How do neural language models acquire a language's structure when trained for next-token prediction? We address this question by deriving theoretical scaling laws for neural network performance on synthetic datasets generated by the random hierarchy model (RHM)—an ensemble of probabilistic context-free grammars designed to capture the hierarchical structure of natural language while remaining analytically tractable. Previously, we developed a theory of representation learning based on data correlations that explains how deep learning models capture the hierarchical structure of the data sequentially, one layer at a time. Here, we extend our theoretical framework to account for architectural differences. In particular, we predict and empirically validate that convolutional networks, whose structure aligns with that of the generative process through locality and weight sharing, enjoy a faster scaling of performance compared to transformer models, which rely on global self-attention mechanisms. This finding clarifies the architectural biases underlying neural scaling laws and highlights how representation learning is shaped by the interaction between model architecture and the statistical properties of data.

Geometric frustration is recognized to generate complex morphologies in self-assembling particulate and molecular systems. In bulk states, frustration drives structured arrays of topological defects. In the dilute limit, these systems have been shown to form a novel state of self-limiting assembly, in which the equilibrium size of multiparticle domains are finite and well defined. In this article we employ Monte Carlo simulations of a recently developed 2D lattice model of geometrically frustrated assembly [Hackney et al., Phys. Rev. X 13, 041010 (2023)] to study the phase transitions between the self-limiting and defect bulk phase driven by two distinct mechanisms: (1) increasing concentration and (2) decreasing temperature or frustration. The first transition is mediated by a concentration-driven percolation transition of self-limiting, wormlike domains into an intermediate heterogeneous network mesophase, which gradually fills in at high concentration to form a quasiuniform defect bulk state. We find that the percolation threshold is weakly dependent on frustration and shifts to higher concentration as frustration is increased, but depends strongly on the ratio of cohesion to elastic stiffness in the model. The second transition takes place between self-limiting assembly at high-temperature or frustration and phase separation into a condensed bulk state at low temperature or frustration. We consider the competing influences that translational and conformational entropy have on the critical temperature or frustration and show that the self-limiting phase is stabilized at higher frustrations and temperatures than previously expected. Taken together, this understanding of the transition pathways from self-limiting to bulk defect phases of frustrated assembly allows us to map the phase behavior of this 2D minimal model over the full range of concentration.

Many real-world networks, ranging from subway systems to polymer structures and fungal mycelia, do not form by the incremental addition of individual nodes but instead grow through the successive extension and intersection of lines or filaments. Yet most existing models for spatial network formation focus on node-based growth, leaving a significant gap in our understanding of systems built from spatially extended components. Here we introduce a minimal model for spatial networks, rooted in the iterative growth and intersection of lines, a mechanism inspired by diverse systems including transportation networks, fungal hyphae, and vascular structures. Unlike classical approaches, our model constructs networks by sequentially adding lines across a domain populated with randomly distributed points. Each line grows greedily to maximize local coverage, while subject to angular continuity and the requirement to intersect existing structures. This emphasis on extended, interacting elements governed by local optimization and geometric constraints leads to the spontaneous emergence of a core-and-branches architecture. The resulting networks display a range of nontrivial scaling behaviors: the number of intersections grows subquadratically; Flory exponents and fractal dimensions emerge consistent with empirical observations; and spatial scaling exponents depend on the heterogeneity of the underlying point distribution, aligning with measurements from subway systems. Our model thus captures the key organizational features observed across diverse real-world networks, establishing a universal paradigm that goes beyond node-based approaches and demonstrates how the growth of spatially extended elements can shape the large-scale architecture of complex systems.

A century after Ising introduced the Ising measure to study equilibrium systems, its relevance has expanded well beyond equilibrium contexts, notably appearing in nonequilibrium frameworks such as the Katz-Lebowitz-Spohn (KLS) model. In this work, we investigate a class of generalized asymmetric simple exclusion processes (ASEPs) for which the Ising measure serves as the stationary state. We show that the average stationary current in these models can display current reversal and other unconventional behaviors, offering insights into transport phenomena in nonequilibrium systems. Moreover, although long-range interaction rates often give rise to long-range interactions in the potential function, our model provides a counterexample: Even with long-range interactions in the dynamics, the resulting potential remains short ranged. Finally, our framework encompasses several well-known models as special cases, including ASEPs, the KLS model, the facilitated exclusion process, the cooperative exclusion process, and the assisted exchange model.

