A transport equation is derived from microscopic considerations, aimed at modeling fractional radial transport in cylindrical-like geometries. The procedure generalizes existing work on one-dimensional Cartesian systems. The transport equation emerges as the fluid limit of an underlying continuous-time random walk (CTRW) that preserves the required symmetries and conservation laws. In the process, appropriate radial fractional operators are identified and defined through their Hankel transforms, providing a smooth interpolation between standard radial differential operators. Finally, propagators for the radial fractional transport equation are obtained in terms of Fox $H$ functions.

A rarefied neutral gas computational model is proposed for efficiently evaluating quasi-steady-state neutral particle density distributions in plasma simulations. The linear evolution approximation of neutral phase space distribution under rarefied flow conditions is introduced and the steady-state solution is connected to the tracking and accumulation of the impulse response using the Duhamel principle. This method is validated against analytical benchmarks and compared with existing classic methods, demonstrating its advantages in resolving steady-state neutral distribution. When coupled with the plasma model, this model achieves accelerated convergence and eliminates transient ionization overshoot. Due to its low computational cost and compatibility with the standard plasma simulation method, this model serves as a practical tool for steady-state plasma research in scenarios with high ionization rates and rarefied neutral flows.

We employ molecular dynamics simulations to investigate the domain morphology and growth kinetics of a phase-separating vapor-liquid system embedded within a complex porous medium. By systematically varying the pore structure, we analyze the scaling behavior of correlation functions, structure factors, and domain growth exponents. The structure factor confirms the breakdown of Porod law and the emergence of fractal-like domain boundaries. Our key finding is the clear crossover in the domain growth law, from the classical power-law behavior observed in bulk fluids to a slower, logarithmic regime in highly confined systems. This transition is driven by screening of the hydrodynamics and energy barriers introduced by the porous geometry, which inhibit coarsening dynamics at later time. We provide a scaling analysis which further confirms this crossover and quantitatively connects the growth behavior with the average pore size.

In this work, the closed Frank model was simulated using a stochastic method based on the Ehrenfest urn on a surface (mathematically expressed as a matrix), considering that the reactions and diffusion processes of the species occur within a predetermined, adjusted neighborhood. It investigated the role of neighborhood size in achieving (or not) the global homochiral state (GHS). All simulations started without enantiomeric excess, with the particles randomly distributed over the surface, and other distributions were also considered. It is shown that, spatially, the surface gradually becomes occupied by domains (of the two enantiomers) that are randomly distributed, and as the size of the neighborhood increases, the number of simulations that reach the GHS also increases, with a corresponding decrease in the simulation time required to reach it. Besides, domains continue to form in this case, but over time, fluctuations (in space and time) in concentrations lead to GHS. When the GHS is not reached, the steady state is characterized by the presence of domains of both species on the surface, as previously mentioned. Still, the reactions continue to occur, and the generation and consumption rates of each species are equal. These occur mainly on the interface of the domains, especially in small neighborhood sizes.

We investigate the transport properties of active particles undergoing a three-state run-and-tumble dynamics in one dimension, induced by nonreciprocal transition rates between self-propelling velocity states $\{-v,0,+v\}$ that explicitly break microscopic reversibility. Departing from conventional reciprocal models, our formulation introduces a minimal yet rich framework for studying nonequilibrium transport driven by internal state asymmetries. Using kinetic Monte Carlo simulations and analytical methods, we characterize the particle's transport properties across the transition-rates space. The model exhibits a variety of nonequilibrium behaviors, including ballistic transport, giant diffusion, and Gaussian or non-Gaussian transients, depending on the degree of asymmetry in the transition rates. We identify a manifold in transition-rate space where long-time diffusive behavior emerges despite the absence of microscopic reversibility. Exact expressions are obtained for the drift, effective diffusion coefficient, and moments of the position distribution. Our results establish how internal-state irreversibility governs macroscopic transport, providing a tractable framework to study nonequilibrium active motion beyond reciprocal dynamics.

