Tipping points (TP) are abrupt transitions between metastable states in complex systems, most often described by a bifurcation or crisis of a multistable system induced by a slowly changing control parameter. An avenue for predicting TPs in real-world systems is critical slowing down (CSD), which is a decrease in the relaxation rate after perturbations prior to a TP that can be measured by statistical early warning signals (EWS) in the autocovariance of observational time series. In high-dimensional systems, we cannot expect a priori chosen scalar observables to show significant EWS, and some may even show an opposite signal. Thus, to avoid false negative or positive early warnings, it is desirable to monitor fluctuations only in observables that are designed to capture CSD. Here we propose that a natural observable for this purpose can be obtained by a data-driven approximation of the first nontrivial eigenfunction of the backward Fokker-Planck (or Kolmogorov) operator, using the diffusion map algorithm.

Multilayer networks offer a powerful framework for modeling complex systems across diverse domains, effectively capturing multiple types of connections and interdependent subsystems commonly found in real-world scenarios. To analyze these networks, embedding techniques that project nodes into a lower-dimensional geometric space are essential. This paper introduces a novel hyperbolic embedding framework that advances the state of the art in multilayer network analysis. Our method, which supports heterogeneous node sets across networks and interlayer connections, generates layer-specific hyperbolic embeddings, enabling detailed intralayer analysis and interlayer comparisons, while simultaneously preserving the global multilayer structure within hyperbolic space—a capability that sets it apart from existing approaches, which typically rely on independent embedding of layers. Through experiments on synthetic multilayer stochastic block models, we demonstrate that our approach effectively preserves community structure, even when layers consist of different node sets. When applied to real brain networks, the method successfully clusters disease-related brain regions from different patients, outperforming layer-independent approaches and highlighting its relevance for comparative analysis. Overall, this work provides a robust tool for multilayer network analysis, enhancing interpretability and offering new insights into the structure and function of complex systems.

We investigate the dynamics of a two-dimensional Sznajd model in which a third ideological position emerges spontaneously from local interactions between agents. Starting from random initial conditions, we observe that this emergent position arises solely as a result of the interaction dynamics, and not from predefined agent states. We analyze the system's behavior as a function of social apathy, modeled as the fraction of nonparticipating agents, and identify a critical apathy threshold that separates distinct regimes of opinion dominance. To quantify the interplay between disorder and structure in the system, we employ the complexity-entropy plane. This analysis reveals that the system is robust to asymmetries in initial conditions; its macroscopic behavior remains qualitatively unchanged. Furthermore, we find that statistical complexity reaches a maximum near the transition region, highlighting the emergence of rich collective dynamics characterized by abrupt topological changes parametrized by apathy.

The “criticality hypothesis,” based on observed scale-free correlations in neural activity, posits that the brain operates at a critical point of transition between two phases. However, the validity of this hypothesis is still debated. Here, employing a commonly used model of cortical dynamics, we find that a phase of long-range order (LRO) in neural activity may be induced by memory (time nonlocality) without invoking criticality. The cortical-dynamics model contains fast and slow time scales that govern the neural and resource (memory) dynamics, respectively. When the resource dynamics are sufficiently slow, we observe an LRO phase, which manifests in avalanche size and duration probability distributions that are fit well by power laws. When the slow and fast time scales are no longer sufficiently distinct, LRO is destroyed. Since this LRO phase spans a wide range of parameters, it is robust against perturbations, unlike critical systems.

The active transfer of phospholipids between membrane leaflets (flip-flop), mediated by (adenosine triphosphate) ATP-dependent enzymes such as flippases and floppases, is a key regulator of membrane asymmetry and curvature. However, the theoretical understanding of curvature generation driven by flip-flop under external perturbations remains incomplete. Here we present a mesoscopic thermodynamic model in which lipid asymmetry couples to membrane curvature via a Landau-type free energy, with curvature as the order parameter and transmembrane voltage as the control parameter. This framework predicts an electrically driven phase transition analogous to a ferroelectric transition. The model reproduces field-induced bistability, critical susceptibility divergence, and hysteresis, with numerical simulations revealing thickness-dependent curvature thresholds and robust curvature-memory effects. These results clarify how electric-field-driven lipid redistribution governs membrane shape and suggest strategies for voltage-controlled nanoscale memory and shape encoding.

