A fundamental problem in the analysis of complex systems is getting a reliable estimate of the entropy of their probability distributions over the state space. This is difficult because unsampled states can contribute substantially to the entropy, while they do not contribute to the maximum likelihood estimator of entropy, which replaces probabilities by the observed frequencies. Bayesian estimators overcome this obstacle by introducing a model of the low-probability tail of the probability distribution. Which statistical features of the observed data determine the model of the tail, and hence the output of such estimators, remains unclear. Here we show that well-known entropy estimators for probability distributions on discrete state spaces model the structure of the low-probability tail based largely on a few statistics of the data: the sample size, the maximum likelihood estimate, the number of coincidences among the samples, and the dispersion of the coincidences. We derive approximate analytical entropy estimators for undersampled distributions based on these statistics, and we use the results to propose an intuitive understanding of how the Bayesian entropy estimators work.

The evolution of a complex multistate system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter $\phi$, which is the number of nonzero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with $D$ vertices (number of system states) and $2\phi D$ edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of nonsparse random generators. We show that the first moment of the eigenvalue distribution scales as $\sim \phi$, while its variance is $\sim \sqrt{\phi }$. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions and clarify the behavior of the spectral gap as a function of $D$. Finally, we analyze complex spacing ratio statistics of ultrasparse generators, $\phi =\mathrm{const}$, and find that starting already at $\phi ⩾2$, spectra of the generators exhibit universal properties typical of Ginibre's orthogonal ensemble.

Prediction of an imminent catastrophic event in a driven disordered system is of paramount importance—from the laboratory scale controlled fracture experiment to the largest scale of mechanical failure, i.e., earthquakes. It has long been conjectured that the statistical regularities in the energy emission time series mirror the “health” of such driven systems and hence have the potential for forecasting imminent catastrophe. Among other statistical regularities, a measure of how unequal avalanche sizes are is potentially a crucial indicator of imminent failure. The inequalities of avalanche sizes are quantified using inequality indices traditionally used in socioeconomic systems: the Gini index $g$, the Hirsch index $h$, and the Kolkata index $k$. It is shown analytically (for the mean-field case) and numerically (for the non-mean-field case) with models of quasi-brittle materials that the indices show universal behavior near the breaking points in such models and hence could serve as indicators of imminent breakdown of stressed disordered systems.

Many empirical time series are genuinely symbolic: Examples range from link activation patterns in network science, to DNA coding or firing patterns in neuroscience, to cryptography or combinatorics on words. In some other contexts, the underlying time series is actually real valued, and symbolization is applied subsequently, as in symbolic dynamics of chaotic systems. Among several time series quantifiers, time series irreversibility—the difference between forward and backward statistics in stationary time series—is of great relevance. However, the irreversible character of symbolized time series is not always equivalent to the one of the underlying real-valued signal, leading to some misconceptions and confusion on interpretability. Such confusion is even bigger for binary time series—a classical way to encode chaotic trajectories via symbolic dynamics. In this paper we aim to clarify some usual misconceptions and provide theoretical grounding for the practical analysis—and interpretation—of time irreversibility in symbolic time series. We outline sources of irreversibility in stationary symbolic sequences coming from frequency asymmetries of nonpalindromic pairs which we enumerate, and prove that binary time series cannot show any irreversibility based on words of length $m<4$, thus discussing the implications and sources of confusion. We also study irreversibility in the context of symbolic dynamics, and clarify why these can be reversible even when the underlying dynamical system is not, such as the case of the fully chaotic logistic map.

Populations of heterogeneous phase oscillators with frustrated random interactions exhibit a quasiglassy state in which the distribution of local fields is volcanoshaped. In a recent work [Phys. Rev. Lett. 120, 264102 (2018)], the volcano transition was replicated in a solvable model using a low-rank, random coupling matrix $\mathbf{M}$. We extend here that model including tunable nonreciprocal interactions, i.e., ${\mathbf{M}}^T\ne \mathbf{M}$. More specifically, we formulate two different solvable models. In both of them the volcano transition persists if matrix elements $M_{jk}$ and $M_{kj}$ are enough correlated. Our numerical simulations fully confirm the analytical results. To put our work in a wider context, we also investigate numerically the volcano transition in the analogous model with a full-rank random coupling matrix.

