We perform large-scale numerical simulations to investigate the critical behavior of $k$-core percolation in two dimensions with an extended interaction range $r$. By systematically varying both the core index $k$ and the interaction range $r$, we construct a comprehensive phase diagram in the $(k,r)$ plane. In contrast to $k$-core percolation in infinite dimensions, no hybrid transition is observed in two dimensions: The phase diagram contains only a continuous transition regime and a strictly first-order regime, separated by a tricritical or critical end point $(k_s,r_s)$. For $k<k_s$ and $r<r_s$, the transition is continuous and belongs to the universality class of standard two-dimensional percolation. For $k>k_s$ and finite $r>r_s$, the transition is discontinuous, with no hybrid features or critical singularities. In this first-order regime, the pseudocritical point approaches the critical point as $1/lnL$, where $L$ is the linear system size, distinct from the $L^{-d}$ scaling typical of conventional thermodynamic first-order transitions in $d$ dimensions. This logarithmic finite-size drift is consistent with a nucleation-driven mechanism, in which rare voids trigger the collapse of the finite-range $k$-core. These results demonstrate that geometric constraints can fundamentally alter the nature of $k$-core percolation found in finite dimensions.

Standard affine jump-diffusion models treat shocks as increments to the system's state, whereas conventional stochastic resetting models reduce their impact to a full reset of the system's state. Neither model is adequate when diffusion along one of the state components persists, while shocks affect another component. Our study examines a linear system perturbed by two independent additive Markovian components. We demonstrate that in this model the solutions to the generalized Feynman-Kac (GFK) equations describing the expected evolution operator admit an exact factorization to solutions corresponding to separate noise components. Factorizable two-component models are applicable in the real world, particularly in systems where an endogenous background noise persists even when exogenous shocks occur. We show that this decoupling enables a comprehensive description of stylized yield-curve shapes. Furthermore, it allows for the prediction of deviations from exponential Moore's Law trajectories for novel technologies.

We introduce a solvable spherical model of coupled oscillators with fully random interactions and distributed natural frequencies. Using the dynamical mean-field theory, we derive self-consistent equations for the stationary response and correlation functions. We show that the finite-temperature spin-glass transition is suppressed for any nonmonodisperse natural-frequency distribution, including continuous, discrete, and asymmetric distributions, because the resulting low-frequency singularity of the correlation function is incompatible with the spherical constraint. At zero temperature, however, a spin-glass phase persists for arbitrary frequency dispersion. This residual zero-temperature glassiness is likely a special feature of the spherical dynamics and would be destroyed by local nonlinearities. The model thus provides a solvable oscillator framework for studying how nonequilibrium perturbations suppress finite-temperature glassy freezing.

Network community detection is usually considered as an unsupervised learning problem. Given a network, the aim is to partition it using some general-purpose algorithm. In this paper, we instead treat community detection as a hypothesis testing problem. Given a network, we examine the evidence for community structure in the observed network compared to a network generated by a null model. To do this, we define an appropriate test statistic, analogous to a z-score, and discuss several null models derived from maximizing entropy under different constraints. These constraints are imposed in expectation, giving a canonical ensemble rather than the usual microcanonical ensemble. We demonstrate the application of this method on real and synthetic data and contrast our method to Bayesian approaches based on the stochastic block model. We demonstrate that this method gives rigorous answers to concrete questions, which can be more useful to analysts than the output of a generic algorithm.

We analyze the protein secondary structure using the Frenet frame to describe the curvature and torsion of the discrete curve formed by the protein $\alpha$-carbons. We show how a simple criterion based on the evaluation of Frenet frame rotation along the discrete curve supports identification of alpha-helices, beta-strands, and selected supersecondary structures in proteins. Moreover, the description of proteins as fixed points of an effective action inspired by a $U\left(1\right)$ gauge model is supported by the analysis of a large dataset of proteins in the Protein Data Bank.

Theoretical expressions used to study the assembly of magnetic particles under the action of external magnetic fields are generalized to incorporate carrier fluids with varying viscosity (shear thinning fluids). The theory is essentially local, and the drag force assigned to each particle considers local viscosity and flow changes due to pseudoplastic behavior. A benchmark is performed with finite element simulations for the case of pairwise interactions and static external fields. General agreement is observed, although simulations reveal the complexity that arises in these systems due to nonlinear hydrodynamic interactions. In the case of external rotating fields, the generalized theory predicts the transition from chains to cluster-like assemblies as the angular frequency of the field increases, similar to what happens in Newtonian media. The difference lies in that particles forming clusters in shear thinning fluids exhibit a higher degree of orientational ordering and are larger than those formed in Newtonian media. An experimental test is performed using magnetite dispersed in a power-law fluid made with xanthan gum. Image analysis reveals that the clusters formed in the power-law fluid are larger than those formed in a reference Newtonian fluid. Further analysis indicates that aggregation in the non-Newtonian fluid is primarily a dynamical process and is little influenced by direct interactions between the magnetite and the polymer.