Perfluorohexane is a biocompatible material that serves as a liquid core for acoustically responsive agents in biomedical applications. Despite its relatively widespread usage, there is a lack of experimental data determining its thermodynamic properties. This challenges numerical simulations to predict the acoustic response of agents developed using this material. In this study, we employ the well-established method of shock compression of materials at relatively high pressures (100–400 MPa) to estimate a kinematic equation of state for perfluorohexane. We use multi-objective optimization to obtain the Noble-Abel stiffened-gas equation of state, which is suitable for hydrodynamic numerical simulations. We then apply the extrapolated equation of state to simulate shock-wave propagation within a perfluorohexane droplet showing excellent agreement with equivalent experiments. This promotes the use of numerical simulations as a valuable tool for understanding the complex acoustic interactions involved in these biomedical agents, ultimately facilitating their translation for clinical purposes.

In standard rotating Hele-Shaw cell flows, an initially circular fluid drop, surrounded by an outer fluid of negligible density and viscosity, is centered at the rotation axis of the cell. The interplay of centrifugal and surface tension forces leads to the emergence of intricate interfacial patterns, markedly characterized by intense competition among the inward-moving fingers of the outer fluid as they penetrate the inner one. In this work, we study a variation of this traditional rotating flow problem, considering that the center of the initially circular drop is at a distance $d$ from the cell's rotation axis. We explore this off-centered situation by employing numerical simulations based on the level set method. Our numerical results show that, at fully nonlinear stages of the flow, the off-center parameter $d$ plays a key role in determining the morphology and dynamic competition among fingers, leading to the development of asymmetric interfacial patterns that drift away from the rotation axis of the cell. The impact of the effective surface tension parameter $B$ (measure of the relative strength of centrifugal and capillary effects) on the main features of these complex, centrifugally driven translating patterns is also discussed.

Fully kinetic simulations are used in this work to study strong shock waves in fully ionized plasmas for charge state $Z$ ranging from 1 to 18. The ion stress tensor amplitude agrees well with classical predictions within the compression layer; however, its spatial structure is noticeably shaped by nonlocal transport effects. The electron stress tensor remains significantly smaller than that of the ions for all values of $Z$ considered. Ion overheating in the compression layer is observed to scale approximately linearly with $Z$. Both findings diverge from hydrodynamic models based on classical electron stress formulations. The electrons exhibit strong nonlocal behavior in the compression layer, with Knudsen numbers approaching unity for $Z\ge 6$.

Residual networks have significantly better trainability and thus performance than feed-forward networks at large depth. Introducing skip connections facilitates signal propagation to deeper layers. In addition, previous works found that adding a scaling parameter for the residual branch further improves generalization performance. While they empirically identified a particularly beneficial range of values for this scaling parameter, the mechanism for the resulting performance improvement and its universality across network hyperparameters remain an open question. For feed-forward networks, finite-size theories have led to important insights with regard to signal propagation and hyperparameter tuning. We here derive a systematic finite-size field theory for residual networks to study signal propagation and its dependence on the scaling for the residual branch. We derive analytical expressions for the response function, a measure for the network's sensitivity to inputs, and show that for deep networks the empirically found values for the scaling parameter lie within the range of maximal sensitivity. Furthermore, we obtain an analytical expression for the optimal scaling parameter that depends only weakly on other network hyperparameters, such as the weight variance, thereby explaining its universality across hyperparameters. Overall, this work provides a theoretical framework to study ResNets at finite size.

An echo state network (ESN) is a type of reservoir computer that uses a recurrent neural network with a sparsely connected hidden layer. Compared with other recurrent neural networks, one great advantage of ESN is the simplicity of its training process. Yet, despite the seemingly restricted learnable parameters, ESN has been shown to successfully capture the spatial-temporal dynamics of complex patterns. Here we build an ESN to model the coarsening dynamics of charge-density waves (CDWs) in a semiclassical Holstein model, which exhibits a checkerboard electron density modulation at half-filling stabilized by a commensurate lattice distortion. The inputs to the ESN are local CDW order parameters in a finite neighborhood centered around a given site, while the output is the predicted CDW order of the center site at the next time step. Special care is taken in the design of couplings between hidden layer and input nodes to ensure lattice symmetries are properly incorporated into the ESN model. Since the model predictions depend only on CDW configurations of a finite domain, the ESN is scalable and transferrable in the sense that a model trained on dataset from a small system can be directly applied to dynamical simulations on larger lattices. Our work opens avenues for efficient dynamical modeling of pattern formations in functional electron materials.

