In this paper, we take a step toward characterizing the physics of the hovering phenomenon in flapping insects and hummingbirds as a class of extremum seeking (ES) feedback systems. By characterizing hovering flight in insects and hummingbirds as a natural hovering ES system, we achieve (1) very simple, (2) stable, (3) model-free, and (4) real-time hovering. More importantly, our hovering ES characterization only needs the natural oscillations of the wing as the ES input. That is, unlike other control techniques in the literature, the natural hovering ES system only needs the natural flapping action built in the system and feedback of local sensations (measurements) related to the altitude where the insect seeks to stabilize itself. Said ES characterization can become an important initial step in starting a new line of research that may succeed in resolving the long-standing gap between model-based control theory and the biologically observed mechanisms that stabilize hovering flight. We provide simulation trials, including comparisons with some approaches from literature, to demonstrate the effectiveness and robustness of our results. We used literature data for hawkmoth, cranefly, bumblebee, dragonfly, hoverfly, and a hummingbird.

We adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane with tilt angle $\gamma$. We explore the dynamics as a function of $\gamma$ and the cone deficit angle $\chi$ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. We find regions of the ($\gamma ,\chi$) plane where, depending on the initial conditions, the trajectories either (a) sample the entire cone base and avoid the apex region, (b) sample only a portion of the base region while again avoiding the apex, or (c) sample the entire cone surface much more uniformly, suggestive of ergodicity. The special case of an untilted cone displays only type (a) trajectories, which form a ring caustic at the distance of closest approach to the apex. However, we observe an intricate transition to chaotic dynamics dominated by type (c) trajectories for sufficiently large $\chi$ and $\gamma$. A Poincaré map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. We then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps.

We analytically uncover the emergence of a universal solidlike phase in one-dimensional hard-sphere systems (Tonks gas). The associated pair distribution function $g\left(r\right)$ decays algebraically as ${(r/\xi )}^{-1/2}$, revealing quasi-long-range order. Beyond that correlation length $\xi \sim \ell /{(1-\varphi )}^2$, an exponential decay regime of the form $e^{-r/\xi }$ sets in, where $\ell$ denotes the mean particle spacing and $\varphi$ the packing fraction. We identify the onset of solidlike behavior near ${\varphi }^{*}\simeq 0.8$ at $\xi \approx \ell$, where the crossover point coincides with the third neighbor shell. Our findings establish a clear theoretical framework for emerging crystallization in one dimension and introduce criteria for detecting incipient solid order in constrained geometries.

In nature, granular materials fail in abrupt avalanches, earthquakes, and other hazardous events, and also creep over time. Proposed failure mechanisms for these systems are broadly framed as friction limited. However, mechanical descriptions of friction in granular system vary, including those that consider the nonlinear, heterogeneous dynamics of grain-contact forces. In order to study granular controls on failure and creep, we imaged contact forces between quasi-2D photoelastic disks at 15-min intervals in experiments over weeks- to 1-month-long observation periods. In the experiments, the particles are distributed in a slope below the angle of repose with a concomitant establishment of the force chain network approximately along the principal static stress directions. The discrete force chain shifts are initially described by an age-weakening Weibull distribution in frequency over time, with the most common clusters of events $<1000$ min apart, but there are long quiescent periods that are not well described by the distribution. Different surface slopes show the same contact aging rate, but the steeper the slope the more likely there is to be a failure event. The postsettling discrete events accompany a longer-term strengthening of the localized stresses. We observe that some events may be related to $2{}^{\circ }\mathrm{C}$ temperature change, but associated ground motions measured by a laboratory-installed seismometer appear to have no correlation with particle displacements or force chain changes. The sum of all ground motion is 4 orders of magnitude smaller than the temperature changes, further ruling out mechanical noise as an appreciable cause of events. The results illustrate that local, grain-scale changes in force chain networks can occur long after a granular pile reaches a mesoscale apparently stable state, sometimes without obvious external forcings or imposed state changes. Such force chain dynamics may underlie transitions in natural granular systems between large-scale failures and creep events.