We present molecular dynamics simulation results for the static nonlocal dielectric function, $\varepsilon \left(k\right)$, and ion-ion correlations in aqueous electrolytes across various concentrations, analyzed within the framework of statistical mechanical theory. We investigate the complicated behavior of $\varepsilon \left(k\right)$ and its influence on the screening patterns of ionic electric fields. At large to moderate values of the wave number $k, \varepsilon \left(k\right)$ is primarily governed by the molecular structure of water, even at concentrations up to 1 M. However, at higher concentrations, ionic structuring begins to diminish water's influence. In agreement with the theoretical analysis presented, both Debye-type plain exponential and exponentially decaying oscillatory contributions to the electrostatic interactions are obtained for low electrolyte concentrations of NaCl in water. The latter originate from the water structure, while the former can be quantitatively described up to at least 2 M solutions by using a Debye length calculated with a concentration-dependent “dielectric constant” obtained by simulations. The theoretical basis for the use of such a “constant” is examined. At higher concentrations, there are only oscillatory contributions to the interactions. In our simulations, we do not have long-range monotonic interactions of the type reported in surface force measurements at concentrations above 1 M NaCl in water, often referred to as “anomalous underscreening.” Instead, we find a Debye-type decay, modified by an electrolyte-concentration-dependent effective dielectric constant. This behavior is seen up to at least 2 M concentrations, with decay lengths significantly shorter than those associated with underscreening.

Transfer entropy is a widely used measure for quantifying directed information flows in complex systems. While the challenges of estimating transfer entropy for continuous data are well known, it has two major shortcomings for data of finite cardinality: it exhibits a substantial positive bias for sparse bin counts, and it has no clear means to assess statistical significance. By computing information content in finite data streams without explicitly considering symbols as instances of random variables, we derive a transfer entropy measure which is asymptotically equivalent to the standard plug-in estimator but remedies these issues for time series of small size and/or high cardinality, permitting a fully nonparametric assessment of statistical significance without simulation.

The linear stability of global nonaxisymmetric modes in differentially rotating, magnetized, nonideal plasma is critical to classifying turbulence and transport phenomena. We investigate the competition between the local magneto-rotational instability (MRI) and the magneto-curvature instability (MCI)—a distinct nonaxisymmetric low-frequency curvature-driven global branch that appears alongside MRI. To accomplish this, we developed a nonideal global spectral method, which is validated against NIMROD code simulations. This spectral approach allows for the direct derivation of an extended effective potential formalism and a resistive Alfvénic resonance condition, providing a framework for direct analysis of energy contributions and confinement mechanisms. Our study reveals that the global, low-frequency MCI persists at low magnetic Reynolds numbers (Rm), whereas the localized, high-frequency MRI is stabilized by diffusive broadening of its structure around its Alfvénic resonances. Consequently, we identify the global MCI branch as the primary onset mechanism for nonaxisymmetric magnetohydrodynamic instability in systems with finite curvature, e.g., astrophysical rotators. We establish distinct parameter regimes for mode dominance: MCI prevails in geometrically moderate-thickness disks with intermediate curvature and radial gaps, while MRI dominates in thin, low-curvature disks with large radial gaps. Mode competition is also highly sensitive to the flow profile, particularly vorticity and its gradient, with nonuniform shear profiles exhibiting more robust instability due to flow curvature (i.e., the second derivative of the flow profile) and shear contributions. A key outcome is the development of spectral diagrams derived from the global spectral method. These diagrams comprehensively map dominant instabilities and their characteristics, offering a predictive tool for critical onset parameters (i.e., flow curvature, magnetic field, and Rm) and facilitating the interpretation of experimental and simulation results. Notably, these diagrams demonstrate that the global MCI is generally the sole unstable mode at the initial onset of nonaxisymmetric instability.

The rotational dynamics of a freely suspended ferromagnetic particle in a viscoelastic fluid subjected to a rotating magnetic field are studied both experimentally and theoretically. Our results reveal that when the characteristic relaxation time of the fluid is much smaller than the inverse of a critical field frequency, the particle's rotational behavior resembles that in Newtonian fluids. Increasing the relaxation time enhances the time-averaged rotation frequency of the particle undergoing asynchronous rotation. Moreover, the critical frequency is shown to scale linearly with the magnetic field intensity and inversely with the fluid's zero-shear viscosity. Our work is expected to guide precise manipulation of ferromagnetic particles in biomedical systems where viscoelastic environments dominate.