We provide a statistical and correlational analysis of the spatial and energetic properties of equilibrium configurations of a few-body system of two to eight equally charged classical particles that are confined on a one-dimensional helical manifold. The two-body system has been demonstrated to yield an oscillatory effective potential, thus providing stable equilibrium configurations despite the repulsive Coulomb interactions. As the system size grows, the number of equilibria increases, approximately following a power-law. This can be attributed to the increasing complexity in the highly nonlinear oscillatory behavior of the potential energy surface. This property is reflected in a crossover from a spatially regular distribution of equilibria for the two-body system to a heightened degree of disorder upon the addition of particles. However, in accordance with the repulsion within a helical winding, the observed interparticle distances in equilibrium configurations cluster around values of odd multiples of half a helical winding, thus maintaining an underlying regularity. Furthermore, an energetic hierarchy exists based on the spatial location of the local equilibria, which is subject to increasing fluctuations as the system size grows.

Temporal coherence—persistent alignment across time—can arise between agents with fundamentally distinct dynamics, yet classical diffusion models (Brownian motion, fractional Brownian motion, generalized Langevin equations with shared noise) struggle under strong heterogeneity and asymmetry. We introduce the coupled memory graph process (CMGP), in which internal memory and directed, distance-gated coupling jointly produce synchronized behavior without reciprocity or common noise. Crucially, CMGP exhibits long-time coherence that reaches far beyond typical inherent memory times: an active particle with long-range memory remains temporally coherent with a subdiffusive partner despite mismatched scaling exponents. We show that this persistence arises from emergent long-range correlations generated by the coupling field rather than direct kernel overlap. Using Bayesian optimization, we identify broad parameter regions that support this “ghost coherence” (coherence without trajectory convergence) while preserving distinct exponents. These results outline a minimal mechanism for coordination in heterogeneous active systems and viscoelastic environments—one that standard stochastic models do not capture under comparable asymmetry unless augmented with explicit common drives or symmetric couplings.

The self-diffusion process of a hard-sphere fluid confined by two parallel plates separated by a distance on the order of the particle diameter is studied. The starting point is a closed kinetic equation for the distribution function that takes into account the effects of the confinement and that is valid in the low-density limit. From it, the Boltzmann-Lorentz equation that describes the dynamics of some tagged particles when the whole system is in equilibrium is derived. An equation that describes the diffusion in the directions parallel to the walls is deduced by applying the Zwanzig-Mori projection technique to the Boltzmann-Lorentz equation, obtaining an explicit expression for the self-diffusion coefficient that depends on the height of the system. A very good agreement between its theoretical prediction and molecular dynamics simulation results is obtained for the whole range of heights.

We study a self-propelled particle moving at a constant speed in three spatial dimensions, where the orientation vector evolves via a rotational Langevin equation with Ornstein-Uhlenbeck-like statistics. This formulation ensures a unit propulsion direction while allowing for fully three-dimensional motion. The orientational noise is implemented orthogonally to both the propulsion axis and a fluctuating auxiliary unit vector, enabling reorientation without affecting speed. We analyze both underdamped and overdamped regimes, deriving analytical results for the particle's dynamics, including the time-dependent mean-squared displacement and alignment of velocity and propulsion direction. In the underdamped case, the dynamics exhibit a ballistic-to-diffusive crossover governed by the interplay between inertial and rotational timescales, independent of the initial velocity. The nonequilibrium nature of the system is characterized through the entropy production rate (EPR), where we derive explicit expressions and demonstrate that the finite misalignment between velocity and propulsion direction leads to a suppression of EPR, distinguishing our model from existing active Ornstein-Uhlenbeck and Brownian particle models. In the presence of harmonic confinement, the overdamped particle exhibits effective diffusion on the surface of a sphere, with a radius set by the interplay between propulsion, friction, and trap stiffness. Numerical simulations confirm our theoretical predictions, supporting the model's relevance for confined active systems in three dimensions.