A key principle underlying most turbulence theories is that the mean energy dissipation rate remains finite even as viscosity vanishes: the so-called zeroth law of turbulence. Although this property has been established for hydrodynamic and magnetohydrodynamic (MHD) turbulence, its validity in weakly collisional plasmas has remained uncertain, where viscous and resistive closures fail. Using fully kinetic numerical simulations, we provide confirmation that the mean dissipation rate approaches a finite, system-independent value when expressed in terms of an effective Reynolds number derived from a heuristic collisionless viscosity. This result demonstrates a finite dissipation anomaly in collisionless plasma turbulence, establishing that the zeroth law extends beyond the collisional MHD regime.

The configuration path integral Monte Carlo (CPIMC) method is employed to perform first-principles simulations of warm dense matter based on atomic orbitals. The system is modeled as an ion sphere with a pointlike nucleus at its center, and the electronic many-body problem is solved using CPIMC without approximations beyond the chosen Hamiltonian, basis truncation, and controllable statistical errors. Within this framework, estimators for the pressure and radial electron density are derived, and simulation schemes for the Gibbs and grand-canonical ensembles are formulated. Motivated by the $\eta$-ensemble concept in path integral Monte Carlo, a practical approach is developed to compute the partition function directly from an ideal grand-canonical reference system. To describe the dynamical coupling between electronic orbitals of the nearest-neighbor ion pairs, “substates” are introduced that combine bonding, antibonding, and continuum-like radial solutions, thereby providing a unified treatment of bound and continuum electrons and a flexible and model-controlled definition of ionization. The implementation is validated by comparing CPIMC simulations for neon using two distinct orbital basis sets, Hartree-Fock-Slater and hydrogen-like orbitals, yielding excellent agreement. The resulting equation of state is further compared with DFT-MD and RPIMC results for neon reported in the literature, showing good agreement in pressure and small deviations in internal energy. Finally, the substate extension is applied to beryllium in the canonical, grand-canonical, and Gibbs ensembles, and the results are compared with methods formulated in both the chemical and physical pictures. These results indicate that CPIMC with atomic orbitals is a promising route toward finite-temperature simulations of real materials in the warm dense regime, particularly for heavier elements.

We develop a long-wavelength theory for the linear stability of a diffuse interface between an active nematic and an isotropic phase. The starting point is a Cahn-Hilliard-Landau-de Gennes description of the coexistence profile, coupled to Brinkman-screened Stokes hydrodynamics. Small perturbations are projected onto three interfacial degrees of freedom: the conserved height mode associated with translations of the coexistence profile, a scalar profile-distortion mode, and a transverse orientational mode corresponding to director rotations. In the screened hydrodynamic regime, activity generates a local analytic contribution, proportional to $q^2$, to the height-sector dynamics. This term competes with the passive conserved capillary relaxation, which enters at order $q^4$, and can destabilize the interface even when the homogeneous active nematic is below its bulk instability threshold. The scalar profile-distortion mode is gapped and can be eliminated at long wavelengths. The transverse orientational mode plays a different role: if its full relaxation rate is positive, it renormalizes the effective scalar height equation; if it remains soft, the long-wavelength instability is described by a coupled height-director mode. The resulting framework separates two issues that are often intertwined in active-interface problems. The first is the transport mechanism, which may be local and analytic for a screened diffuse interface or nonanalytic when long-ranged outer flow produces terms such as $\left|q\right|$. The second is the modal structure of the unstable branch, which may be predominantly heightlike or intrinsically coupled to director rotations.

We present a framework for chemical learning based on competitive dimerization networks (CDNs)—systems in which multiple molecular species, e.g., proteins or DNA-RNA oligomers, reversibly bind to form dimers. We show numerically that these networks can, in principle, be trained in vitro through directed evolution, enabling the implementation of complex learning tasks such as multiclass classification without digital hardware or prior knowledge of all microscopic association constants. Each molecular species functions analogously to a neuron, with binding affinities acting as tunable synaptic weights. A training protocol involving mutation, selection, and amplification of DNA-based components allows CDNs to robustly discriminate among noisy input patterns. The resulting classifiers exhibit strong output contrast and high mutual information between input and output, especially when guided by a contrast-enhancing loss function. Comparative analysis with in silico gradient descent training reveals closely correlated performance. These results establish CDNs as a promising platform for analog physical computation, bridging synthetic biology and machine learning, and advancing the development of adaptive, energy-efficient molecular computing systems.

