Scaling Behavior Research Articles (Page 1)

Backgroundα-Synuclein (α-syn) is a prominent protein associated with neurodegenerative conditions such as Parkinson's disease (PD), dementia, and multiple system atrophy, and is a key player in synucleinopathies. Despite its significance, the specific changes in α-syn fibril conformations during the progression of PD remain a subject of uncertainty.ObjectiveThis study investigates the structural alterations in α-syn aggregation from cerebrospinal fluid samples at different PD stages (pre-PD, mid-PD, and late-PD).MethodsIn the present study, we used multifractal detrended fluctuation analysis (MFDFA) and persistent homology. The analysis involves constructing protein contact networks for major and minor α-syn polymorphs. The subsequent application of MFDFA to vertex degree, vertex clustering coefficients, and vertex closeness centrality on this time series data reveals multifractal properties and scaling behaviors. Simultaneously, topological analyses, including Rips complexes, Alpha complexes, and Betti numbers, uncover essential structural features and connectivity patterns in α-syn networks.ResultsThis study illuminates α-syn multifractal dynamics and topological characteristics, providing valuable insights into disease-related protein aggregation and network alterations in the progression of PD.ConclusionsThis study provides unique information on MFDFA and persistent homology of α-syn aggregates across disease stages.

The emergence of Large Language Model (LLM)-based agents marks a significant step towards more capable Artificial Intelligence. However, the effectiveness of these agents is fundamentally constrained by the static nature of their internal knowledge. Tool use has become a critical paradigm to overcome these limitations, enabling agents to interact with dynamic data, execute complex computations, and act upon the world. This paper provides a comprehensive survey of the methods, challenges, and future directions in empowering LLM-based agents with tool-use capabilities. Through a systematic literature review, we synthesized the current state of the art, charting the evolution from foundational agent architectures and core invocation mechanisms like function calling to advanced strategies such as dynamic tool retrieval and autonomous tool creation. Our analysis revealed several critical challenges that impede the deployment of robust agents, including knowledge conflicts between internal priors and external evidence, significant performance degradation in long-context scenarios, non-monotonic scaling behaviors in compound systems, and novel security vulnerabilities. By mapping the current research landscape and identifying these key obstacles, this survey proposes a research agenda to guide future efforts in building more capable, secure, and reliable AI agents.

We study the quantum chromodynamics (QCD) phase transitions in the complex chemical potential plane via the Dyson-Schwinger equation approach, incorporating a constant gluonic background field that represents the confining dynamics. We solve the quark gap equation and the background field equation self-consistently, which allows us to directly explore the confinement phase transition and furthermore, evaluate the impact of the back-coupling of confinement on chiral symmetry breaking. Moreover, within such a coupled framework toward the complex chemical potential region, we demonstrate the emergence of Roberge-Weiss (RW) symmetry and investigate the trajectory of Lee-Yang edge singularities (LYESs). Our analysis reveals that the LYESs scaling behavior is similar to our previous findings without the background field condensate. However, a significant difference from our earlier work is that the trajectory of LYESs terminates when the imaginary part of the singularity becomes 1 / 3 π T . We elaborate that this cutoff behavior is caused by the RW symmetry that is symmetric to the imaginary chemical potential Im μ = 1 / 3 π T .

Abstract Global society is facing growing risks from extreme rainfall events. Obtaining accurate sub-daily rainfall data to support flood-related engineering designs can mitigate such risks, yet it is difficult to obtain worldwide. Temporal scaling offers a method to infer sub-daily extremes from available daily observations. The scaling behaviour is described by the parameter $$\:\beta\:$$ ( $$\:{\beta\:}_{o\:}$$ for observed, $$\:{\beta\:}_{p}$$ for model-projected, and $$\:{\beta\:}_{f}$$ for future values). Using a global gridded precipitation dataset, this study presents a 0.1° resolution model for estimating extreme rainfall scaling from daily to sub-daily durations under present (1979–2020) and future (2071–2100, RCP4.5 and 8.5) climate conditions worldwide. We first assess the influence of geographical and climatic variables—latitude, longitude, altitude, distance to coast, and Köppen–Geiger class—on $$\:{\beta\:}_{o\:}$$ . Then we use four machine learning models to estimate $$\:{\beta\:}_{p}$$ with Random Forest achieving the best performance ( r 2 > 0.95 across all climate types) and greatly outperforming the baseline linear model ( r 2 = 0.13). The model was further applied to estimate 4-hour, 30-year return period rainfall intensities across eight global cities under present and future climate. Results show stronger time-scaling (lower $$\:{\beta\:}_{o\:}$$ ) at higher latitudes, with KG classification being a key predictor. Under future RCP8.5 climate scenarios, projected intensities rise by 20–62% at illustrative sites. This is the first global, high-resolution study using daily rainfall and geographic data to estimate sub-daily extremes, offering a practical tool for assessing flood risks and guiding infrastructure design in ungauged regions.

