Shannon entropy is not the only entropy that is relevant to machine-learning datasets, nor possibly even the most important one. Traditional entropies such as Shannon entropy capture information represented by elements' frequencies, but not the richer information encoded by their similarities and differences. Capturing the latter requires similarity-sensitive entropy: “sentropy.” Sentropy can be measured using either the recently developed Leinster-Cobbold-Reeve framework (LCR) or the newer Vendi score (VS). This raises the practical question of which one to use: LCR or VS. Here we address this question theoretically and numerically, using 53 large and well-known imaging and tabular datasets. We find that LCR and VS values can differ by orders of magnitude and are complementary, except in limiting cases. We show that both LCR and VS results depend on how similarities are scaled and introduce the notion of “half distance” to parametrize this dependence. We prove that VS provides an upper bound on LCR for all non-negative values of the Rényi-Hill order parameter, as well as for negative values in the special case that the similarity matrix is full rank. We conclude that VS is preferable only when a dataset's elements can be usefully interpreted as linear combinations of a more fundamental set of “ur-elements” or when the system that the dataset describes has a quantum mechanical character. In the broader case where one simply wishes to capture the rich information encoded by elements' similarities and differences as well as their frequencies, we propose that LCR should be favored; nevertheless, for certain half distances, the two methods can complement each other.

In the study of complex systems, evaluating physical observables often requires sampling representative configurations via Monte Carlo techniques. These methods rely on repeated evaluations of the system's energy and force fields, which can become computationally expensive. To accelerate these simulations, deep learning models are increasingly employed as surrogate functions to approximate the energy landscape or force fields. However, such models introduce epistemic uncertainty in their predictions, which may propagate through the sampling process and affect the simulation's macroscopic behavior. In our work, we present the penalty ensemble method to quantify epistemic uncertainty and mitigate its impact on Monte Carlo sampling. Our approach introduces an uncertainty-aware modification of the Metropolis acceptance rule, which increases the rejection probability in regions of high uncertainty, thereby enhancing the reliability of the simulation outcomes.

We analytically and numerically study the effect of finite spatial boundaries on the Takayasu model of diffusing and aggregating particles with steady monomer input in one dimension. Exact expressions are derived for the steady-state density profile, two-point correlation functions, and mean-squared density under both open and periodic boundary conditions. The single-site mass distribution exhibits a crossover from a bulk power law $P\left(m\right)\sim m^{-4/3}$ to an edge power law $P\left(m\right)\sim m^{-5/3}$, occurring near the boundaries or the condensate that forms in periodic systems. The equivalence between the two boundary conditions is shown to break down in the case of multipoint probability distributions near the edge. The exact solution identifies a distinct boundary layer and shows that the edge anomaly arises when spatial mass currents, which scale as $\mathcal{O}\left(L\right)$, dominate over the $\mathcal{O}\left(1\right)$ constant flux in mass space. We further generalize these results, using scaling arguments, to mass-dependent diffusion.

Driven by transport processes in natural and man-made systems, we examine a locally disordered totally asymmetric simple exclusion process with Langmuir kinetics in a resource-constrained environment. The disorder is in the form of a defect that may bind to or unbind from a particular site and slows down the particle movement, when present on the lattice. Using a mean-field approach, we analyze the steady-state behavior of the model by computing density profiles and constructing phase diagrams in the $\alpha –\beta$ parameter space, thereby revealing a rich quantitative and qualitative phase structure. The impact of finite resources and Langmuir kinetics on the stationary properties of the system is analyzed by varying the filling factor and binding constant. Upon varying the filling factor, the results uncover several critical values where the phase diagram changes qualitatively and the resulting phase complexity varies nonmonotonically. As the binding constant is increased from small values, the phase structure evolves from a limited set of phases to maximal diversity at moderate values, before settling into a simplified regime at large values. An obstruction factor is introduced to capture the combined effects of defect density and the slowdown rate in order to incorporate the role of the dynamic defect. Owing to this combined effect, increasing the obstruction factor simplifies the phase diagram, suppressing Meissner-type phases and promoting high density and shock regimes. The mean-field predictions are verified through Monte Carlo simulations implemented via Gillespie algorithm.

