Search Results (46)

Path-dependent dynamic systems are often analysed through separate metric, stochastic, econometric or nonlinear methods, while their integration within a unified architecture remains limited. This study develops and validates a Hybrid Dynamic Modelling Framework (HDMF) for path-dependent stochastic systems, using EU-27 digital public service trajectories during 2017–2022 as an empirical application. The framework combines DTW-TimeSeriesKMeans clustering, Markov transition analysis, PVAR, SEM and MLP approximation. The findings answer the four research questions sequentially. First, EU digital public service development is represented by three distinct trajectory regimes rather than a single convergence path. Second, the Markov layer shows strong diagonal dominance, limited mobility and practical non-ergodicity over the finite observation horizon, indicating persistent state dependence. Third, the PVAR layer confirms dynamic stability and bounded feedback within the institutional–digital system. Fourth, the SEM and MLP layers identify governance as a coherent latent institutional structure and reveal nonlinear predictive heterogeneity, suggesting possible threshold-type behaviour. Overall, the findings support the HDMF as a coherent multilayer architecture for modelling complex path-dependent systems and demonstrate its empirical usefulness through the EU digital public service case. Full article

Sparse ergodic control provides a natural framework for long-run stochastic decision-making under resource constraints. Existing formulations, however, are typically restricted to control-affine systems with control-independent diffusion. When the diffusion coefficient depends explicitly on the control input, the associated ergodic Hamilton–Jacobi–Bellman (HJB) equation becomes non-separable through the term $\mathrm{t}\mathrm{r}\left[a\left(x, u\right){\nabla }^{2}V\right]$, so classical arguments based on control-affine separability no longer apply directly. In this work, we study sparse ergodic control of stochastic systems with control-dependent diffusion and nonlinear dynamics within a viscosity-solution and learning-based framework. To address the discontinuous ${\mathsf{\ell }}_0$-type sparsity penalty, we introduce smooth non-convex sparsity approximations that preserve differentiability while retaining sparse threshold behavior. Within a viscosity-solution framework, we analyze the existence and uniqueness properties of the associated ergodic pair and establish localized approximation error estimates for the smooth approximation. We further characterize a quasi-threshold sparse structure of the resulting optimal feedback policies in non-affine stochastic systems with control-dependent noise. On the computational side, we develop a Physics-Informed Neural Network (PINN)-based solver with adaptive residual-driven sampling for high-dimensional sparse ergodic HJB equations, together with a distributed monotone-inspired iterative scheme for weakly coupled multi-agent systems. Numerical experiments on multi-robot swarm navigation and renewable-integrated smart-grid control demonstrate that the proposed methods produce sparse control policies while preserving stable long-run performance under stochastic disturbances. Full article

The paper deals with the asymptotic behavior of a widely used correlation characteristic in large quantum systems. The correlation is quantum entanglement, the characteristic is entanglement entropy, and the system is an ideal gas of lattice fermions. If the one-body Hamiltonian of fermions is an ergodic finite difference operator with an exponentially decaying spectral projection, then the large-block form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified consists of discrete Schrödinger operators with random potential. In this paper, we prove the area law for several classes of Schrödinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move to a highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying chaotic phenomena. The corresponding asymptotic study requires involved spectral analysis, which therefore constitutes the bulk of the paper. Specifically, we prove uniform localisation of the eigenfunctions for the Maryland model and exponential decay of the eigenfunction correlator for various models. We believe these properties are of significant independent interest. Full article

This paper studies a black-box methodology for optimizing ergodic stochastic systems, focusing on the construction of scalar measures that reliably indicate progress toward optimality. Our starting point is a state-value quantity that inherently exhibits oscillatory behavior and does not converge under standard conditions. We show that, despite its fluctuations, this quantity admits a recursive representation derived from a one-step-ahead fixed-local-optimal policy. The approach relies on identifying a Lyapunov-like function whose evolution reflects the long-run behavior of the system without requiring explicit knowledge of its internal dynamics. Such a function provides a monotonic indicator—non-increasing over time—that remains valid for any initial probability distribution. Whenever an optimal trajectory of the Markov chain exists, the proposed method guarantees convergence to it. We also provide a constructive procedure for obtaining the Lyapunov-like function and validate the methodology through theoretical analysis and numerical simulations. Full article

Sol–gel phase transitions are complex far-from-equilibrium processes characterized by limited reproducibility, whose origin remains poorly understood and rarely quantified. We investigated the thermally induced sol–gel transition of agar using turbidimetry. A phenomenological model was applied to extract key kinetic parameters (maximum absorbance, maximum rate, and characteristic times) from 96 independent replicates. Variability was quantified and compared with that of an enzymatic reaction exhibiting similar sigmoidal kinetics, allowing for separation of experimental, intrinsic, and nonergodic contributions. Agar gelation displays markedly higher variability. The total variability (CV ≈ 16%) exceeds both the experimental error (1–2%) and the nonergodic contribution (≈2%), demonstrating that it predominantly arises from intrinsic process dynamics. Variability increases sharply during early stages of gelation and then evolves more gradually, indicating that stochastic nucleation and network formation pathways drive divergent kinetic trajectories despite identical initial conditions. Variability in gelation is therefore not a measurement artifact but an intrinsic hallmark of the sol–gel transition. This inherent stochasticity limits the predictive power of deterministic models, particularly at meso- and microscopic scales, and should be considered a fundamental feature of gel-forming systems. Our approach provides a quantitative framework for characterizing variability in phase transitions and may be extended to more complex biological and soft matter systems. Full article