Abstract The width of the coastal upwelling zone and the depth from which the upwelling originates are two fundamental features of eastern boundary upwelling systems as they play a significant role in determining the state of the upper ocean in these regions. Here, we aim to characterize the width and depth scales associated with the coastal upwelling zone and to identify a set of empirical scaling laws for these length scales. We estimate the length scales using high-resolution, regional ocean simulations of an idealized coastal upwelling system. Our results indicate, though, that there are two distinct horizontal length scales related to the upwelling zone: first, the width of the offshore region where upwelling occurs, which scales like ND s / f , and second, the width of the offshore region where isopycnals deform significantly from horizontal, which scales like , where τ is the wind stress, N is the stratification, D s is the upwelling source depth, and ρ 0 is the reference density. We also find that the scaling for the mean upwelling source depth does not depend on f and instead scales like . We emphasize that while these scaling behaviors provide a strong fit to the numerical results, they are incomplete as they involve coefficients that are not order one nor dimensionless. These scalings are consistent with the results of our numerical simulations as well as self-consistent with the derivation of the mean source depth that depends on the proposed length scale for isopycnal deformation.

ABSTRACT The entanglement spectrum serves as a powerful tool for probing the structure and dynamics of quantum many‐body systems, revealing key information about symmetry, topology, and excitations. While the entanglement entropy (EE) of ground states typically follows an area law, highly excited states obey a volume law, leading to a striking contrast in their scaling behavior. In this paper, we investigate the crossover between these two regimes, focusing on the role of quasi‐particles (QPs) in mediating this transition. By analyzing the energy dependence of EE in various many‐body systems, we explore how the presence of long‐lived QPs influences the entanglement structure of excited states. We present numerical results for spinless fermions, a spin chain near a many‐body localization transition, and the Sachdev‐Ye‐Kitaev (SYK) model, which lacks a conventional QP description. Our findings are complemented by a theoretical model based on Fermi liquid theory, providing insight into the interaction‐dependent scaling of EE and its consistency with numerical simulations. We find that a hallmark of QPs is a linear dependence of the EE on energy, which breaks down at high energies and in the limit of strong interaction. The slope of this linear dependence reflects the QP weight, which decreases with interaction strength.

Unified power-law scaling behavior of collapse mobility and deposition morphology of granular columns composed of frictional-pentagonal grains

Intrinsically disordered proteins (IDPs) exhibit highly dynamic and heterogeneous conformational ensembles that are strongly influenced by sequence features. While global properties such as chain compaction and scaling behavior have been widely studied, they often fail to resolve the fine-grained, sequence-specific structural variation that underlies IDP function. Here, we perform long-time scale atomistic simulations of 47 representative IDP sequences from the yeast proteome to systematically investigate the relationship between sequence composition and conformational ensemble. To analyze the high-dimensional structural data, we apply uniform manifold approximation and projection (UMAP), a nonlinear dimensionality reduction technique that preserves local structural relationships. The resulting low-dimensional embeddings effectively differentiate IDP ensembles and reveal a novel descriptor─local compactness asymmetry─that quantifies directional differences in chain organization. This metric, denoted , captures conformational features orthogonal to traditional global measures such as radius of gyration and end-to-end distance. We show that correlates with sequence-level asymmetries in charge and hydropathy, and that conformational dynamics preferentially occur in the more extended region of the chain. The simulation data set generated in this work also provides a valuable resource for training machine learning models and developing improved coarse-grained force fields for disordered proteins.

Monte Carlo methods provide a probabilistic framework that connects mathematical modeling with physical systems. This study chooses the two-dimensional Ising model as the research object, with Python-based simulation as the main tool. A simple-sampling baseline generates random spin configurations and applies Boltzmann re-weighting, while an importance-sampling scheme based on the Metropolis algorithm samples configurations directly from the Boltzmann distribution. The simulations demonstrate that importance sampling achieves lower variance and greater computational efficiency, particularly at low temperatures where simple sampling becomes inefficient. Moreover, the analysis highlights how algorithmic rules encode physical laws through acceptance probabilities, showing the deep connection between computation and statistical mechanics. It also explains the influence of burn-in, autocorrelation, and finite-size effects on estimator reliability, helping to understand the practical limitations of numerical experiments. The conclusions emphasize the accuracy and broad applicability of Monte Carlo methods while acknowledging difficulties in critical regimes and suggesting future improvements. Beyond these two approaches, alternative strategies such as the microcanonical Monte Carlo method provide another route to explore configuration space under energy constraints. In the vicinity of the critical point, Monte Carlo simulations can also be combined with renormalization group techniques to investigate scaling behavior and critical exponents. Finite-size systems exhibit rounded transitions that can be systematically described using scaling laws near the critical temperature.