Turing patterns, typical examples of self-organization phenomena, are widely observed in the natural world. From Turing's pioneering work [Philos. Trans. R. Soc. Lond. B 237, 37 (1952)] to recent advances by Nakao et al. [Nat. Phys. 6, 544 (2010)], a fundamental framework has been established for analyzing pattern formation in network-organized reaction-diffusion systems. Notably, most existing studies assume identical diffusion coefficients for all network links to find the conditions for Turing patterns emergence. This naturally raises a question: is pattern formation possible if only a subset of network links has diffusion coefficients that fulfill the requirements? Links characterized by diffusion coefficients satisfying these conditions are termed as activated links. This paper investigates the emergence of Turing patterns through a partial link activation approach, encompassing both random and targeted activation schemes. The mean-field theory is employed to derive the conditions for pattern formation when network links are randomly activated. Numerical simulations are conducted to compare the efficiency of random and targeted activation schemes in triggering Turing patterns. The results demonstrate that the partial link activation approach can indeed generate Turing patterns, offering a more efficient mechanism. And the targeted activation scheme exhibits enhanced performance in promoting pattern formation relative to a random activation scheme. This study provides new perspectives on the role of local regulation in governing global network dynamics.

Although atoms in a supercooled liquid appear randomly arranged, local crystal-like structural motifs can emerge. Such motifs are assumed to reduce the kinetic stability of the liquid against crystallization. In this study, the relation between local structure and kinetic stability of a supercooled liquid alloy is investigated through molecular dynamics simulations, using a model alloy that forms Laves phases. To detect subtle structural differences relevant to crystallization, the liquid structure is analyzed using the $p_3$ code. This method encodes each local environment as a graph with or without species labels. Unexpectedly, our analysis shows that liquids with more crystalline structural units can remain metastable for a longer time before crystallization. This apparent paradox is resolved by showing that kinetic stability increases when crystalline units do not easily connect to form extended clusters. Further analysis shows that clusters with the same Voronoi index can yield distinct graphs that exhibit different correlations with kinetic stability, revealing limitations of the traditional Voronoi index. Even clusters with the same graph can correlate differently due to variations in species labeling, highlighting the importance of chemical ordering. To capture medium-range order undetected by conventional analysis based on crystalline units, the concept of loose-crystalline units is introduced. Analysis of loose-crystalline configurations suggests that the previously observed fivefold twin-like motifs in the glassy state are remnants of crystal seeds frozen before they could fully develop. Our approach applies to various systems that form Laves phases.

When considering effective interactions in a many-body or thermodynamic system, it is important to recognize that the inherent character of the effective forces between two bodies already incorporates many-body effects that appear unified in a nontrivial way. Working backward to disentangle these effects into the contributions of pairs, triplets, or larger groups is generally complicated and seldom accomplished. However, this could offer essential insights into the structural architecture and self-assembly of complex systems, such as soft materials. In this contribution, we tackle this problem by employing a simulation-based method, here termed contraction of the bare forces (CBF), which, on the one hand, permits a precise evaluation of the effective interactions between colloids [de los Santos-López et al., J. Chem. Phys. 157, 074903 (2022)] and, on the other hand, makes it possible to quantify the many-body contributions to these effective interactions. The CBF approach is applied to calculate the depletion forces in asymmetric binary mixtures of hard spheres; the approach is sufficiently sensitive to quantify the effects of higher-order terms on the depletion forces between large particles. To assess the accuracy of our approach, we first explore and confirm the long-established prediction founded on purely geometric considerations, which states that for the particle size ratio, $q$, smaller than $q<q_3=0.1547$, the depletion interaction between large colloids does not depend on the concentration of large particles, provided that the chemical potential of the smaller particles is kept constant. In contrast, for mixtures with less size asymmetry, such effects become considerably influential. Specifically, we have conducted an in-depth examination of scenarios with size asymmetries values of $q=0.45$ and 0.60. We directly compare our results for the depletion forces with those obtained from the integral equation framework for effective interactions. Through the use of a naive approximation formulated at the level of the bridge functions, this comparison has further enabled us to identify higher-order contributions to the depletion forces between large colloids, thereby improving and sharpening the underlying theoretical approximations. More importantly, in this work, we explicitly disentangle the many-body components of the depletion forces and assess how significantly they affect the contact attraction strength as a function of both the size ratio and the concentration of the large colloids.