We theoretically investigate the dynamics, bifurcation structure, and stability of dark localized states emerging in Kerr cavities in the presence of positive second- and fourth-order dispersion. In this previously unexplored regime, dark states form through the locking of uniform wave fronts, or domain walls, connecting two coexisting stable uniform states, and undergo a generic bifurcation structure known as collapsed homoclinic snaking. We characterize the robustness of these states by computing their stability and bifurcation structure as a function of the main control parameter of the system. Furthermore, we show that by increasing the dispersion of fourth order, bright localized states can be also stabilized.

Saccharomyces cerevisiae and Candida albicans, the two well-known human pathogens, can be found in all three morphologies, i.e., yeast, pseudohyphae, and true hyphae. The cylindrical daughter-bud (germ tube) grows very long for true hyphae, and the cell cycle is delayed compared to the other two morphologies. The place of the nuclear division is specific for true hyphae determined by the position of the septin ring. However, the septin ring can localize anywhere inside the germ tube, unlike the mother-bud junction in budding yeast. Since the nucleus often migrates a long path in the hyphae, the underlying mechanism must be robust for executing mitosis in a timely manner. We explore the mechanism of nuclear migration through hyphae in light of mechanical interactions between astral microtubules and the cell cortex. We report that proper migration through constricted hyphae requires a large dynein pull applied on the astral microtubules from the hyphal cortex. This is achieved when the microtubules frequently slide along the hyphal cortex so that a large population of dyneins actively participate, pulling on them. Simulation shows timely migration when the dyneins from the mother cortex do not participate in pulling on the microtubules. These findings are robust for long migration and positioning of the nucleus in the germ tube at the septin ring.

How the architecture of gene regulatory networks shapes gene expression patterns is an open question, which has been approached from a multitude of angles. The dominant strategy has been to identify nonrandom features in these networks and then argue for the function of these features using mechanistic modeling. Here we establish the foundation of an alternative approach by studying the correlation of network eigenvectors with synthetic gene expression data simulated with a basic and popular model of gene expression dynamics: Boolean threshold dynamics in signed directed graphs. We show that eigenvectors of the network adjacency matrix can predict collective states (attractors). However, the overall predictive power depends on details of the network architecture, namely the fraction of positive 3-cycles, in a predictable fashion. Our results are a set of statistical observations, providing a systematic step towards a further theoretical understanding of the role of network eigenvectors in dynamics on graphs.

Developing microscopic understanding of the thermal properties of liquids is challenging due to their strong dynamic disorder, which prevents characterization of the atomic degrees of freedom. There have been significant research interests in the past few decades to extend the normal mode analysis for solids to instantaneous structures of liquids. However, the nature of normal modes that arise from these unstable structures is still elusive. In this paper, we explore the instantaneous eigenmodes of dynamical matrices of various Lennard-Jones argon liquid and gas systems at high temperatures and show that the normal modes can be interpreted as an interpolation of $T\to \infty$ (gas) and $T=0$ (solid) mode descriptions. We find that normal modes become increasingly collisional and translational, recovering atomistic gaslike behavior rather than vibrational with increase in temperature, suggesting that normal modes in liquids may be described by both solidlike and gaslike modes.

We study by numerical simulation the impact of a one-dimensional composite projectile, composed of two superposed homogeneous parts, on an infinitely rigid and massive wall. The coefficient of restitution and the contact time are systematically measured as functions of the contrasts of mass and stiffness between the two parts. For purely elastic parts, these quantities show complex trends associated with different dynamics of the deformation waves propagating inside the projectile. A significant portion of the initial kinetic energy can be trapped in the deformation modes: the coefficient of restitution is lowest, about 0.2, when there is a strong stiffness contrast between the two parts and the stiff and soft parts are at the leading and trailing edges of the projectile respectively. In this case, we highlight the presence of multiple bounces, whose number increases as the proportion of the soft part increases. Finally, viscoelastic parts can be implemented in the same numerical framework to successfully recover the results obtained in real composite projectile impact experiments [D'Angelo et al., Phys. Rev. E 103, 053005 (2021)].