We investigate the scaling behavior of high harmonic generation (HHG) driven by magnetic dynamics in spin-orbit coupled systems. In contrast to optically driven HHG-where the harmonic cutoff scales as \(\omega^{-3}\) with the driving frequency \(\omega\)-our time-dependent quantum transport simulations reveal a qualitatively distinct scaling law for magnetically driven HHG in the presence of Rashba spin-orbit interaction: the harmonic cutoff \(n_{\mathrm{max}}\) scales as \(\omega^{-1}\). This fundamental difference arises from distinct excitation mechanisms-namely, spin-flip transitions driven by vectorial magnetic precession, as opposed to scalar electric fields. Furthermore, we demonstrate that the precession cone angle \(\theta\) serves as a crucial control parameter. Increasing \(\theta\) broadens the harmonic bandwidth, with peak emission achieved for nearly in-plane magnetic dynamics. Our findings establish magnetically driven HHG as a robust and tunable mechanism for nonlinear spin transport, governed by unique scaling laws with potential applications in ultrafast spintronic technologies.

Thermoelectric metamaterials featuring width modulation through constrictions (constricted geometries) have emerged as a promising approach for improving heat management and thermoelectric performance. Through a combination of theoretical calculations, analytical formalism, and validation against experimental data, it is shown that thermoelectric performance in such geometries is governed by two fundamental mechanisms of pure geometrical origin: (i) a characteristic scaling behavior of resistance with Transmissivity and (ii) the critical formation of the Constriction Thermal Resistance. Hourglass-shaped thermoelectric legs—identified as optimal in recent experiments—are found to exhibit the same underlying transport mechanisms observed in other constricted profiles, including single and multiple sharp constrictions. The commonly used Geometric Parameter is found to be insufficient for capturing the full influence of geometry on transport, whereas Transmissivity serves as a robust descriptor of constricted geometry, independent of material choice or device-operating conditions. A universal scaling formalism is derived linking electrical and thermal resistances, along with key thermoelectric performance metrics, to the Transmissivity. A unified optimization framework is also developed for composite legs, incorporating both constricted material and contact electrodes. This framework indicates that previously reported performance gains may be largely attributed to contact resistance, rather than geometry alone. Transmissivity is established as a key geometric descriptor, enabling generalized design principles and global optimization criteria for enhancing thermoelectric power generation. This analysis elucidates new avenues in the design of thermoelectric metamaterials for efficient energy conversion.

Reservoir computing is a powerful framework for real-time information processing, characterized by its high computational ability and quick learning, with applications ranging from machine learning to biological systems. In this paper, we investigate how the computational ability of reservoir recurrent neural networks (RNNs) scales with an increasing number of readout neurons. First, we demonstrate that the memory capacity of a reservoir RNN scales sublinearly with the number of readout neurons. To elucidate this observation, we develop a theoretical framework for analytically deriving memory capacity that incorporates the effect of neuronal correlations, which have been ignored in prior theoretical work for analytical simplicity. Our theory successfully relates the sublinear scaling of memory capacity to the strength of neuronal correlations. Furthermore, we show this principle holds across diverse types of RNNs, even those beyond the direct applicability of our theory. Next, we numerically investigate the scaling behavior of nonlinear computational ability, which, alongside memory capacity, is crucial for overall computational performance. Our numerical simulations reveal that as memory capacity growth becomes sublinear, increasing the number of readout neurons successively enables nonlinear processing at progressively higher polynomial orders. Our theoretical framework suggests that neuronal correlations govern not only memory capacity but also the sequential growth of nonlinear computational capabilities. Our findings establish a foundation for designing scalable and cost-effective reservoir computing, providing insights into the interplay among neuronal correlations, linear memory, and nonlinear processing.