The phase-field lattice Boltzmann (LB) method is a powerful tool for simulating two-phase flows. However, for gas-liquid-solid systems with large density ratios, the accurate and systematic treatment of three-dimensional curved solid surfaces remains a challenge. In this work, we propose a wetting boundary scheme within the phase-field LB framework to address this problem. First, a phase-field LB model is developed for simulating multiphase flows with the density ratio up to 1000. Then, based on the free-energy approach, a method is introduced to determine the location of solid boundaries and their normal vectors. Finally, depending on the spatial position of the solid surface, the wetting condition is imposed using different interpolation schemes. The proposed method is validated through three benchmark cases: a droplet resting on a spherical surface, capillary rise in cylindrical tubes, and self-propelled droplet on conical fibers. Simulation results demonstrate that the method achieves high accuracy in both static and dynamic processes and is capable of handling arbitrarily complex curved surfaces in three-dimensional space. Moreover, to the best of our knowledge, two new power-law behaviors are observed in the third case, indicating the potential of the model for discovering novel wetting dynamics.

Over the past seven decades, the classical Monte Carlo method has played a huge role in the fields of rarefied gas flow and micro/nanoscale heat transfer, but it also has shortcomings: the time step and cell size are limited by the relaxation time and mean free path, making it difficult to efficiently simulate multiscale heat and mass transfer problems from the ballistic to diffusion limit. To overcome this drawback, a unified gas-kinetic wave-particle (UGKWP) method is developed for solving the phonon Boltzmann transport equation (BTE) in all regimes covering both ballistic and diffusive limits. This method is built upon the space-time coupled evolution model of the phonon BTE, which provides the framework for constructing a multiscale flux at the cell interfaces. At the same time, in order to capture nonequilibrium transport efficiently, the multiscale flux comprises two distinct components: a deterministic part for capturing the near-equilibrium or diffusive transport, and a statistical particle part for recovering nonequilibrium or ballistic transport phenomena. The UGKWP method exhibits remarkable multiscale adaptability and versatility, seamlessly bridging the gap between the diffusive and ballistic transport phenomena. In the diffusive limit, the present method naturally converges to Fourier's law, with the diminishing particle contribution, whereas in the ballistic limit, the nonequilibrium flux is fully described by the free-streaming particles. This inherent adaptability not only allows for precise capturing of both equilibrium and nonequilibrium heat transfer processes, but it also guarantees that the model adheres strictly to the underlying physical laws in each phonon transport regime. A series of numerical tests fully demonstrate the excellent performance of the UGKWP method in all Knudsen regimes, where the time step and cell size are not constrained by the relaxation time and mean free path in the diffusive regime. The present method is an efficient and accurate computational tool for simulating multiscale nonequilibrium heat transfer, and offering significant advantages over traditional methods in terms of numerical performance and physical applicability.

We study the impact of an external alignment field on the structure formation and polarization behavior of low-density dipolar active particles in three dimensions. Performing extensive Brownian dynamics simulations, we characterize the interplay between long-range dipolar interactions, field alignment, and self-propulsion. We find that the competition between activity (favoring bond breaking) and the field's orientational constraint (promoting bond formation) gives rise to a rich variety of self-assembled, actuated structures. At low to intermediate field strengths, disordered fluids composed of active chains and active percolated networks can emerge, whereas strong fields drive the formation of polarized columnar clusters. Counterintuitively, low activity levels significantly extend the range of field strengths over which percolated networks persist. This structural evolution manifests in the polarization response of strongly dipolar systems, which exhibit a transition from super-Langevin to sub-Langevin behavior with increasing activity, as a result of the coupling between structure formation and activity-induced bond breaking.

We investigate the effect of polydispersity on fluctuation-induced interactions between colloids trapped at a fluid interface. Using the scattering-matrix formalism, we calculate the interaction energy in two- and three-body systems under three mechanical boundary conditions. We find that size asymmetry can either suppress or enhance the many-body interaction compared to a monodisperse system, with the outcome depending on colloids mobility and separation. For fixed colloids, the interaction is suppressed at short separations but enhanced at large separations. In contrast, for mobile colloids, the interaction is predominantly enhanced at large distances but exhibits a competitive behavior in the near-field. This culminates in a large, multi-order-of-magnitude amplified sensitivity to size asymmetry for bobbing and tilting colloids at long range, highlighting a complex interplay between geometry and colloids mobility with significant implications for self-assembly.