Tracking the movement of tracer particles has long been a strategy for uncovering complex structures. Here, we study discrete-time random walks on finite Cayley trees to infer key parameters such as tree depth and geometric bias toward the root or leaves. By analyzing first-passage properties, we show that the first two first-passage-time factorial moments (FPTFMs) uniquely determine the tree structure. However, if the random walker experiences waiting phases—due to sticky branch walls or presence of traps—then this identification becomes nontrivial. We demonstrate that the generating function of the first-passage-time (FPT) distribution decomposes into contributions from the waiting-time distribution and the random walk without waiting, leading to a nonlinear system of equations relating the factorial moments of the waiting-time distribution and the FPTFMs of random walks with and without waiting. For geometrically distributed waiting times, additional moment measurements do not suffice, but unique determination of the structure is achieved by varying initial conditions or fitting the Fourier transform of the FPT distribution to measured data. The latter method remains effective also for power-law waiting-time distributions, where higher-order FPTFMs are undefined. These results provide a framework for reconstructing treelike networks from FPT data, with applications in biological transport and spatial networks.

Motility is important for bacteria for their survival, adaptation, and pathogenesis, especially in complex environments such as mucus, tissues, and biofilms. Although bacterial motility has been extensively studied based on imaging and tracking bacterial cell bodies, how the behavior and dynamics of the flagellar filaments of bacteria are affected by complex environments remains largely unexplored. To address this knowledge gap, we exploited site-directed mutagenesis and specific fluorescence labeling and directly visualized the flagellar filaments of Escherichia coli (E. coli) bacteria in polyethylene glycol-based hydrogel under fluorescence microscopy. We observed and classified three distinct types of flagellar motions of the E. coli bacteria in hydrogels: swimming (SWIM), trapped (TRAP), and stalled (STALL). Additionally, we quantified and compared the shapes and behaviors of the bacterial flagella of the three types using various shape quantifiers and descriptors. We found that these shape descriptors and quantifiers reliably and consistently reported the behaviors of the bacterial flagellar filaments. We examined the correlation of bacterial motility and flagellar dynamics for the three types, and found that the interactions of the bacterial flagella and hydrogel polymers/mesh reduced such correlation. Last, we inspected the flagellar filaments in more detail and identified their abnormalities due to hydrogel confinement and entanglement of the flagellar filaments with hydrogel polymers. The methods and analyses from this study are expected to support future efforts to develop and optimize hydrogels for more effective trapping of motile bacteria, thereby informing improved contamination-control strategies in hydrogel-related biomedical applications.

Proteins are composed of chains of amino acids that fold into complex three-dimensional structures. Several key features, such as the radius of gyration, fraction of core amino acids $f_{\mathrm{core}}$, packing fraction $\langle \varphi \rangle$ of core amino acids, and structure factor $S\left(q\right)$ define the structure of folded proteins. It is well-known that folded proteins are compact with a radius of gyration $R_g\left(N\right)\sim N^{\nu }$ that obeys power-law scaling with the number of amino acids $N$ and $\nu \sim 1/3$, $f_{\mathrm{core}}\approx 0.09$, and $\langle \varphi \rangle \approx 0.55$. We also investigate the internal scaling of the radius of gyration $R_g\left(n\right)$ versus the chemical separation $n$ between amino acids for subchains of length $n$ and show that it does not obey simple power-law scaling with $\nu \sim 1/3$. Instead, $R_g\left(n\right)\sim n^{{\nu }_{1,2}}$ with a larger exponent ${\nu }_1>1/3$ for small $n$ and a smaller exponent ${\nu }_2<1/3$ for large $n$. To develop a minimal model for proteins that recapitulates these defining structural features, we carry out collapse simulations for a series of coarse-grained models with increasing complexity. We show that a model, which coarse-grains amino acids into a single spherical backbone bead and several variable-sized side-chain beads and enforces bend- and dihedral-angle constraints for the backbone, recapitulates $R_g\left(n\right)$, $f_{\mathrm{core}}$, $\langle \varphi \rangle$, and $S\left(q\right)$ for more than 2500 x-ray crystal structures of proteins.