Fractals are complex, repeating, and infinitely self-similar patterns. While natural fractals such as snowflakes, coastlines, and other shapes have been reported, emergent work shows that fractal growth may underlie biomolecular self-assembly. However, biomolecular fractals have been observed to be of limited range and, owing to physical realities, prone to exhibiting imperfections. We herein explore Sierpiski fractal-like self-assembly of citrate synthase, a fractal captured recently with cryo-electron microscopy experiments. We first atomistically remodel the basic unit of growth, the initial dimer, and further investigate the geometric, energetic, and dynamical principles underlying the hierarchical self-assembly of citrate synthase oligomers. Using coarse-grained molecular dynamics and subsequent analyses, we quantify deviations from ideal fractality and fluctuations with an increasing fractal level. We analyze solvent-accessible regions lacking amino acid occupancy to characterize the hallmark void of the Sierpiski fractal architecture using a custom algorithm. Since fractals exhibit scaling, we uncover the underlying power law exponents of the structural and thermodynamic signatures of growth. This work provides a framework to understand how hierarchical assembly, structural fluctuations, and scaling behavior contribute to the stability of protein fractal architectures and lend insights into plausible factors that limit higher ordered fractal growth in biomolecular systems.

In light of recent simulations showing the importance of hydration in the thermodynamic properties and structural organization of polyelectrolytes in solution, we revisit small-angle neutron scattering measurements examining the influence of charge valence, along with polymer concentration and mass, on the structure of model salt-free polystyrene sulfonate (PSS) solutions and basic scattering features, such as the polyelectrolyte peak and the radius of gyration and persistence length lp of the PSS molecules. In previous companion work, we showed that the replacement of monovalent by divalent counterions leads to a reduced low-angle scattering intensity related to a reduction of chain association, while the interchain correlation length within the clusters becomes larger, indicating that the divalent counterions give rise to a weakening of the interchain attractive interactions and a reduction in the slow mode relaxation time arising from ion solvation rather than enhanced molecular association. In the present work, we show that the scaling exponent describing the polymer concentration dependence of the polyelectrolyte peak position q* correspondingly changes from a magnitude near 1/2 to a value near 1/3 over a wide polymer concentration range below 1M. This change of scaling behavior with counterion valence raises questions about the hypothesis that salt-free long polyelectrolyte chains exhibit a rod-like conformation in the dilute limit. Direct contrast-matching neutron scattering measurements indicate that the polyelectrolyte chains exhibit substantially enhanced chain flexibility with the replacement of monovalent by divalent counterions. Moreover, lp was found correspondingly to scale with the salt concentration with exponents 1/2 to a value near 1/3, providing further evidence of the sensitivity of the polyelectrolyte properties to counterion valence. Collectively, these properties generally point to the importance of counterion valence and ion and polymer hydration in the properties of polyelectrolyte solutions.

The motion of dopants in magnetic spin lattices has received tremendous attention for at least four decades due to its connection to high-temperature superconductivity. Despite these efforts, we lack a complete understanding of their behavior, especially out of the equilibrium and at nonzero temperatures. In this paper, we take a significant step towards a much deeper understanding based on state-of-the-art matrix-product-state calculations. In particular, we investigate the nonequilibrium dynamics of a dopant in two-leg t−J ladders with antiferromagnetic XXZ spin interactions. In the Ising limit, we find that the dopant is for all investigated temperatures due to an emergent disordered potential, with a localization length controlled by the underlying correlation length of the spin lattice, which increases exponentially with decreasing temperature. The dopant, hereby, only delocalizes asymptotically in the zero temperature limit. This greatly generalizes the localization effect discovered recently in Hilbert space fragmented models [, ]. In the presence of spin-exchange processes at rate α, the dopant diffuses with a diffusion coefficient, Dh, depending nonmonotonically on α. It initially increases linearly as Dh∝α for α≪1 before dropping off as α−1 for α>1. Moreover, we show that the underlying spin dynamics at infinite temperature behaves qualitatively the same, albeit with important quantitative differences. We substantiate these findings by showing that the dynamics features self-similar scaling behavior, which strongly deviates from the Gaussian behavior of regular diffusion, especially for weak spin exchange. Finally, we show that the diffusion coefficient Dh follows an Arrhenius relation at high temperatures, whereby it is exponentially suppressed upon cooling.

This study investigates the scaling process in landfill leachate collection and transportation pipelines. A system dynamics model was established to predict pipeline scaling, employing a combined approach of orthogonal experiments， and simulation techniques to account for the nonlinear coupling effects of crystallisation kinetics and thermodynamics, among other factors. Different parameters were adjusted to study variations in scaling behaviour. The results show that the model predictions of scale formation changes are basically consistent with the experimental scale formation changes. The scale formation shows a rapid increase followed by a decrease， and then gradually stabilises and fluctuates. The water quality factors are ranked as Ca2+ > COD > pH > Humus substances. The effect of temperature on scale formation can be divided into two stages. This study provides new insights into pipeline scale formation research.