We study the out-of-equilibrium dynamics of dipolar bosons and fermions after a sudden change in the interaction strength from zero to a finite repulsive value. We simulate the interaction quench on the initial state which is the ground state of harmonic potential with noninteracting bosons and fermions. We solve the time-dependent many-boson Schrödinger equation exactly using numerical methods. To understand the many-body dynamics we analyze several measures of many-body information entropy, monitoring their time evolution and assessing their dependence on interaction strength. We establish that for weak interaction quench the dynamics is statistics independent, both dipolar bosons and fermions do not relax, whereas it is significantly different for dipolar bosons from that of dipolar fermions in the stronger interaction quench. While dipolar bosons exhibit concurrent signature of relaxation in all entropy measures, dipolar fermions fail to relax, exhibiting modulated oscillation in all entropy dynamics. For dipolar bosons and larger interaction quench, the many-body information entropy measures dynamically approach the value predicted for the Gaussian orthogonal ensemble of random matrices, implying statistical relaxation and ergodicity. The entire relaxation process and signature of ergodicity are established by three unique features: dynamical delocalization in Hilbert space, violent orbital fragmentation, and saturation of all entropy measures to the maximum entropy states in the long-time dynamics. It highlights the importance of many-body effects with a possible exploration in quantum simulation with ultracold atoms.

We study the fluctuation properties of the local time density, ${\rho }_T=\frac{1}{T}{\int }_0^T\delta \left(r\left(t\right)-1\right)dt$, spent by a $d$-dimensional Brownian particle at a spherical shell of unit radius, where $r\left(t\right)$ denotes the radial distance from the particle to the origin. In the large observation time limit, $T\to \infty$, the local time density ${\rho }_T$ obeys the large deviation principle, $P({\rho }_T=\rho )\sim e^{-TI\left(\rho \right)}$, where the rate function $I\left(\rho \right)$ is analytic everywhere for $d\le 4$. In contrast, for $d>4, I\left(\rho \right)$ becomes nonanalytic at a specific point $\rho ={\rho }_c^{\left(d\right)}$, where ${\rho }_c^{\left(d\right)}=d(d-4)/(2d-4)$ depends solely on dimensionality. The singularity signals the occurrence of a first-order dynamical phase transition in dimensions higher than four. Such a transition is accompanied by temporal phase separations in the large deviations of Brownian trajectories. Finally, we validate our theoretical results using a rare-event simulation approach.

The so-called eigenstate thermalization hypothesis (ETH), which has been tested in various many-body models by numerical simulations, supplies a way of understanding eventual thermalization and is believed to be important for understanding processes of thermalization. Two functions play important roles in the application of ETH, one for averaged diagonal elements and the other for the variance of off-diagonal elements of an observable addressed by ETH on the energy basis. For the former function, a semiclassical expression is known of the zeroth order of $\hslash$, while, little is known analytically for the latter. In this paper, a semiclassical expression is derived for the former diagonal function, which includes higher-order contributions of $\hslash$. And, another semiclassical expression is derived for the scale of the height of a central platform of the latter off-diagonal function. These analytical results are checked numerically in the Lipkin-Meshkov-Glick model. Possible relevance of the latter result for future investigation is discussed.

Despite intensive studies on epidemic spreading in metapopulation networks, the infection process is normally assumed to be a Markovian process without considering time-dependent infectivity. In this paper, a reaction-diffusion SIRS epidemic model with non-Markovian infection processes is constructed on metapopulation networks, where the infection rate function associated with the sojourn time in the infected state is obtained by constructing the generation time distribution of the emerging infection process. Meanwhile, the recurrent mobility patterns are incorporated by the metapopulation model. By Lyapunov stability analysis, the epidemic outbreak condition is derived theoretically. The results show that the non-Markovian infection characteristics can significantly alter the transient behavior of epidemic spreading under the same basic reproduction number. Furthermore, with the increase in the average time for an infected individual to generate secondary cases in the non-Markovian process, the impact of mobility rate on transient epidemic spreading behavior will be reduced. Finally, mobility can have a suppressing effect on epidemic spreading dynamics when the population distribution exhibits significant asymmetry. The results can sharpen our understanding of the role of real-world non-Markovian infection processes in epidemic outbreaks.