The spatial profile of several types of focusing two-dimensional shock waves (weak shock regime) created in a thin liquid gap are imaged using a shadowgraph with strobe photography. Their profiles before and after focusing are shown to constitute approximately a Hilbert transform pair, regardless of the range of directional angles involved in the focusing and of whether the focusing is at a single point or an extended caustic. This Hilbert transform relation, observable here from a single image or a pair of images, can be explained in terms of the $\pi /2$ Gouy phase shift for focused waves in two dimensions, as was shown elsewhere with experiments involving terahertz pulses in the time domain.

We present a symmetry-based framework for equilibrium statistical mechanics that formulates a single Lie group combining conventional spacetime symmetries with a recently identified phase-space gauge-shifting invariance [Müller et al., Phys. Rev. Lett. 133, 217101 (2024)]. Using Noether's theorem, we obtain a set of general Ward identities together with previously unexplored cross-relations arising from the noncommutation of different symmetry generators. The approach extends standard many-body symmetries–such as translations, rotations, Galilean boosts, dilations, and particle exchange–by incorporating an internal gauge-shift symmetry within a unified group structure. The resulting Lie algebra suggests a hierarchy of exact identities that encompass established sum rules and indicate possible cross-coupling relations between distinct response and correlation functions. We also identify a Wigner-Eckart-Ward reduction that simplifies tensor-hyperforce correlators to two scalar radial spectra in isotropic fluids, and we outline an equivariant gauge-constrained-DFT formulation whose Euler-Lagrange equations are constructed to satisfy the corresponding Ward and cross-Ward constraints. This framework provides a consistent organizational basis for phenomena in liquids, mixtures, and interfaces, and may offer a symmetry-based perspective connecting structure, mechanics, and dynamics in many-body systems.

We study the ${\varphi }^6$ model and derive two broad classes of lattice discretizations that admit static, translationally invariant kinks; that is, stationary kink profiles that can be centered at an arbitrary position relative to the lattice. These discretizations are constructed using a one-dimensional map, ${\varphi }_{n+1}=F\left({\varphi }_n\right)$, which provides a direct and systematic algorithm for generating such models. Numerical computations for two representative cases show that the discrete kinks do not possess internal modes, consistent with the continuum theory, although an additional high-frequency mode may appear above the upper edge of the phonon band. We also show that generic discretizations of the ${\varphi }^6$ model do not support static kink solutions. Instead, the resulting dynamics produce autotraveling and self-accelerating kinks that propagate at the maximal group velocity while continuously emitting radiation.

In this work, we investigate the elastic properties of deflated vesicles and their shape dynamics in uniaxial extensional flow. By analyzing the Helfrich bending energy and viscous flow stresses in the limit of highly elongated shapes, we demonstrate that all stable vesicle configurations are metastable. For vesicles with small reduced volume, we identify the type of bifurcation at which the stationary state is lost, leading to unbounded vesicle elongation in time. We show that the stationary vesicle length remains finite at the critical extension rate. The critical behavior of the stationary vesicle length and of the growth rates of small perturbations is obtained analytically and confirmed by direct numerical computations. The beginning stage of the unbounded elongation dynamics is simulated numerically, in agreement with the analytical predictions.

We investigate the dynamics of an inertial active Ornstein-Uhlenbeck particle suspended in a non-Markovian environment. The particle is additionally subjected to external forces, such as harmonic confinement and a magnetic field. Motivated by the importance of understanding the non-Markovian behavior of complex environments, we examine the impact of a viscoelastic medium by employing the Jeffreys fluid framework for modeling the particle motion, which effectively captures both viscous and elastic contributions of the environment. Within this model, we explicitly derive the corresponding Fokker-Planck equation for each case. Building on this, we extend the analysis to the general non-Markovian framework and derive the corresponding generalized Fokker-Planck equation for a free active particle. Furthermore, we obtain the probability distribution function valid for an arbitrary memory kernel under various conditions, including both free and confined motion with and without a magnetic field. This formulation provides a solid basis for analyzing the dynamics of an active particle in a non-Markovian environment, such as mucus and polymer solutions, and further allows the study of relaxation in confined geometries and responses to external fields.

