We derive the asymptotic first-passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\alpha \left(x\right)$ varies with position. On a bounded interval with an absorbing and a reflecting boundary, we show that the survival probability decays as $\mathrm{\Psi }\left(t\right)\sim C{t}^{-{\alpha }_{*}}/{[ln(t/\tau )]}^{\nu }$, where ${\alpha }_{*}$ is the minimum value of the fractional exponent and $\nu$ is determined by the location and shape of the minimum. The exponent $\nu$ is fixed by the location and order of the minimum, taking the value $1/k$ at a $k\mathrm{th}$-order interior minimum, 1 at the reflecting boundary and 2 at the absorbing boundary. For a constant fractional exponent, $\nu =0$, so the logarithmic factor distinguishes a continuously varying order field from a homogeneous medium. We recover both exponents from simulated first-passage times by a direct linear fit, with no amplitude or geometric input. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $\alpha \left(x\right)$.

The protein ion channels inserted in cell membranes can act as bioelectrical diodes that control ionic transport. The nonlinear dynamics inherent to voltage-gated channels is useful to process temporal patterns and we explore the response of these channels to low-frequency voltage oscillations, which are observed in most organisms as internal state integrators. To this end, we employ experimentally motivated current-voltage curves in parallel networks of voltage-gated ion channels performing the time-series analysis of a low-frequency oscillatory rhythm. The simulations suggest how channel diversity can contribute to the system-wide internal control, which should be of interest to biophysics and neuromorphic computing.

Beyond temporary trapping, particles may become permanently fixed due to irreversible trapping in biomedical systems and chemical catalysis systems, which is named permanent immobilization. To describe anomalous diffusion processes in which particles become permanently immobilized with a certain probability, this paper proposes an immobilization model based on a non-normalized jump-length distribution under the continuous-time random walk framework. By classifying particles into active and immobilized states, we derive a generalized Montroll-Weiss equation incorporating the immobilization mechanism and investigate the diffusion behaviors for Gaussian and power-law jump-length distributions. In the long-time limit, for Gaussian jump lengths, the system evolves toward a stationary state characterized by a Laplace distribution, whereas for power-law jump lengths, the stationary distribution retains a Lévy-type power-law tail. Furthermore, we derive the Fokker-Planck equations governing the particle's displacement distributions. These equations show that the immobilization mechanism introduces only an additional term, $P(x,t)-\delta \left(x\right)$, weighted by the immobilization probability, without changing the fractional derivative operators. We also conduct a detailed comparison on the common quantities between the immobilization mechanism and stochastic resetting due to the similarities between them. This paper provides a theoretical perspective for studying transport processes under irreversible immobilization in complex media.

Brownian particles in complex media often exhibit diffusion-caging-diffusion dynamics: short-time diffusion, an intermediate plateau due to diffusion hindrance, and long-time diffusion, usually attributed to cage-release events. We analyze an alternative mechanism that reproduces the same phenomenology: a perfectly confining cage undergoing stochastic motion that transports the particle without allowing escape. Despite their distinct physical origins, both mechanisms yield identical MSD behavior with the canonical three-regime structure. In contrast, the non-Gaussian parameter (NGP) provides a clear discriminator between them. For the two models presented, we provide analytical results for the NGP that conceptually support the main findings obtained from the simulation. These results offer diagnostic tools for microrheology and refine the interpretation of dynamical signatures near the glass transition.