Extensive behavioral experiments reveal that conditional cooperation is a prevalent phenomenon. Previous game-theoretical studies have predominantly relied on hard-manner models, where cooperation is triggered only upon reaching a specific threshold. This approach contrasts with the observed flexibility of human behaviors, where individuals adapt their strategies dynamically based on their surroundings. To capture this adaptability, here we introduce a soft form of conditional cooperation by integrating the $Q$-learning algorithm from reinforcement learning. In this form, players not only reciprocate cooperation but may also defect in highly cooperative environments or cooperate in less-cooperative settings to maximize rewards. To explore the effects of hard and soft conditional cooperators, we examine their interactions in two scenarios: structural mixture (SM) and probabilistic mixture (PM), where the two behavioral modes are fixed and probabilistically adopted, respectively. In SM, hard conditional cooperators enhance cooperation when the threshold is low but hinder it otherwise. Surprisingly, in PM, the cooperation prevalence exhibits two first-order phase transitions as the probability is varied, leading to high, low, and vanishing levels of cooperation. Analysis of $Q$ tables offers insights into the “psychological shifts” of soft conditional cooperators and the overall evolutionary dynamics. Model extensions confirm the robustness of our findings. These results highlight the novel complexities arising from the diversity of conditional cooperators.

Real-world networks exhibit universal structural properties such as sparsity, small worldness, heterogeneous degree distributions, high clustering, and community structures. Geometric network models, particularly random hyperbolic graphs (RHGs), effectively capture many of these features by embedding nodes in a latent similarity space. However, networks are often characterized by specific connectivity patterns between groups of nodes—i.e., communities—that are not geometric, in the sense that the dissimilarity between groups does not obey the triangle inequality. Structuring connections only based on the interplay of similarity and popularity thus poses fundamental limitations on the mesoscale structure of the networks that RHGs can generate. To address this limitation, we introduce the random hyperbolic block model (RHBM), which extends RHGs by incorporating block structures within a maximum-entropy framework. We demonstrate the advantages of RHBM through synthetic network analyses, highlighting its ability to preserve community structures where purely geometric models fail. Our findings emphasize the importance of latent geometry in network modeling while addressing its limitations in controlling mesoscale mixing patterns.

We derive a nonlocal Ostrovsky equation to describe two internal waves generated at distinct locations and times, together with their correlations and interactions. When the initial conditions are $\widehat{P}\widehat{T}$ symmetry invariant, the internal waves can either exhibit cnoidal wave structures that are largely insensitive to rotational effect, or, in the case of solitary waves, evolve into nonlinear wave packets under rotation. In this scenario, the two waves possess antiphase amplitudes, resulting in a nodal surface of zero displacement at the middepth layer. In contrast, under the $\widehat{P}\widehat{T}$ symmetry breaking initial conditions, the two internal waves develop snoidal waveforms, with rotation producing a pronounced asymmetry with two nonequivalent crest heights within each wave period. In this case, the two waves exhibit amplitude anticorrelation and phase lag, causing partial destructive interference. Furthermore, the results demonstrate that stronger rotation cannot only accelerate the attenuation of internal solitary waves, but also enhance the peak asymmetry of the snoidal waveforms, whereas introducing shear flow can partially mitigate rotational effect.

Deep autoencoder neural networks can generate highly accurate, low-order representations of turbulence. We design a family of autoencoders which are a combination of a “dense-block” encoder-decoder structure [Page et al., J. Fluid Mech. 991, A10 (2024)], an "implicit rank minimization" series of linear layers acting on the embeddings [K. Zeng et al., Mach. Learn.: Sci. Technol. 5, 025053 (2024)], and a full $\mathrm{discrete}+\mathrm{continuous}$ symmetry reduction. These models are applied to two-dimensional turbulence in Kolmogorov flow for a range of Reynolds numbers $25\le \text{Re}\le 400$ and used to estimate the dimension of the chaotic attractor, $d_{\mathcal{A}}\left(\text{Re}\right)$. We find that the dimension scales like $\sim {\text{Re}}^{1/3}$—much weaker than known bounds on the global attractor, which grow like ${\text{Re}}^{4/3}$. In addition, two-dimensional maps of the latent space in our models reveal a rich structure not seen in previous studies, including multiple classes of high-dissipation events at lower Re, which guide bursting trajectories. We visualize the embeddings of large numbers of “turbulent” unstable periodic orbits, which the model indicates are distinct (in terms of features) from any flow snapshot in a large turbulent dataset, suggesting their dynamical irrelevance. This is in sharp contrast to their appearance in more traditional low-dimensional projections, in which they appear to lie within the turbulent attractor.