This study focuses on forecasting and optimizing the performance of a real-world object modelled by a multi-server queueing system that processes two types of users: primary (new) users and repeating users. The repeating users are those who succeeded in entering processing upon arrival and then decided to repeat it. These users have privilege and can enter processing when they wish once at least one device is idle. The primary user is admitted to the system only if the number of occupied devices is less than some threshold value and the quantity of repeating users residing in the system does not exceed certain thresholds. Repeating users are impatient and non-persistent. Arrivals of primary users are described by the Markovian arrival process. Processing times of primary and repeating users have distinct phase-type distributions. Utilizing the concept of the generalized phase–time distributions, the dynamics of this queueing system are formally characterized by the multidimensional Markov chain, which is examined in this paper. The ergodicity condition is derived. The relation of the key performance characteristics of the system and the thresholds defining the policy of the primary user’s admission is numerically highlighted. Optimal threshold selection is demonstrated numerically. Full article

Polycystic Ovary Syndrome (PCOS) is a widespread hormonal disorder affecting women of reproductive age, often leading to infertility and associated complications. This study presents a comprehensive stochastic mathematical framework to analyze the dynamics of PCOS with a particular focus on infertility and treatment outcomes. Here, the transitions between compartments represent progression of women through clinical states of PCOS (risk, diagnosis, treatment, recovery) rather than infection or transmission, since PCOS is a non-communicable disorder. The model incorporates probabilistic elements to break the symmetric and predictable assumptions inherent in deterministic approaches. This allows it to reflect the randomness and asymmetry in hormonal regulation and ovulation cycles, enabling a more realistic representation of disease progression. By utilizing stochastic differential equations, the study evaluates the impact of treatment adherence on fertility restoration. We establish the conditions for disease extinction versus the existence of an ergodic stationary distribution, which represents a form of long-term statistical symmetry. The results emphasize the importance of early diagnosis and consistent treatment. Furthermore, the proposed approach provides a valuable tool for clinicians to predict patient-specific trajectories and optimize individualized treatment plans, accounting for the asymmetric nature of patient responses. Full article

This study develops a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck (OU) processes for both transmission rate $\beta \left(t\right)$ and quarantine release rate $k\left(t\right)$; this is distinct from existing non-white-noise stochastic epidemic models, most of which focus on single-parameter perturbation or only stability analysis. It synchronously embeds OU dynamics into two core epidemic parameters to capture asynchronous fluctuations between infection spread and control measures. It adopts a rare measure solution framework to derive rigorous infection extinction conditions, linking OU’s ergodicity to long-term ${\beta }^{+}\left(t\right)$ averages. It obtains the explicit probability density function of the four-dimensional SIQR system, filling the gap of lacking quantifiable density dynamics in prior studies. Simulations validate that $R_0^d<1$ ensures almost sure extinction, while $R_0^e>1$ leads to stable stochastic persistence. Full article

Side-chain engineering is versatile for tuning the chain mobility of graft polymers and governs their thermal stability. However, it remains elusive to predict the graft effect on chain mobility, especially for competitive side-chain types. Here, relying on molecular dynamics simulation and energy landscape theory, we introduce a three-stage virtual pipeline to sequentially refine the screening of graft chain mobility while minimizing computation cost, by taking the example of grafting similar side-chain types (hydroxyethyl methacrylate (HEMA), methyl methacrylate (MMA), and vinyl acetate (VAC)) onto amorphous polypropylene (PP). Ascribed to their structural similarity, these graft systems exhibit a non-evident chain mobility distinction, with the atom displacement—governing the local “roughness” in potential energy landscape (PEL)—exhibiting only weak-to-modest correlation with their initial atomic energy, volume, and stress. This necessitates the subsequent-stage screening for broader PEL navigation, which confirms a stability and roughness rank of VAC ≥ MMA > HEMA > PP, with their chain activation energy revealing that these side chains enhance the PEL roughness through a counterbalance between possibly lowering the overall energy barrier but extensively wrinkling the landscape. Overall, the three-stage screening establishes a state-of-the-art efficient strategy to evaluate thermal stability of graft polymers in stepwise higher precision from local to ergodic roughness inspection. Full article