We study a three-terminal thermoelectric engine, focusing on the role of quantum coherence and information flow. A double dot connects two reservoirs at different chemical potentials, while a third dot monitors their occupation via Coulomb interaction and can be interpreted as an autonomous Maxwell demon. Within the parameter range where the device operates as an engine, we identify conditions under which this interpretation holds, namely, when a finite information flow toward the monitoring dot is accompanied by negligible energy flow. The system dynamics is described within a Redfield master equation that allows us to identify two distinct dynamical regimes with steady states well captured by suitable Lindblad approximations. For large interdot tunneling, energy-basis coherences vanish, and the steady state reduces to that of a classical stochastic master equation associated with a fully secular approximation. In contrast, for small tunneling, coherence persists in both energy and local eigenbases, and a Lindblad description derived from a partial secular approximation to the Redfield equation must be employed. These two regimes define a first quantum-to-classical transition controlled by the interdot tunneling strength. We further consider the effect of a phonon bath coupled to the double dot, which induces a second quantum-to-classical transition by generating incoherent transport and decoherence in the small interdot tunneling regime. We find a competition between these effects: incoherent transport enhances both particle and information flows, whereas decoherence suppresses them, and this competition governs the effectiveness of the demon. In particular, we identify a parameter region where phonon-induced decoherence suppresses both the coherent transport contribution and the information flow toward the monitoring dot, suggesting that coherence can enhance the demon mechanism in this regime. By tracking information and transport properties across these crossovers, our model shows how coherent tunneling, decoherence, and incoherent phonon-assisted transport compete in an autonomous information engine while clarifying which thermodynamic Lindblad description is appropriate in each regime.

We study the transition from order to chaos in an effective Hamiltonian three-body system governed by a static, position-dependent effective beyond-Newtonian potential for rotating compact primaries derived from the Fodor–Hoenselaers–Perjés formalism. The model constitutes a perturbed extension of the classical Circular Restricted Three-Body Problem that retains an exact Jacobi integral, incorporating effective beyond-Newtonian multipole corrections while preserving its conservative character. In this formulation, the potential is purely spatial; velocity-dependent gravitomagnetic spin-orbit couplings are not included. This static formulation, having an exact Jacobi integral, thereby enables a systematic large-scale survey of phase space, allowing geometric and orbit-based diagnostics to be applied uniformly to more than ${10}^5$ initial conditions. Equilibrium points are determined using a Newton–Raphson scheme, and the associated basins of convergence are analyzed via the basin entropy ($S_b$) and boundary basin entropy ($S_{bb}$), which quantify global unpredictability and interfacial mixing in phase space. The Smaller Alignment Index (SALI) is employed to probe local dynamical stability and chaotic diffusion, while convergence statistics reveal the role of long-lived transients near fractal basin boundaries. The system displays a distinctly nonmonotonic evolution of dynamical complexity. Both entropy measures and SALI exhibit correlated maxima within an intermediate range of the control parameter $\epsilon$, associated with transitions from regular to strongly chaotic motion driven by changes in the equilibrium structure. The coincidence between the maxima of basin entropy and the peak fraction of chaotic orbits detected by SALI indicates that geometric basin complexity and dynamical instability arise from the same restructuring of the equilibrium configuration of the effective beyond-Newtonian potential. At higher values of the control parameter, the annihilation of equilibrium points alters the potential topology, leading to smoother basin structures and a partial restoration of order. In contrast to earlier studies that focused mainly on equilibrium locations and linear stability, these results establish a quantitative and unified link between potential geometry, equilibrium multiplicity, basin topology, entropy growth, and orbit-based dynamical instability within a well-defined effective Hamiltonian system. They demonstrate how controlled multipole corrections reorganize phase-space structure and govern the onset, amplification, and suppression of chaos in this class of perturbed conservative systems.

Age-structured population equations avoid simulating every neuron, but convex spike hazards make first-moment closures systematically biased. We develop a predictive joint-Gaussian closure for adaptive stochastic escape-rate neurons. It evolves age-conditioned voltage and adaptation means, variances, and covariance, while Gaussian exponential tilting supplies spike-selected moment loss and conservative reinjection at age zero. Against matched microscopic Monte Carlo (MC) ground truth, mean-only, prescribed-voltage-variance, and dynamic-joint closures have baseline $50\text{−}\mathrm{ms}$-smoothed root-mean-square errors (RMSEs) of 1.017, 0.274, and 0.090 Hz. The dynamic closure predicts the empirical conditional hazard with log-RMSE 0.002 03 without using MC moments as inputs. Its RMSE remains 0.091–0.108 Hz across nine voltage-gain/noise combinations and 0.095–0.103 Hz over a fourfold adaptation-strength sweep. Its total finite-size error scales as $N^{-0.492}$ with a fitted bias floor statistically consistent with zero, and closed-loop normalized RMSE remains below one for normalized feedback $g\le 0.321$. For a 2-s benchmark at $1\mathrm{ms}$ age resolution, the joint partial differential equation  reaches 0.0131 Hz error in $6.16\mathrm{s}$, whereas $N=5\times {10}^4$ MC reaches 0.0334 Hz in $25.9\mathrm{s}$. The less expensive prescribed correction remains useful at moderate accuracy. The joint closure does not remove step-response latency error and loses finite-moment stability at stronger feedback, providing explicit boundaries on its validity.