Stochastic geometry provides a powerful analytical framework for evaluating interference-limited cellular networks with randomly deployed base stations (BSs). While prior studies have examined limited channel state information at the transmitter (CSIT) and low-resolution analog-to-digital converters (ADCs) separately, their joint impact in multi-user multiple-input multiple-output (MIMO) systems remains largely unexplored. This paper investigates a downlink cellular network in which BSs are distributed according to a homogeneous Poisson point process (PPP), employing zero-forcing beamforming (ZFBF) with limited feedback, and receivers are equipped with one-bit ADCs. We derive a tractable approximation for the achievable spectral efficiency that explicitly accounts for both the quantization error from limited feedback and the receiver distortion caused by coarse ADCs. Based on this approximation, we determine the optimal feedback rate that maximizes the net spectral efficiency. Our analysis reveals that the optimal number of feedback bits scales logarithmically with the channel coherence time but its absolute value decreases due to coarse quantization. Simulation results validate the accuracy of the proposed approximation and confirm the predicted scaling behavior, demonstrating its effectiveness for interference-limited multi-user MIMO networks.

Rheology aims at quantifying the response of materials to mechanical forcing. However, standard rheometers provide only global macroscopic quantities, such as viscoelastic moduli. They fail to capture the heterogeneous flow of soft amorphous materials at the mesoscopic scale, arising from the rearrangements of the microstructural elements, that must be accounted for to build predictive models. To address this experimental challenge, we have combined shear rheometry and time-resolved X-ray micro-tomography on 3D liquid foams used as model soft jammed materials, yielding a unique access to the stresses and contact network topology at the bubble scale. We reveal a universal scaling behavior of the local stress build-up and relaxation associated with topological modifications. Moreover, these plastic events redistribute stress non-locally, as if the foam were an elastic medium subjected to a quadrupolar deformation. Our findings clarify how the macroscopic elastoplastic behavior of amorphous materials emerges from the spatiotemporal stress variations induced by microstructural rearrangements.

We present a comprehensive information-theoretic evaluation of three widely used rigid water models (TIP3P, SPC, and SPC/) through systematic analysis of water clusters ranging from single molecules to 11-molecule aggregates. Five fundamental descriptors—Shannon entropy, Fisher information, disequilibrium, LMC complexity, and Fisher–Shannon complexity—were calculated in both position and momentum spaces to quantify electronic delocalizability, localization, uniformity, and structural sophistication. Clusters containing 1, 3, 5, 7, 9, and 11 molecules (denoted 1 M, 3 M, 5 M, 7 M, 9 M, and 11 M) were selected to balance computational tractability with representative scaling behavior. Molecular dynamics simulations validated the force fields against experimental bulk properties (density, dielectric constant, self-diffusion coefficient), while statistical analysis using Shapiro–Wilk normality tests and Student’s t-tests ensured robust discrimination between models. Our results reveal distinct scaling behaviors that correlate with experimental accuracy: SPC/ demonstrates superior electronic structure representation with optimal entropy–information balance and enhanced complexity measures, while TIP3P shows excessive localization and reduced complexity that worsen with increasing cluster size. The transferability from clusters to bulk properties is established through systematic convergence of information-theoretic measures toward bulk-like behavior. The methodology establishes information-theoretic analysis as a useful tool for comprehensive force field evaluation.

We investigate the penetrative probe diffusion in a model amphiphilic polymer co-network (APCN) synthesized via a hetero-complementary coupling reaction between 2-(4-nitrophenyl)-benzoxazinone-terminated tetra-poly(ε-caprolactone) (t-PCL) and amino-terminated tetra-poly(ethylene glycol) (t-PEG) using isorefractive dynamic light scattering (DLS). We employ spherical silver nanoparticles (AgNPs) and esterified dextrans of varying molecular weights in the APCN swollen in toluene to get insights about the diffusion-governing length scales, namely the correlation length and the hydrodynamic screening length of the network. The diffusion data are analyzed using hydrodynamic and obstruction models, with the hydrodynamic model proving more suitable for such networks. Our results reveal scaling laws for the correlation length as a function of the polymer volume fraction, matching previous theoretical simulations and experimental findings, alongside the determination of the hydrodynamic screening length, marking the transition from the Rouse to the Zimm regime. Additionally, we demonstrate how structural length scales evolve with swelling, offering more profound insights into the structure-property relationships of APCNs. Comparative diffusion measurements in non-crosslinked t-PEG/t-PCL solutions reveal that network crosslinking significantly affects both the characteristic length scales and the scaling behavior of diffusion.