Mass disorder attenuates solitary waves traveling through Hertzian chains in a way that resembles viscous damping, even though the microscopic dynamics remains Hamiltonian and conserves energy. We show that this attenuation depends not only on the strength of the disorder, but also on its spatial organization. While the leading pulse remains identifiable, simulations reveal an approximately exponential decay, with an attenuation rate proportional to the mass variance in the weak-disorder regime. At fixed variance, spatial correlations are found to considerably reduce the attenuation rate. We describe this suppression using a spectral-overlap framework, in which attenuation depends on how the disorder spectrum couples to the response of the solitary wave. This response is measured directly using sinusoidally graded chains: long-wavelength variations interact weakly with the pulse, whereas variations at shorter spatial scales scatter it more strongly. The apparent damping therefore reflects a redistribution of energy from the coherent pulse into secondary excitations and an irregular wake, rather than true dissipation.

In this paper, we use the Feingold-Peres model, a well-known paradigm of quantum chaos, as an example. Using semiclassical analysis and numerical simulations, we investigate the statistical properties of observables in few-body systems with chaotic classical limits and the emergence of random-matrix-theory universality. More specifically, we focus on (1) the applicability of the eigenstate thermalization hypothesis in few-body systems and the dependence of its form on the effective Planck constant and (2) the existence of a universal random-matrix-theory description of observables when truncated to a small microcanonical energy window. Our results provide insights into the established field of few-body quantum chaos and help bridge it to modern perspectives, such as the general eigenstate thermalization hypothesis.

We study the thermalization properties of a fully nonlinear lattice model originally derived from the two-dimensional cubic defocusing nonlinear Schrödinger equation (NLS) using analytical and numerical methods. The model conserves both energy and norm, whose densities define a microcanonical energy-norm parameter space, while the nonlinear nearest-neighbor coupling is controlled by a parameter $D$. Within this space, our analysis identifies broad parameter regimes in which the dynamics is ergodic not only within but also outside the standard Gibbs region, indicating the need for a modified statistical description. At higher energies, the system instead exhibits long-lived compacton-mediated localization and signatures of weak nonergodicity, as evidenced by finite-time variances, excursion-time statistics, and probability distributions of local amplitudes. We show that stronger coupling $D$ enhances fluctuations and accelerates the crossover of the finite-time variance of the local norm density from an initial $\sim T^{-1/2}$ decay toward the faster $\sim T^{-1}$ decay characteristic of ergodic thermalization, where $T$ denotes the averaging time. In the high-energy regime, weak coupling $D\le 1$ favors persistent single-site compacton localization, whereas stronger coupling $D>1$ yields long-lived two-site localization. Our results provide insights into the interplay between thermalization, localization, and non-Gibbs statistical behavior in genuinely nonlinear systems.

Accurate reliability assessment of interconnected and interdependent systems requires explicit representation of interaction mechanisms arising from shared components, load redistribution, feedback coupling, and correlated degradation. Classical system reliability theory is formulated using real-valued probabilities and combinatorial aggregation and therefore lacks an intrinsic mathematical structure capable of encoding interaction strength, synchronization, and pathway competition. As a result, redundancy and vulnerability in highly coupled systems are often mischaracterized. This study introduces a quantum system reliability (QSR) framework that reformulates system reliability as a complex-valued field. In the proposed representation, the magnitude of each component amplitude corresponds to postevent functionality, while the phase encodes interaction induced by connectivity, redistribution, constraint release, and dependency. Because classical reliability contains no inherent phase information, this work rigorously defines reliability phase. It develops two independent extraction strategies: (1) a spectral-domain formulation based on discrete Fourier analysis of reliability fields, and (2) a quantum-field formulation derived from divergence, curl, and geometry-induced flux operators. Component reliabilities are encoded as complex amplitudes, and system reliability emerges through interference-aware aggregation rules that generalize classical series, parallel and inclusion-exclusion principles. The formulation preserves exact classical reliability under neutral-phase conditions while enabling constructive and destructive interference in the presence of dependencies. The framework is validated using benchmark networks, including cascading IEEE-118 data and interdependent graph models, demonstrating improved alignment with functional system behavior compared to classical connectivity-based metrics.