We propose a particle acceleration mechanism driven by large-amplitude Alfvén waves in a strong magnetic field. The acceleration process proceeds through multiple stages triggered by counterpropagating wave-particle resonant acceleration (CWRA) via decay instability. Initially, parent and daughter Alfvén waves resonantly accelerate particles perpendicular to the ambient magnetic field. The resultant modulational instability generates electrostatic fields within the wave packet, which are locally amplified by the ponderomotive force of the Alfvén wave packet. These fields subsequently drive further acceleration within a few relativistic gyroperiods via gyroresonant surfing acceleration (GRSA). During this, the $\mathbf{v}\times \mathbf{B}$ force facilitates momentum transfer from the perpendicular to the parallel direction. In the later stage, particles become trapped by the parent wave and gain additional energy through single wave resonant acceleration (SWRA). Furthermore, the accumulation of accelerated particles induces electrostatic trailing fields behind and at the tail of the wave packet, which drive trailing-field acceleration (TFA) of other electrons. The combined effects of these mechanisms, CWRA followed by GRSA and SWRA, result in highly relativistic electron energy. The electron energy accelerated through the above process is higher than that accelerated through TFA. This multistage acceleration process provides insights into the generation of high energy cosmic rays in astrophysical environments.

We identify the local structural defects that control the nonaffine displacement fields in jammed disk packings subjected to athermal, quasistatic simple shear. While complex nonaffine displacement fields typically occur during simple shear, isolated effective quadrupoles are also observed and their probability increases with increasing pressure. We show that the emergence of an isolated effective quadrupole requires the breaking of an interparticle contact that is aligned with low-frequency, spatially extended vibrational modes. Since the Eshelby inhomogeneity problem gives rise to quadrupolar displacement fields in continuum materials, we reformulate and implement Eshelby's equivalent inclusion method (EIM) for jammed disk packings. Using EIM, we show that we can reconstruct the nonaffine displacement fields for jammed disk packings in response to applied shear as a sum of discrete Eshelby-like defects that are caused by mismatches in the local stiffnesses of triangles formed from Delaunay triangulation of the disk centers.

We investigate forward signal propagation and gradient back propagation in deep, randomly initialized transformers, and we use our analysis to propose simple and geometrically meaningful criteria for hyperparameter initialization that ensures trainability of deep transformers. Our approach leverages a dynamical perspective in which we regard the token vectors as particles evolving according to the layerwise map of the transformer. We derive simple update equations for the ensemble-averaged geometry of this many-particle configuration starting from a permutation-symmetric simplex. Our analysis reveals an order-chaos phase transition in the forward direction, which corresponds to converging/diverging token angles, and another transition in the backwards direction between exploding/vanishing gradients. We derive the phase boundaries as a function of the initialization hyperparameters, such as the variance of the weights and the strength of attentional and MLP residual branches. We analytically derive two Lyapunov exponents: a token angle exponent that governs the departure from the edge of chaos and a gradient exponent that governs the rate of exponential growth or decay of backpropagated gradients. We show through experiments that the final test loss at the end of training is correlated with these two exponents at the beginning of training, suggesting that simultaneous vanishing of these two exponents constitute simple criteria for achieving minimal test loss.

Active polymers under specific boundary constraints are a central topic in both biophysics and polymer physics. In this work, we perform a comparative study of polar active polymers with one end fixed, highlighting how the choice of the fixed point (tail vs head) leads to dramatically distinct conformational and dynamical behaviors. Tail fixation induces nonuniform stretching along the polymer and results in a nonmonotonic dependence of conformational relaxation on chain activity. In contrast, head fixation causes uniform chain compression, forming a yarn-ball-like structure and giving rise to a distinct conformational relaxation dynamics. We identify several scaling relations that have not been discussed previously, including the dependence of polymer size on chain length $R_{\mathrm{g}}\sim N^{-0.38}$, the conformational relaxation time on chain length ${\tau }_{\mathrm{r}}^{*}\sim N^{0.6}$ and on activity ${\tau }_{\mathrm{r}}^{*}\sim {\mathrm{Pe}}^{-1.35}$. These results provide fundamental insights into the interplay between activity and boundary conditions in determining the structural and dynamical behaviors of polar active polymers, offering potential implications for guiding the design of bioinspired active materials.