Jerky active particles are Brownian self-propelled particles which are dominated by “jerk,” the change in acceleration. They represent a generalization of inertial active particles. In order to describe jerky active particles, a linear jerk equation of motion which involves a third-order derivative in time, Stokes friction, and a spring force is combined with activity modeled by an active Ornstein-Uhlenbeck process. This equation of motion is solved analytically and the associated mean-square displacement (MSD) is extracted as a function of time. For small damping and small spring constants, the MSD shows an enormous superballistic spreading with different scaling regimes characterized by anomalous high dynamical exponents 6, 5, 4, or 3 arising from a competition among jerk, inertia, and activity. When exposed to a harmonic potential, the gigantic spreading tendency induced by jerk gives rise to an enormous increase of the kinetic temperature and even to a sharp localization-delocalization transition, i.e., a jerky particle can escape from harmonic confinement. The transition can be either first or second order as a function of jerkiness. Finally it is shown that self-propelled jerky particles governed by the basic equation of motion can be realized experimentally both in feedback-controlled macroscopic particles and in active colloids governed by friction with memory.

We consider a homogeneous chain of spheres linked by liquid bridges under tension. The rupture of a single liquid bridge leads to a fragmentation cascade driven by the inverse relation between the capillary force and the sphere distances. The initial length of the liquid bridges determines the number and size of the fragments and the velocity of the fragmentation front.

Recently, thanks to the development of artificial intelligence (AI), there is increasing scientific attention in establishing the connections between theoretical physics and AI. Traditionally, these connections have been focusing mostly on the relation between string theory and image processing and involve important theoretical paradigms such as holography. Recently, G. Bianconi has formulated the Gravity from Entropy (GfE) approach to quantum gravity in which gravity is derived from the geometric quantum relative entropy (GQRE) between two metrics associated with the Lorentzian spacetime. Here, it is demonstrated that the famous Perona-Malik algorithm for image processing is the gradient flow of the GfE action in its simple warm-up scenario. Specifically, this algorithm is the outcome of the maximization of the GfE action between two Euclidean metrics: the one in support of the image and the one induced by the image. As the Perona-Malik algorithm is known to preserve sharp contours, this implies that the GfE action does not, in general, lead to uniform images upon iteration of the gradient flow dynamics as it would be intuitively expected from entropic actions maximizing classical entropies. Rather, the outcome of the maximization of the GfE action is compatible with the preservation of complex structures. These results provide the geometrical and information theory foundations for the Perona-Malik algorithm and might contribute to establish deeper connections between GfE, machine learning, and brain research.

Systems driven far from equilibrium may exhibit anomalous density fluctuations: active matter with orientational order displays giant density fluctuations at a large scale, while systems of interacting particles close to an absorbing phase transition may exhibit hyperuniformity, suppressing large-scale density fluctuations. We show that these seemingly incompatible phenomena can coexist in nematically ordered active systems, provided activity is conditioned to particle contacts. We characterize this unusual state of matter and unravel the underlying mechanisms simultaneously, leading to spatially enhanced (on large length scales) and suppressed (on intermediate length scales) density fluctuations. Our work highlights the potential for a rich phenomenology in active matter systems in which the particles' activity is triggered by their local environment, and calls for a broader exploration of absorbing phase transitions in orientationally ordered particle systems.

The reversible adsorption of a polymer chain to an attractive surface is an important problem in materials science and biophysics. The location of this transition ($T_c$) is sensitive to both polymer flexibility ($l_p$) and the range of the attractive surface potential ($\lambda$) and, for long chains, simple scaling arguments predict $T_c\sim {\left(l_p\right)}^a{\left(\lambda \right)}^b$ with different power-law exponents for different regimes of $l_p$ and $\lambda$. Verification of these scaling laws for semiflexible polymers via computer simulation is challenging due to the long chain lengths ($N$) required to reach the asymptotic scaling regime. Here we propose a finite-size-scaling method using partition function zeros to obtain adsorption transition temperatures in the long-chain limit from simulations of chains of moderate length. By combining the real and imaginary parts of the leading partition function zeros it is possible to eliminate the size- and flexibility-dependent scaling function that describes the $N\to \infty$ approach of these leading zeros to the critical point in the complex inverse-temperature plane. Our model polymer is a flexible tangent-hard-sphere chain (sphere diameter $\sigma$) with a local bond angle restriction that sets a persistence length $l_p$. The chain is end-tethered to a flat surface that has a square-well attractive potential of range $\lambda \sigma$. We use a Wang-Landau simulation algorithm to obtain the density of states for chains up to length $N=2560$ with $1\le l_p/\sigma \le 13,100$ and $0.01\le \lambda \le 20$. We distinguish three distinct scaling regimes over this wide parameter space: (i) worm-like-chain behavior for $l_p/\sigma >max(10,10\lambda )$, (ii) expanded-coil behavior for $\lambda >1$ with $l_p/\sigma <\lambda$, and (iii) a single-bead-interaction region for $\lambda <1$ with $l_p/\sigma <10$, and we find scaling laws consistent with simple scaling expectations for each of these regions. In the rigid-rod limit ($l_p\to \infty$), an exact solution of the model shows that $T_c^{*}\to \infty$ for $N\to \infty$.