We formulate a canonical quantization of equilibrium thermodynamics by applying Dirac's theory of constrained systems. Thermodynamic variables are treated as conjugate pairs of coordinates and momenta, allowing extensive and intensive quantities to be promoted to operators in a Hilbert space. The formalism is applied to the ideal gas, the van der Waals gas, and the photon gas, illustrating both first- and second-class quantization procedures. For the ideal gas, a Schrödinger-like equation emerges in which entropy plays the role of time, and the wave function acquires a phase determined by the internal energy. A pseudo-Hermitian framework restores Hermiticity of the temperature operator and establishes the equivalence among constraint realizations. The approach naturally leads to thermodynamic uncertainty relations and suggests extensions to quantum and topological phase transitions, as well as black-hole and nonequilibrium thermodynamics.

Extended Self-Similarity (ESS) is a widely used tool for uncovering universal power-law scaling in systems dominated by nonlinear interactions. This work demonstrates that ESS scaling can also emerge in a system governed by purely linear physics: the propagation of coherent light in a multimode fiber. The system produces complex speckle patterns arising solely from deterministic linear mode interference. We analyze the intensity structure functions of these speckles and observe a robust extended scaling range. The measured scaling exponents align with the classical Kolmogorov scaling exponents. This finding establishes that the statistical signatures captured by ESS are not exclusive to nonlinear systems, revealing a broader applicability of this scaling framework to complex linear systems.

The Sherrington-Kirkpatrick (SK) model is a foundational model for understanding spin glass systems. It is based on the pairwise interaction between each two spins in a fully connected lattice with quenched disordered interactions. The SK model, as an infinite-range model, simplifies the study of this system by eliminating fluctuations. An advanced SK model, known as the $p$-spin model, introduces higher-order interactions that involve the interaction of P spins. This study focuses on the general Hamiltonian of the spin glass model with infinite-range interactions, referred to as the mixed $p$-spin glass model, which consists of adding all $p$-spin interaction terms. This study aims to derive the equations of state for this Hamiltonian, formulate the equations of state within the framework of the first replica symmetry breaking, and determine both the stability condition of the replica symmetric solution and the stability of the replicas belonging to the same group in the first step of replica symmetry breaking. Additionally, this study considers two special cases. The first case is a mixed $p$-spin glass model in which all interactions are normally distributed around zero. The second case focuses on the SK model compound with perturbative higher-order interactions to investigate how these interactions influence the phase transition.

The 2025 Bondi Beach mass shooting of Jews was perpetrated by individuals inspired by ISIS (Islamic State) propaganda that increasingly featured anti-Semitic hate content following the October 2023 start of the Israel-Palestine war. There is an urgent need to get ahead of future threats by understanding how and when a newly created piece of hate content will spread systemwide online. We present a two-species coalescence-fragmentation model with susceptible-infected-recovered dynamics that incorporates the following published empirical features: (1) New pieces of hate content tend to be generated and promoted by a subset of in-built communities on less regulated platforms. (2) These ‘‘hate’’ communities create links (hyperlinks) with each other and with nonhate communities across all platforms to form dynamically evolving clusters (i.e., coalescence) across which new hate content can then spread. (3) These clusters can get broken up by moderator shutdowns (i.e., fragmentation). We present numerical solutions and derive two levels of approximate mean-field theory: effective medium theory and beyond effective medium theory. Both numerical and analytic solutions reveal that systemwide spreading is governed by reentrant threshold phases: as the fraction of hate communities varies, the system can transition from spreading to no spreading and back to spreading. The derived analytic formulas give explicit insight into how these phase boundaries might be manipulated to prevent systemwide spreading. More broadly, the reentrant phase behavior warns that policies which steadily reduce the number of hate communities can initially succeed but then backfire if pushed further, suggesting that blanket requirements for platforms to simply do ‘‘more’’ are oversimplistic.

Turing patterns are a well-studied model of reaction-diffusion equations for developmental patterning. Their applicability has often been limited by the difficulty in identifying candidate molecules that satisfy the requisite criteria for patterning. Here, we build on recent work on geometric models to describe Turing patterning as a potential flow. We show how the universal dynamics of Turing patterning is described by a landscape, largely independent of the underlying reaction-diffusion equations. We apply our framework to three-component systems and demonstrate that we can accurately capture the dynamics of any given component. We extend our framework to larger networks and to models of Turing patterns coupled with external morphogens that provide positional information. We provide a quantitative description of the dynamics of chosen markers and apply it to the dynamics of SOX9 expression during digit patterning.