We study the nonlinear wave dynamics of one-dimensional chains of polycatenated rings. These interlocked structures support amplitude-dependent nonlinear wave propagation driven by tensile activation and internal structural flexibility, unlike traditional granular crystals. Through dynamic impact experiments, finite-element modeling, and discrete-particle simulations of vertical chains, we observe and model a distinctive nonlinear wave morphology. It results from energy partitioning between the chain's two deformation channels: nonlinear indentation of the inter-ring contacts, which carries a compact leading wave front, and bending of the individual rings, which trails it as a persistent oscillatory train. Further, we demonstrate that the system's nonlinearity is not a fixed material constant. By altering the rings' geometric aspect ratio and contact angles, we can tune the effective contact exponent and the amplitude scaling of the wave speed. This work builds upon nonlinear wave propagation in classical granular crystals and establishes polycatenated systems as a highly tunable and designable platform to study and control nonlinear dynamics.

From the firing of neurons to the formation of social consensus, the interplay between conformist and contrarian behaviors regulates collective dynamics. We unravel how adaptation timescales induce distinct synchronization behaviors in mixed populations of conformists and contrarians. While slow adaptation leaves contrarians in a persistent state of incoherence, fast adaptation traps them into a scaffold formed by conformists, stabilized by rapid feedback from the evolving network structure, leading to the emergence of synchronized clusters. Notably, contrarians are entrained only within an intermediate coupling window and require a conformist majority. We analytically reduce a low-dimensional description using a collective coordinate approach and demonstrate that a conformist majority and a threshold adaptation rate are necessary conditions for contrarians to synchronize in adaptive networks, which agrees well with numerical simulations. Our results highlight the critical role of adaptation timescales in reshaping collective behaviors in mixed populations, offering insights into consensus formation and polarization in adaptive social networks.

In network science, collective dynamics of complex systems are typically modeled as (nonlinear, often including many-body) vertex-level update rules evolving over a graph interaction structure. In recent years, frameworks that explicitly model such higher-order interactions in the interaction backbone (e.g., hypergraphs) have been advanced, somehow shifting the imputation of the effective nonlinearity from the dynamics to the interaction structure. In this work, we discuss such structural-dynamical representation duality, and we investigate how and when a nonlinear dynamics defined on the vertex set of a graph allows an equivalent representation in terms of a linear dynamics defined on the state space of a sufficiently richer, higher-order interaction structure. Using Carleman linearization arguments, we show that finite polynomial dynamics defined in the $\left|V\right|$ vertices of a graph admit an exact representation as linear dynamics on the state space of an hb-graph of order $\left|V\right|$, a combinatorial structure that extends hypergraphs by allowing vertex multiplicity, where the specific shape of the nonlinearity indicates whether the hb-graph is either finite or infinite (in terms of the number of hb-edges). For more general analytic nonlinearities, exact linear representation always require an hb-graph of infinite size, and its finite-size truncation provides an approximate representation of the original nonlinear graph-based dynamics.

The shortest path multiplicity, here denoted by $\mu$, is an important metric of complex networks. For real networks $\mu$ is high and it correlates with the network community structure. Since local network evolution induces network communities, it is possible that a high shortest path multiplicity is the natural expectation of local evolution rules. Here I demonstrate, by means of numerical simulations and exact results, that this is indeed the case. For random graphs with arbitrary degree distributions $p_k$, $\langle \mu \rangle \sim \langle k(k-1)\rangle /(\langle k\rangle e)$, growing with the network size when $p_k\sim k^{-\gamma }$ and $\gamma \le 3$. For networks generated by stochastic local rules, $\langle \mu \rangle$ increases with the network size and it does so faster than what is observed in their randomized versions. Furthermore, the number of communities increases with the network size and the correlation with $\langle \mu \rangle$ follows.