Dropout is a regularization technique widely used in training artificial neural networks to mitigate overfitting. It consists of dynamically deactivating subsets of the network during training to promote more robust representations. Despite its widespread adoption, dropout rates are often selected heuristically, and theoretical explanations of its success remain sparse. Here we analytically study dropout in two-layer neural networks trained with online stochastic gradient descent. In the high-dimensional limit, we derive a set of ordinary differential equations that fully characterize the evolution of the network during training and capture the effects of dropout. We obtain a number of exact results describing the generalization error and the optimal dropout probability at short, intermediate, and long training times. Our analysis shows that dropout reduces detrimental correlations between hidden nodes and mitigates the impact of label noise and that the optimal dropout probability increases with the level of noise in the data. Our results are validated by extensive numerical simulations.

Steric clashes pose a challenge when exploring dense protein systems using conventional explicit-chain methods. A minimal example is a single lattice protein confined on a minimal grid, with no free sites. Finding its minimum energy is a hard optimization problem, with similarities to scheduling problems. It can be recast as a quadratic unconstrained binary optimization (QUBO) problem amenable to classical and quantum approaches. We show that this problem in its QUBO form can be swiftly and consistently solved for chain length 48, using either classical simulated annealing or hybrid quantum-classical annealing on a D-Wave system. In fact, the latter computations required about 10 s. We also test linear and quadratic programming methods, which work well for a lattice gas but struggle with chain constraints. All methods are benchmarked against exact results obtained from exhaustive structure enumeration, at a high computational cost.

Optical generation of energetic particle bunches requires high-power laser facilities operating in picosecond or femtosecond temporal domain. It is therefore preferable to use also short laser pulses in all-optical platforms providing quasistationary electromagnetic fields for guiding and focusing of such particle beams, increasing their brightness and decreasing their angular divergence. In this work, theoretical and experimental results are discussed for a novel electromagnetic guiding setup based on a shaped spiral-like “snail” target with a relatively large focusing aperture in comparison with the previous studies. Due to the diameter increased to a submillimeter scale, wider particle bunches may be efficiently focused, as shown with theoretical modeling using the field distributions, consistent with the presented experimental data.

Diffusion-triggered convection can occur in a gravitationally stable system of two superimposed mixtures. As instability develops, two distinct patterns may emerge: double-diffusive (DD) or diffusion-layer convection (DLC). Traditionally, nonsymmetric patterns above and below the interface were thought to require chemical reaction. We show that symmetry can be broken by composition-dependent diffusion, with or without cross diffusion. Furthermore, only the composition-dependent cross diffusion can lead to a range of coexisting patterns and provide new insights into staircase instability.

Phase field crystal (PFC) models constitute central tools for a microscopic understanding of the dynamics of complex systems in soft-matter physics. They have found widespread application in the modeling of the uniaxial orientational ordering of liquid crystals. However, only very limited progress has been made in applying them to the more complex cases of biaxial phases and biaxial particles. Here we discuss the microscopic derivation of PFC models for biaxial liquid crystals. We illustrate it by presenting two models, one involving four scalar orientational order parameters relevant for the dynamics of biaxial particles and one involving two scalar order parameters and all three directors to describe biaxial phases in a three-dimensional molecularly uniaxial nematic liquid crystal. These models allow for an efficient simulation of spatially inhomogeneous biaxial orientational ordering dynamics. We also combine our microscopic with a macroscopic approach to extract model coefficients for a full biaxial model from the microscopic derivation for a simple special case. This universal method also enables to perform derivations for other low-symmetry particles where, due to the complexity of the more general cases, this has not been previously attempted.

The macroscopic behavior of active matter arises from nonequilibrium microscopic processes. In soft materials, active stresses typically drive macroscopic shape changes, which in turn alter the geometry constraining the microscopic dynamics, leading to complex feedback effects. Although such mechanochemical coupling is common in living matter and associated with biological functions such as cell migration, division, and differentiation, the underlying principles are not well understood due to a lack of minimal models that bridge the scales from the microscopic biochemical processes to the macroscopic shape dynamics. To address this gap, we derive tractable coarse-grained equations from microscopic dynamics for a class of mechanochemical systems, in which biochemical signal processing is coupled to shape dynamics. Specifically, we consider molecular interactions at the surface of biological cells that commonly drive cell-cell signaling and adhesion, and obtain a macroscopic description of cells as signal-processing droplets that adaptively change their interfacial tensions. We find a rich phenomenology, including multistability, symmetry-breaking, excitability, and self-sustained shape oscillations, with the underlying critical points revealing universal characteristics of such systems. Our tractable framework provides a paradigm for how soft active materials respond to shape-dependent signals, and suggests novel modes of self-organisation at the collective scale. These are explored further in our companion manuscript [T. Dullweber et al., Phys. Rev. Lett. 135, 128404 (2025)].