A slitted base cavity of constant depth with a varying filling ratio $0\le R_f\le 100\%$ is experimentally investigated to reduce the form drag of a three-dimensional blunt body (the so-called squareback Ahmed body) at a Reynolds number $\mathrm{Re}=2.89\times {10}^5$. The drag reduction is achieved by a decrease of base suction (or, equivalently, the increase of pressure at the base). The plain cavity ($R_f=100\%$) reduces the base suction by 22% compared to the case with no cavity ($R_f=0$). All intermediate filling ratio are obtained by the enlargement of the slits, initially having a zero width for the plain cavity case. It is shown that the gradual base suction change can be related to the level of the entrainment flux of the free shear layers developing from the rear separation and to the suppression of the transverse steady asymmetric instability of the wake. The model of the vortex formation region length of Gerrard [J. Fluid Mech. 25, 401 (1966)] is shown to provide an insightful interpretation of the drag reduction mechanism using ventilated base cavities.

We examine the conjecture of equivalence of nonequilibrium ensembles for turbulent flows in two dimensions in a dual-cascade setup. We construct a formally time-reversible Navier-Stokes equation in two dimensions by imposing global constraints of energy and enstrophy conservation. A comparative study of the statistical properties of its solutions with those obtained from the standard Navier-Stokes equations clearly shows that a formally time-reversible system is able to reproduce the features of a two-dimensional turbulent flow. Statistical quantities based on one- and two-point measurements show an excellent agreement between the two systems for the inverse- and direct-cascade regions. Moreover, we find that the conjecture holds very well for two-dimensional turbulent flows with both conserved energy and enstrophy at finite Reynolds number.

We study the stochastic spatial Lotka-Volterra model for predator-prey interaction subject to a periodically varying carrying capacity. The Lotka-Volterra model with on-site lattice occupation restrictions (i.e., finite local carrying capacity) that represent finite food resources for the prey population exhibits a continuous active-to-absorbing phase transition. The active phase is sustained by the existence of spatiotemporal patterns in the form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional lattice are utilized to investigate the effect of seasonal variations of the environment on species coexistence. The results of our simulations are also compared to a mean-field analysis in order to specifically delineate the impact of stochastic fluctuations and spatial correlations. We find that the parameter region of predator and prey coexistence is enlarged relative to the stationary situation when the carrying capacity varies periodically. The (quasi-)stationary regime of our periodically varying Lotka-Volterra predator-prey system shows qualitative agreement between the stochastic model and the mean-field approximation. However, under periodic carrying capacity-switching environments, the mean-field rate equations predict period-doubling scenarios that are washed out by internal reaction noise in the stochastic lattice model. Utilizing visual representations of the lattice simulations and dynamical correlation functions, we study how the pursuit and evasion waves are affected by ensuing resonance effects. Correlation function measurements indicate a time delay in the response of the system to sudden changes in the environment. Resonance features are observed in our simulations that cause prolonged persistent spatial correlations. Different effective static environments are explored in the extreme limits of fast and slow periodic switching. The analysis of the mean-field equations in the fast-switching regime enables a semiquantitative description of the (quasi-)stationary state.

Elastic constants of zero-temperature amorphous solids are given as the difference between the Born term, which results from a hypothetical affine deformation of an amorphous solid, and a correction term, which originates from the fact that the deformation of an amorphous solid due to an applied stress is, at the microscopic level, nonaffine. Both terms are non-negative and thus it is a priori not obvious that the resulting elastic constants are non-negative. In particular, theories that approximate the correction term may spuriously predict negative elastic constants and thus an instability of an amorphous solid. Here we derive alternative expressions for elastic constants of zero-temperature amorphous solids that are explicitly non-negative. These expressions provide a useful blueprint for approximate theories for elastic constants and sound damping in zero-temperature amorphous solids.

Evolution of resonant three-wave interaction is governed by quadratic nonlinearities. While propagating localized modes and inverse scattering mechanisms have been studied, transient states such as rogue waves and breathers are not fully understood. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. Here we study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT). While nonlinearity governs the actual dynamics, the range of wave numbers for modulation instability remains a useful indicator. Depending on the stability characteristics of the fundamental mode and the higher-order harmonics (“sidebands”), “regular” and “staggered” FPUT patterns can arise. A “cascading mechanism” provides analytical verification, as the fundamental and sideband modes attain the same magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also computed, similar to experimental observations of optical pulses. Such spectra can elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of “effective energy” is examined and the eigenvalues of the inverse scattering mechanism are computed. Both approaches are utilized to correlate with the occurrence of regular or staggered FPUT. These numerical and analytical studies can enhance our understanding of wave interactions in fluid mechanics and optics.