In populations with community structure, the formation of consensus requires both alignment within and diffusion of beliefs across groups, processes that evolve on distinct timescales. How do modularity, asymmetry, and polarization shape this process? We study a variant of the voter model in which a population is divided into two cliques. At each time step, a pair of nodes is selected; if their binary opinions differ, then each agent adopts the opinion of the other with a certain probability. The type of pair selected depends on a network structure parameter—modularity that is central in this paper. We analyze how this coupling strength, population imbalance, and initial polarization jointly determine the time to consensus. Formation of consensus generally starts with interclique interactions rapidly synchronizing the two cliques' opinion fractions, after which consensus is reached through a slower diffusion along the synchronized curve. This slow stage is largely insensitive to modularity except when the cliques are nearly disconnected. To analyze these dynamics, we derive time-continuous approximations in the large-population limit and assess their accuracy against the discrete model. While modularity primarily affects the synchronization stage, initially polarized and asymmetric populations exhibit nontrivial effects, including regimes in which an intermediate level coupling minimizes consensus time. A small-clique scaling analysis reveals that this optimum arises from a competition between fast alignment drift and noise amplification in the smaller group.

When searching for communities in networks, domain experts may have some prior expectations about the size of communities. Yet, community detection methods normally do not optimize communities under cluster size constraints. Multiresolution techniques allow users to indirectly control the average community size through changing a resolution parameter, but this practice does not control the size of individual communities. We here study the problem of size-constrained community detection, where the size of all communities is limited to a user-specified range of values, in the context of modularity optimization. We propose a heuristic for modularity optimization under community-size constraints. To demonstrate the reliability of our proposed heuristic, we also formulate an exact integer optimization model and use its results as a baseline. Our analysis based on synthetic benchmarks and real networks demonstrate the issues with the currently common practice of changing resolution parameters and reveal the advantages of the proposed methods as a principled way of obtaining size-constrained communities.

Cell proliferation and cell movement are fundamentally stochastic processes which lead to variability in the growth and spatial structure of cell populations in many biological settings, such as cell invasion, wound healing, and tumor growth. Understanding how variability in cell-cycle progression and local spatial constraints interact to impact population-level behavior requires modeling frameworks that explicitly represent individual-cell behavior and elucidate how these affect emergent tissue-level behavior. We develop stochastic, on-lattice agent-based models (ABMs) which incorporate volume exclusion, random movement, and multistage representations of the cell cycle. The multistage framework enables a more realistic representation of true cell-cycle time distributions. We also introduce a form of myopic behavior, in which cells sense their local environment when attempting to proliferate. For each ABM, we derive a corresponding continuum partial differential equation (PDE) description under the mean-field approximation. Using numerical simulations, we investigate how different proliferation mechanisms influence population-level dynamics in both the discrete and continuum models. In particular, we consider biologically relevant contexts of growth-to-confluence assays (using uniform initial conditions) and traveling wave behavior associated with cell invasion. We examine how the PDE solutions compare with the behavior of the corresponding ABMs averaged over many realizations.

Pancreatic $\beta$-cells play a central role in maintaining glucose homeostasis through the pulsatile secretion of insulin. This essential function relies not only on intracellular regulatory mechanisms but also on coordinated interactions among $\beta$-cells within the islets of Langerhans. Disruptions in this intercellular coordination are increasingly implicated in metabolic disorders such as type II diabetes. In this work, we employ a computational framework to investigate the collective dynamics of a network of coupled $\beta$-cells interacting through a nonlocally coupled ring topology that incorporates both electrical and metabolic coupling pathways. This topology provides an idealized representation of short- and longer-range interactions and allows their dynamical consequences to be studied systematically. Numerical simulations reveal a variety of emergent behaviors, including synchronization, traveling waves, and traveling chimera states, in which coherent and incoherent domains coexist and propagate across the network. These findings provide new insights into the mechanisms governing coordinated $\beta$-cell activity and the regulation of pulsatile insulin secretion. By clarifying how coupling structure and intercellular communication shape islet-wide dynamics, this work contributes to a deeper understanding of the dysfunctions underlying diabetes.

