for instance, as confinements of living cells or in loudspeakers. When applying a net force, the resulting flows in an unbounded two-dimensional incompressible low-Reynolds-number fluid or displacements in a two-dimensional linearly elastic solid seem to diverge logarithmically with the distance from the force center, which has led to some debate. Recently, we have demonstrated that such divergences cancel when the total (net) force vanishes. Here, we illustrate that if a net force is present, the boundaries play a prominent role. Already a single no-slip boundary regulates the flow and displacement fields and leads to their decay to leading order inversely in distance from a force center and the boundary. In other words, it is the boundary that stabilizes the system in this situation, unlike the three-dimensional case, where an unbounded medium by itself is able to absorb a net force. We quantify the mobility and displaceability of an inclusion as a function of the distance from the boundary, as well as interactions between different inclusions. In the case of free-slip boundary conditions, a kinked boundary is necessary to achieve stabilization.

We study the dynamics of active nematic films on a substrate driven by active flows with or without the incompressible constraint. Through simulations and theoretical analysis, we show that arch patterns are stable in the compressible case, while they become unstable under the incompressibility constraint. For compressible flows at high enough activity, stable arches organize themselves into a smecticlike pattern, which induce an associated global polar ordering of $+1/2$ nematic defects. By contrast, divergence-free flows give rise to a local nematic order of the $+1/2$ defects, consisting of antialigned pairs of neighboring defects, as established in previous studies.

We present three-dimensional discrete element method simulations of bidisperse granular packings to investigate their jamming densities ${\varphi }_J$ and dimensionless bulk moduli $K$ as functions of the size ratio $\delta$ and the concentration of small particles $X_{\mathrm{S}}$. We determine the partial and total bulk moduli for packings near their jamming densities, including a second transition that occurs for sufficiently small $\delta$ and $X_{\mathrm{S}}$ when the system is compressed beyond its first jamming transition. While the first transition is sharp, exclusively with large-large contacts, the second is rather smooth, carried by small-large interactions at densities much higher than the monodisperse random packing baseline, ${\varphi }_J^{\mathrm{mono}}\approx 0.64$. When only nonrattlers are considered, all the effective transition densities are reduced, and the density of the second transition emerges rather close to the reduced baseline, ${\tilde{\varphi }}_J^{\mathrm{mono}}\approx 0.61$, due to its smooth nature. At size ratios $\delta \le 0.22$ a concentration $X_{\mathrm{S}}^{*}$ divides the diagram—either with most small particles nonjammed or jammed jointly with large ones. For $X_{\mathrm{S}}<X_{\mathrm{S}}^{*}$, the modulus $K$ displays different behaviors at first and second jamming transitions. Along the second transition, $K$ rises relative to the values found at the first transition; however, is still small compared to $K$ at $X_{\mathrm{S}}^{*}$. Explicitly, for our smallest $\delta =0.15$, the discontinuous jump in $K$ as a function of $X_{\mathrm{S}}$ is obtained at $X_{\mathrm{S}}^{*}\approx 0.21$ and coincides with the maximum jamming density where both particle species mix most efficiently. Our results will allow tuning or switching the bulk modulus $K$ or other properties, such as the wave speed, by choosing specific sizes and concentrations based on a better understanding of whether small particles contribute to the jammed structure or not, and how the micromechanical structure behaves at either transition.

We consider the penetration dynamics of a vertical cylinder into a dry granular medium subjected to successive impacts. The depth of the impactor below the free surface $z_N$ first evolves linearly with the impact number $N$ and then follows a power-law evolution $z_N\propto N^{1/3}$. The depth reached by the cylinder after a given number of impacts is observed to increase with the impact energy, but to decrease with its diameter and the density of the granular medium. We develop a model that accounts for the quasistatic and inertial granular forces applying on the cylinder to rationalize our observations. This approach reveals the existence of two intrusion regimes for large and small impact numbers, allowing all data to be rescaled on a master curve. Then, we extend the study to the effect of sidewalls on the dynamics of the impactor. We show that lateral confinement changes the dependence of the impactor depth on the impact number $z_N\left(N\right)$. This effect is accounted for by considering the increase of the granular drag with the lateral confinement.