We study wave turbulence in systems with two special properties: a large number of fields (large $N$) and a nonlinear interaction that is strongly local in momentum space. The first property allows us to find the kinetic equation at all interaction strengths—both weak and strong, at leading order in $1/N$. The second allows us to turn the kinetic equation—an integral equation—into a differential equation. We find stationary solutions for the occupation number as a function of wave number, valid at all scales. As expected, on the weak-coupling end, the solutions asymptote to Kolmogorov-Zakharov scaling. On the strong-coupling end, they asymptote to either the widely conjectured critical balance scaling or a Kolmogorov-like scaling exponent.

A laser induced fluorescence (LIF) scheme is investigated to probe singly ionized iodine ions (${\mathrm{I}}^{+}$) in electric propulsion devices. The transition $6s{}^{5}S_2^{\mathrm{o}}\to 6p{}^{5}P_3$ at 516.261 nm is considered and the theoretical computation of the weights and positions of the hyperfine structure is carried out. First, the measurement of the fluorescence signal is performed in the ionization chamber of an rf gridded ion engine, the PEGASES thruster, demonstrating that the chosen LIF scheme is well suited for the study of electric propulsion devices operating with iodine. Second, spatial measurements of the temperature, velocity, and density are taken in this plasma source for several conditions of iodine pressure from 0.5 to 10 mTorr. First, an increase of the temperature of the ionic excited population close to the antenna is observed. Second, the analysis of the Doppler shift of the LIF profile shows the acceleration of the ions in the source. Third, the comparison of the measured LIF profiles with previous measurements shows that the density of the probed lower excited state is proportional to that of the ground state in the conditions of the experiment.

We develop a statistical theory of temporal learnability in recurrent neural networks, quantifying the maximal temporal horizon ${\mathcal{H}}_N$ over which gradient-based learning can recover lag-dependent structure at finite sample size $N$. The theory is built on the effective learning rate envelope $f\left(\ell \right)$, a function that captures how gating mechanisms and adaptive optimizers jointly shape the coupling between state-space dynamics and parameter updates during Backpropagation Through Time. Under heavy-tailed ($\alpha$-stable) fluctuations, where empirical averages concentrate at rate $N^{-1/{\kappa }_{\alpha }}$ with ${\kappa }_{\alpha }=\alpha /(\alpha -1)$, the interplay between envelope decay and statistical concentration yields explicit scaling laws for the growth of ${\mathcal{H}}_N$: logarithmic, polynomial, and exponential temporal learning regimes emerge according to the decay law of $f\left(\ell \right)$. These results identify envelope decay as the key determinant of temporal learnability. Slower attenuation of $f\left(\ell \right)$ enlarges ${\mathcal{H}}_N$, while heavy-tailed fluctuations compress it by weakening statistical concentration. Moreover, envelope geometry outweighs dataset size: slowing the envelope's decay enlarges ${\mathcal{H}}_N$ more than adding data, so more complex architectures that realize slower-decaying envelopes can be more data-efficient than simpler ones. Experiments across multiple gated architectures and optimizers corroborate these structural predictions.

Using a minimal aggregation-based model, we address the efficient information transfer observed in natural flocks during collective turns. In our system, the turning signal emitted by a single individual propagates throughout the collective primarily at a constant speed. To characterize the source of this capability, we analyze the system in the continuum limit. Thus, we demonstrate that wave-propagating features arise solely from the nonreciprocal nature of interactions. We explore spectral anomalies that emerge due to this mechanism and provide an explanation through a perturbative analysis. Remarkably, within this scenario, the description of velocity fluctuations is found to follow a structure analogous to a Born approximation. Our model can be extended to study other physical contexts exhibiting polar order, and our arguments can be tested in natural systems.

The origin of segregated axial bands for size-bidisperse particles in long rotating tumblers is a long-standing question in granular flows and a problem of both fundamental and practical interest. Using DEM simulations, we consider the streamwise and time-dependent development of the underlying instability as well as performing a broad parametric study over a wide range of particle size ratios, tumbler lengths and diameters, species fractions, and fill levels to determine the conditions under which axially segregated bands appear. We show that initial radial segregation of the small and large particles in the surface flowing layer results in a layer of mixed particles with a higher bulk density due to more efficient packing over a layer of nearly monodisperse small particles with a lower bulk density. This leads to a granular Rayleigh-Taylor-like instability that produces recirculation cells driving axial band formation. The appearance of a layer of high-density mixed particles over a layer of lower-density monodisperse particles can be directly connected to the operating conditions, explaining why axially segregated bands appear only under certain conditions. Typically, increasing the particle size ratio improves packing and increases the density of the mixed layer, enhancing the Rayleigh-Taylor instability and accelerating axial segregation.