Memory effects in amorphous materials have been widely studied because of their possible widespread future applications. We show here that ultrastable glasses can exhibit a transient reversible memory effect when subjected to both a local driving force via run-and-tumble active particles and global shear. We investigate the system's response across different yielding regimes by selectively switching the shear direction at different strains. We analyze how changes in shear direction influence yielding, postyield behavior, and structural evolution in active amorphous solids. Our model active system exhibits an enhanced anisotropic response, displaying both conventional and inverse Bauschinger effects, depending on the deformation history. The results indicate that activity-induced shear band networks create structural memory, enabling the system to heal upon shear reversal due to the transient nature of this phenomenon. Additionally, shear softening under cyclic loading produces irreversible and less branched networks with increasing cycles, a structural evolution underlying the transition from inverse Bauschinger effect to conventional Bauschinger effect. These findings provide novel insights into how activity and shear collectively contribute to mechanical response, including memory formation in ultrastable disordered systems.

Embryonic development relies on the formation of sharp, precise gene expression boundaries. In the fruit fly Drosophila melanogaster, boundary formation has been proposed to occur at a dynamical critical point. Yet, in the paradigmatic case of the hunchback (hb) gene, evidence suggests that boundary formation occurs in a bistable regime, not at the dynamical critical point. We develop a minimal model for hb expression and identify a single parameter that tunes the system from its monostable regime to its bistable regime, crossing the critical point in between. We find that boundary precision is maximized when the system is weakly bistable—near, but not at, the critical point—optimally negotiating the trade-off between two key effects of bistability: sharpening the boundary and amplifying its noise. Incorporating the diffusion of Hb proteins into our model, we show that boundary precision is maximized simultaneously at an optimal degree of bistability and an optimal diffusion strength. Our work elucidates design principles of precise boundary formation and has general implications for pattern formation in multicellular systems.

Chiral active materials are those that break both time-reversal symmetry and parity microscopically. This is usually a result of the average rotation of the material's complex particles around their center-of-mass (c.m.), which drives the system far from equilibrium. In this paper we show that the nonvanishing spin angular momentum brings about a difference between the c.m. momentum and the total momentum, where the latter accounts for the momentum of all atoms that compose the complex rotating particles. This is in stark contrast to equilibrium fluids where the c.m. stress and the total momentum are essentially equivalent. In fact, we find that generally the c.m. dynamics are insufficient to describe the dynamics of a chiral active material. The total momentum, other than being experimentally accessible in simple rheological experiments, also imposes a constraint—its stress must be allowed to be written in a symmetric way. We find that this requirement imposes a relation between possible central-force interactions and spin-spin interactions, and constraints the amount of odd viscosities in the system to the well-known odd (Hall) viscosity and the odd pressure.

Pairwise nonreciprocal interactions are known to drive nonequilibrium collective phenomena. However, there also exist interactions that are reciprocal for a pair of objects but break reciprocity through multibody effects. In this case, nonreciprocity emerges spontaneously, without having to use special objects or tailor the interactions. Here, we demonstrate this with three identical spheres acoustically levitated in air that become excited into novel dynamical states through nonreciprocal multibody interactions induced by sound. This three-sphere system allows detailed interrogation of the competition between wave scattering and viscous microstreaming in generating self-organized, dynamic steady states that exhibit collective limit cycles and self-propulsion. Crucially, energy from the acoustic field is redirected to power these dynamics when the configuration of the three particles breaks inversion symmetry. These results introduce a minimal model system for emergent activity and open up new possibilities for self-assembly, where multibody interactions not only determine the resulting structure but also drive spontaneously emerging dynamics.

The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability $p$, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as $p$ increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range $p_{c_1}<p<p_{c_2}$, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At $p_{c1}$ and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At $p_{c2}$, the fractal dimension of clusters and the scaling exponent along $p$ direction are estimated as $y_{h2}=1.9580\left(6\right)$ and $y_{p2}=0.552\left(9\right)$, of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.