We investigate the Taylor-Aris dispersion resulting from oscillatory squeeze flow (OSF) of an Oldroyd-B viscoelastic fluid in the gap between two hydrophobic disks. The slippage between the fluid and the surfaces of both disks is modeled using a dynamic slip boundary condition, which accounts for periodic motion and the fluid's rheological properties. The fluid motion is induced by the periodic vertical movement of one of the disks. Using the lubrication approximation, we simplify and solve the governing equations to determine the flow field. We then derive key variables, including the velocity field, pressure distribution, and the force and power associated with OSF. To analyze mass transport, we used the convection-diffusion equation, which we solved using the multiple-timescale homogenization method. This study highlights the influence of various dimensionless parameters that govern hydrodynamics and Taylor dispersion, specifically the Womersley number, two Deborah numbers related to the fluid's relaxation and retardation times, and a slip parameter. Our research focuses on how the slip boundary condition and the fluid's rheology affect the dispersion process and other related variables. A key aspect of our work is to identify conditions that maximize the effective dispersion coefficient under the given assumptions. Our findings indicate that the slip condition generally reduces the dispersion coefficient in an OSF. Furthermore, when comparing the effect on the effective dispersion coefficient using the Oldroyd-B and Maxwell models, we observe that the Maxwell model tends to overestimate the effective dispersion coefficient.

Walkers are oil drops surfing on a vibrated oil surface and driven by their self-generated capillary waves. Since some of the first measurements on walkers seemingly showed quantum-like behavior, walkers have been considered a model system for a hydrodynamic pilot-wave system. An early experiment showed that the passage of the walker over a submerged barrier resulted in what was coined “unpredictable walker tunneling” in analogy with quantum tunneling. Together with the double slit experiment, tunneling is one of the fundamental phenomena that show the probabilistic nature of quantum mechanics. A later experiment that refined the measurement of position and timing of the walker narrowed the region where transmission changed from 0 to 1 by a factor 10 but still found “unpredictable tunneling." However, at a certain small distance close to the rim of the barrier, predictability sets in, and separation between reflected and transmitted trajectories is observed. After careful analysis of identified noise sources, the authors find that neither noise nor insufficient knowledge of initial conditions alone could explain the unpredictability. They therefore conclude that it is more likely due to changes in the drop's vertical dynamics arising when it interacts with the barrier. They also analyzed a quantum particle impinging on a barrier by step-wise collapsing the wave function and showed a similar behavior for this. Here we revisit the “tunneling” of walkers. Enforcing a stringent control of temperature in all aspects of the experiment results in the observation of a much sharper transition, much like that of a classical system, reducing the unpredictability by another factor of 10. Apart from varying the velocity of the walker by changing the drop size, we varied it by changing the amplitude of the drive. We furthermore found that a micron-size change in fluid height changes transmission from 0 to 1. The observed sharpness of the transition suggests that in a noise-free environment the transition would be truly abrupt, at least in the regions of phase-space for the drop that have been investigated so far.

Integrating machine learning (ML) techniques in established numerical solvers represents a modern approach to enhance computational fluid dynamics simulations. Within the lattice Boltzmann method (LBM), the collision operator serves as an ideal entry point to incorporate machine learning (ML) techniques to enhance its accuracy and stability. In this work, an invariant neural network is constructed, acting on an equivariant collision operator, optimizing the relaxation rates of nonphysical moments. This optimization enhances robustness to symmetry transformations and ensures consistent behavior across geometric operations. The proposed neural collision operator (NCO) is trained using forced isotropic turbulence simulations driven by spectral forcing, ensuring stable turbulence statistics. The desired performance is achieved by minimizing the energy spectrum discrepancy between direct numerical simulations and underresolved simulations over a specified wave number range. The loss function is further extended to tailor numerical dissipation at high wave numbers, ensuring robustness without compromising accuracy at low and intermediate wave numbers. The NCO's performance is demonstrated using three-dimensional Taylor-Green vortex (TGV) flows, where it accurately predicts the dynamics even in highly underresolved simulations. Compared to other LBM models, such as the BGK and KBC operators, the NCO exhibits improved accuracy while maintaining stability. In addition, the operator shows robust performance in alternative configurations, including turbulent three-dimensional cylinder flow. Finally, an alternative training procedure using time-dependent quantities is introduced. It is based on a reduced TGV model along with newly proposed symmetry boundary conditions. The reduction in memory consumption enables training at significantly higher Reynolds numbers, successfully leading to stable yet accurate simulations.