The ergodicity and ageing phenomena in a vibrational motor system driven by a periodic external force are investigated. Within the tailored parameter regime, the amplitude and frequency demonstrate contrasting effects on ergodicity. An increase of amplitude induces a transition from non-ergodic to ergodic behavior, whereas a higher driving frequency leads to a transition from ergodic to non-ergodic dynamics. These transitions are attributed to the enhanced ability of larger amplitudes to overcome potential energy barriers and the improved responsiveness of the system to external variations at lower frequencies. Moreover, pronounced ageing effects are observed at low amplitudes or high frequencies. These findings offer new insights into the intrinsic dynamical mechanisms of vibrational motor systems and provide a theoretical foundation for predicting their long-term operational performance. Full article

In engineering, the study of the dynamic response of structures subjected to non-deterministically variable loads is particularly important, especially when considering the damage that such loads can cause due to fatigue phenomena. This is the case, for example, of the vibrations that a satellite must withstand during the launch phase. In the preliminary design phases, it is very useful to have semi-analytical calculation methodologies that are sufficiently reliable but, at the same time, simple. In the technical literature, there are numerous publications that deal with the study of the random dynamic response of beam models. In general, the presented studies are rather complex, and the dynamic solutions are often obtained in the time domain. The case of a linear elastic uniform cantilever beam model is considered here, for which the analytical expressions of the transfer functions for acceleration, displacement, bending moment, and bending stress are calculated, taking as input the acceleration assigned to the root section or an external lateral load. Knowing the spectral density of the input loads, the spectral densities of all the above-mentioned variables are calculated along the beam axis, assuming stationary and ergodic random processes. Using the spectral density of each output variable, the effective value (RMS) is obtained via integration, which allows for a preliminary estimate of the severity of the working conditions of the beam. The spectral density of the responses also allows us to quickly highlight the contribution of each natural vibration mode as the spectrum of the load varies. The results were obtained using simple spreadsheets available to the reader. Full article

We examine the Rosencrantz coin that can “stick” in states for extended periods. Non-ergodic dynamics is highlighted by logarithmically growing block lengths in sequences. Traditional entropy decomposition into predictable and unpredictable components fails due to the absence of stationary distributions. Instead, sequence structure is characterized by block probabilities and Stirling numbers of the second kind, peaking at block size $n/\log n$. For large n, combinatorial growth dominates probability decay, creating a deterministic-like structure. This approach shifts the focus from predicting states to predicting temporal horizons, providing insights into systems beyond traditional equilibrium frameworks. Full article

We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic systems, have recently regained interest for their role in non-ergodic dynamics, as they retain memory of their initial states. We elucidate these features of quantum scars within the same framework of this toy model. The integrable part of the model consists of two large spins, with a classical counterpart, which we combine with a random matrix to induce ergodic behavior. Scarred states can be selectively generated from the integrable spin Hamiltonian by protecting them from the ergodic states using a projector method. Deformed projectors mimic the ‘quantum leakage’ of scarred states, enabling tunable mixing with ergodic states and thereby controlling the degree of scarring. In this simple model, we investigate various properties of quantum scarring and shed light on different aspects of many-body quantum scars observed in more complex quantum systems. Notably, the underlying classicality can be revealed through the entanglement spectrum and the dynamics of ‘out-of-time-ordered correlators’. Full article

In this investigation, we explored the microrheological characteristics of dilute hydrogels composed exclusively of Poly(vinyl) alcohol (PVA), Phytagel (PHY), and a blend of the two in varying concentrations. Each of these polymers has established applications in the biomedical field, such as drug delivery and lens drops. This study involved varying the sample concentrations from 0.15% to 0.3% (w/w) to assess how the concentration influenced the observed rheological response. Two probe sizes were employed to examine the impact of the size and verify the continuity hypothesis. The use of two polymer blends revealed their immiscibility and tendency to undergo phase separation, as supported by the existing literature. Exploring the microrheological structure is essential for a comprehensive understanding of the molecular scale. Dynamic light scattering (DLS) was chosen due to its wide frequency range and widespread availability. The selected dilute concentration range was hypothesized to fall within the transition from an ergodic to a non-ergodic medium. Properly identifying the sample’s nature during an analysis—whether it is ergodic or not—is critical, as highlighted in the literature. The obtained results clearly demonstrate an overlap in the results for the storage (G’) and loss moduli (G″) for the different probe particle sizes, confirming the fulfillment of the continuum hypothesis. Full article

The random changes in the environment play a crucial role in the sustainability of ecosystems. Usually, the construction of stochastic models does not take into account the non-linear growth of intrinsic growth rate. In addition, prey only considers the collective response of the population when encountering predators and ignores the role of individual prey. To address this issue, we contemplate the dynamics of a stochastic prey–predator model with Smith growth rate and cooperative defense. The population density of prey is measured by mass, and the growth limitations are based on the proportion of unused available resources. Additionally, the grazing pattern of the predator incorporates cooperative characteristics into the functional response. We carry out existence and uniqueness analysis for the global positive solution. Then, we construct sufficient conditions for the existence of an ergodic stationary distribution of positive solutions for investigating whether prey and predator populations continue to survive. Numerical examples indicate that the Smith growth rate, cooperative defense and environmental disturbance play crucial roles in the coexistence of interacting populations. Full article