The rich thermotropic behavior of lipid bilayers is addressed using phenomenological theory informed by many experiments. The most recent experiment not yet addressed by theory has shown that the tilt modulus in DMPC lipid bilayers decreases dramatically as the temperature is lowered toward the main transition temperature $T_M$. It is shown that this behavior can be understood by introducing a simple free energy functional for tilt that couples to the area per molecule. This is combined with a chain melting free energy functional in which the area is the primary order parameter that is the driver of the main transition. Satisfactory agreement with experiment is achieved with values of the model parameters determined by experiments, but the transition is directly into the gel phase. The theory is then extended to include the enigmatic ripple phase by making contact with the most recent experimentally determined ripple structure.

Rheology of macroscopic particle-laden interfaces, called “granular rafts,” has been experimentally studied in the simple shear configuration. The shear-stress relation obtained from a classical rheometer exhibits the same behavior as a Bingham fluid, and the viscosity diverges with the surface fraction according to evolutions similar to 2D suspensions. The velocity field of the particles that constitute the granular raft has been measured in the stationary state. These measurements reveal nonlocal rheology similar to dry granular materials. Close to the walls of the rheometer cell, one can observe regions of large local shear rate while in the middle of the cell a quasistatic zone exists. This flowing region, characteristic of granular matter, is described in the framework of an extended kinetic theory showing the evolution of the velocity profile with the imposed shear stress. Measuring the probability density functions of the instantaneous local shear rate, we provide evidence of a balance between positive and negative instantaneous local shear rate. This behavior is the signature of a quasistatic region inside the granular raft.

In some parameter and solution regimes, a minimally coupled nonrelativistic quantum particle in one dimension is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in three dimensions with ratio of bending modulus to linear density ${(\hslash /2m)}^2$. For $m=m_e$, this quantity is comparable to that of a microtubule. Axial forces and torques applied to the rod play the role of scalar and vector potentials, respectively, and rod inextensibility plays the role of normalization. We show how an uncertainty principle $\mathrm{\Delta }x\mathrm{\Delta }{p}_x\gtrsim \hslash$ governs transverse deformations propagating down the inextensible, force and torque-free rod, and how orbital angular momentum quantized in units of $\hslash$ or $\hslash /2$ (depending on calculation method) emerges when the force and torque-free inextensible rod is formed into a ring. For torqued rings with large wave numbers, a “twist quantum” appears that is somewhat analogous to the magnetic flux quantum. These and other results are obtained from a purely classical treatment of the rod, i.e., without quantizing any classical fields.

We conduct an experimental study on the flow structures and dynamics of turbulent Rayleigh-Bénard convection in an annular cell with radius ratio $\eta \simeq 0.5$ and aspect ratio $\mathrm{\Gamma }\simeq 4$. The working fluid is water with a Prandtl number of $\text{Pr}\simeq 5.4$, and the Rayleigh number (Ra) ranges from $5.05\times {10}^7$ to $5.05\times {10}^8$. The multithermal-probe method and the particle image velocimetry technique are employed to measure the temperature profiles and the velocity fields, respectively. Two distinct states with multiroll standing waves are observed, which are the quadrupole state (QS) characterized by a four-roll structure and the sextupole state (SS) by a six-roll structure. The scaling exponents of Reynolds number Re with Ra are different for the two states, which are 0.56 for QS and 0.41 for SS. In addition, the standing waves become unstable upon tilting the cell by $1^{\circ }$ in relation to the horizontal plane, and they evolve into traveling waves. At relatively high Ra, for instance, $\text{Ra}⩾2.55\times {10}^8$, it is observed that the traveling wave state SS undergoes a transition to the traveling wave state QS. However, the opposite transition from QS to SS is not observed in our experiments. Our findings provide insights into the flow structures and dynamics in the convection flow with rotation symmetry.