A single dust particle confined in an asymmetric ratchet potential is periodically driven by two oppositely directed laser beams, forming an underdamped dusty plasma rocking ratchet. We experimentally investigate the transport dynamics of the particle under varying driving amplitudes and frequencies. Depending on the driving conditions, the particle exhibits positive, zero, or negative net currents, and current reversal is observed when the driving parameters cross critical thresholds. To interpret these transport behaviors, we develop a simplified model based on the competition between the driving force and the ratchet confinement. The model reveals that directional transport is governed by two requirements: the driving force must exceed the depinning threshold, and the duration of a driving semicycle must be longer than the uphill escape time from a ratchet well. The resulting dynamic phase diagram quantitatively reproduces the experimentally observed transport regimes and current reversals. These results demonstrate dusty plasma as a versatile platform for investigating nonequilibrium transport phenomena of underdamped particles in rocking ratchets.

The Koopman-von Neumann equation describes the evolution of wave functions associated with autonomous ordinary differential equations and can be regarded as a quantum physics-inspired formulation of classical mechanics. The main advantage compared to conventional transfer operators such as Koopman and Perron-Frobenius operators is that the Koopman-von Neumann operator is unitary even if the dynamics are non-Hamiltonian. Projecting this operator onto a finite-dimensional subspace allows us to represent it by a unitary matrix, which in turn can be expressed as a quantum circuit. We will exploit relationships between the Koopman-von Neumann framework and classical transfer operators to derive numerical methods to approximate the Koopman-von Neumann operator and its eigenvalues and eigenfunctions from data. Furthermore, we will show that the choice of basis functions and domain are crucial to ensure that the operator is well-defined. We will illustrate the results with the aid of guiding examples, including simple undamped and damped oscillators and the Lotka-Volterra model.

As Moore's law reaches its limits, Ising machines offer a promising alternative computing approach for difficult optimization problems. However, many analog, time-continuous Ising machines rely on gradient-descent-like dynamics to find solutions, which can limit speed and robustness. We investigate whether momentum and Adam optimization can improve these systems. Since these optimizers are traditionally formulated in discrete time, we derive continuous-time versions suitable for analog, time-continuous Ising-machine dynamics. On Max-Cut benchmarks, we find that Adam-based dynamics substantially reduce time to target and improve solution quality compared with gradient-descent- and momentum-based dynamics. We further introduce a first-order continuous-time approximation of Adam that is intended as a simpler starting point for future physical implementations and while performing better than the full Adam formulation in a continuous-time setting. We also study a purely algorithmic discrete-time setting, where the performance gap is reduced on easier problem instances, while the Adam-based update rule performs best on harder weighted problem instances. These results identify continuous-time Adam dynamics as a powerful design principle for analog Ising machines.

Quenched disorder in the environment can fundamentally alter transport dynamics in both active and passive systems. We explore how disordered arrays of filaments govern the distribution of intermittently moving particles which switch between diffusive and processive transport. Motivated by the mixed-polarity arrangements of parallel microtubules observed in mammalian dendrites, we show that such arrays tend to result in localization of particles at regions of convergent filament orientation. In the rapid attachment-detachment limit, the disordered system can be described by a noisy one-dimensional effective energy landscape, whose structure is approximated by a random walk. The depth and width of wells on this landscape are expressed as a function of the transport kinetics and system geometry. Localization is shown to be strongest at intermediate run lengths, where biased transport persists long enough to sense the quenched filament polarity but not so long as to facilitate escape from local traps. These results demonstrate robust localization of particles moving on random filament networks, highlighting the emergent spatial organization that arises from an interplay of active transport and quenched disorder.