We present a graph theoretical approach to the configurational statistics of random treelike objects, such as randomly branching polymers. In particular, for ideal trees we show that Prüfer labeling provides (i) direct access to the exact configurational entropy as a function of the tree composition, (ii) computable exact expressions for partition functions and important experimental observables for tree ensembles with controlled branching activity, and (iii) an efficient sampling scheme for corresponding tree configurations and arbitrary static properties.

Experiments using a rotational rheometer have demonstrated that the apparent viscosity becomes negative under the electric-field-induced turbulent state of conductive nematic liquid crystals [Orihara et al., Phys. Rev. E 99, 012701 (2019); F. Kobayashi et al., Phys. Rev. E 101, 022702 (2020)]. When the upper rotating plate of the rheometer is left free, spontaneous rotation—that is, spontaneous shear flow—has also been observed. In this study, we reproduce these phenomena through three-dimensional simulations based on continuum theory. The simulations reveal characteristic velocity, director, and stress fields in the negative-viscosity state. Furthermore, they clarify the interplay between topological defects (disclinations) and space charges which drive the turbulence.

Quantum walks on networks are a paradigmatic model in quantum information theory. Quantum-walk algorithms have been developed for various applications, including spatial-search problems, element-distinctness problems, and node-centrality analysis. Unlike their classical counterparts, the evolution of quantum walks is unitary, so they do not converge to a stationary distribution. However, for many applications, it is important to understand the long-time behavior of quantum walks and the impact of network structure on their evolution. In the present paper, we study localization of continuous-time quantum walks on networks. We derive an analytical expression for the long-time inverse participation ratio that depends on combinations of eigenvectors of the quantum-walk Hamiltonian. We then demonstrate how localization emerges in highly clustered networks that we construct by recursively attaching triangles. Building on the insights from this example, we then show that localization also occurs in Kleinberg navigable small-world networks and Holme–Kim power-law cluster networks. Our results illustrate that local clustering, which is a key structural feature of networks, can induce localization of quantum walks.

In this Letter, we derive a full scaling law for describing the space charge oscillation frequency in electron emission diodes based on the Buckingham $\pi$-theorem and the similarity method. The scale-invariant nature of the oscillation phenomena is unraveled and demonstrated via first-principles particle-in-cell simulations. By combining the concepts of similarity and scaling laws, two types of scaling networks are established to correlate space charge oscillations at different states, which are essential for designing high-current electron-emitting systems in various fields, particularly for miniaturized electronic devices.

Stressed under a constant load, materials creep with a final acceleration of deformation and for any given applied stress and material, the creep failure time can strongly vary. We investigate creep on sheets of paper and confront the statistics with a simple fiber bundle model of creep failure in a disordered landscape. In the experiments, acoustic emission event times $t_j$ were recorded, and both this data and simulation event series reveal sample-dependent history effects with log-normal statistics and non-Markovian behavior. This leads to a relationship between $t_j$ and the failure time $t_f$ with a power law relationship, evolving with time. These effects and the predictability result from how the energy gap distribution develops during creep.

In purely coherent transport on finite networks, destructive interference can significantly suppress transfer probabilities, which can only reach high values through careful fine-tuning of the evolution time or tailored initial-state preparations. We address this issue by investigating excitation transfer on a ring, modeling it as a locally monitored continuous-time chiral quantum walk. Chirality, introduced through time-reversal symmetry breaking, imparts a directional bias to the coherent dynamics and can lift dark states. Local monitoring, implemented via stroboscopic projective measurements at the target site, provides a practical detection protocol without requiring fine-tuning of the evolution time. By analyzing the interplay between chirality and measurement frequency, we identify optimal conditions for maximizing the asymptotic detection probability. The optimization of this transfer protocol relies on the spectral properties of the Perron-Frobenius operator, which capture the asymptotic nonunitary dynamics, and on the analysis of dark states. Our approach offers a general framework for enhancing quantum transport in monitored systems.