Mutually repelling particles spontaneously form ordered clusters when forced into confinement. The clusters may adopt similar spatial arrangements even if the underlying particle interactions are contrastingly different. Here, we demonstrate with both simulations and experiments that it is possible to induce particles of very different types to self-assemble into the same ordered geometric structure by simply regulating the ratio between the repulsive and confining forces. This is the case for both long- and short-ranged potentials. This property is initially explored in systems with two-dimensional circular symmetry and subsequently demonstrated to be valid throughout the transition to one-dimensional structures through continuous elliptical deformations of the confining field. We argue that this feature can be utilized to manipulate the spatial structure of confined particles, thereby paving the way for the design of clusters with specific functionalities.

We investigate density fluctuations in three-dimensional chiral active fluids by using a simple model of helical self-propelled particles. Helical motion is generated by a constant angular velocity (or chiral torque) acting on the self-propelled force. The chiral torque is assumed to have the same direction and magnitude for all particles. Due to the helical nature of the particle motion, the system is generically anisotropic even when it is spatially homogeneous. Numerical simulations demonstrate that the helicity induces an anisotropic pattern and a singularity in the static structure factor (the density correlation function in Fourier space) in the low-wave-number limit. Moreover, the system in the limit of infinite persistence time exhibits hyperuniformity in the direction perpendicular to the chiral torque, while giant density fluctuations emerge along the parallel direction. We then construct a fluctuating hydrodynamic theory for the system to describe the singular behavior. A linear analysis of the resulting equations yields an analytical expression for the static structure factor, which qualitatively agrees with our numerical findings.

We investigate a one-dimensional waterlike lattice model with van der Waals and hydrogen-bond interactions, allowing for particle number fluctuations through a chemical potential. The model, defined on a chain with periodic boundary conditions, exhibits three ground-state phases, gas, bonded liquid, and dense liquid, separated by sharp phase boundaries in the chemical potential and temperature plane. Using the transfer-matrix method, we derive exact analytical results within the grand-canonical ensemble and examine the finite-temperature behavior. The system exhibits clear pseudotransition features, including sharp but analytic changes in entropy, density, and internal energy, along with finite peaks in specific heat and correlation length. To assess the role of thermodynamic constraints, we consider the behavior under fixed density through a Legendre transformation. This constrained analysis reveals smoother anomalies, such as entropy kinks and finite jumps in specific heat, contrasting with the sharper grand-canonical signatures. These results underscore the ensemble dependence of pseudotransitions and show how statistical constraints modulate critical-like behavior. We also verify that the residual entropy continuity criterion holds in the grand-canonical ensemble but is violated when the system is constrained. Our findings illustrate how even a simple one-dimensional model can mimic waterlike thermodynamic anomalies.

Boolean networks, inspired by gene regulatory networks, were developed to understand the complex behaviors observed in biological systems, with network attractors corresponding to biological phenotypes or cell types. In this article, we present a proof for a conjecture by Williadsen, Triesch, and Wiles about upper bounds for the stability of basins of attraction in Boolean networks. We further extend this result from a single basin of attraction to the entire network. Specifically, we demonstrate that the asymptotic upper bound for the robustness and the basin entropy of a Boolean network are negatively linearly related.

Assemblies of magnetic beads, also called magnetostructures, exhibit interesting mechanical properties, adapting orientations of the dipoles to minimize the dipolar energy. A ring made of $N$ spherical magnetic beads behaves like an elastic annulus. This elastic-like property is due to the dipolar nature of the particles. When submitted to gravity and as a function of its size, the initial circular shape of a magnetic ring is seen to experience flattening. This capsule-like shape appears when the number $N$ of magnets reaches a critical point scaling with the Bond number $N_f\propto {\mathrm{Bo}}^{-1/3}$. When the number of magnets increases more and more, the ring starts to buckle and a flat object appears at a second critical point $N_z\propto {\mathrm{Bo}}^{-1}+b$. There, a zipping state corresponding to the attraction of two opposite sides is formed. We propose a theoretical approach to capture these scaling laws in agreement with experimental data. All shapes are also numerically obtained in a discrete element model confirming our findings.