The Cahn-Hilliard equation, as a classical diffusion-interface method of phase field, has been extensively employed for simulating two-phase fluid dynamics. However, it suffers from a key challenge in the simulation process, specifically the volume conservation of each phase cannot be guaranteed. To address this issue, in this paper, a modified Cahn-Hilliard equation for two-phase flow modeling is first introduced, and the basic idea of this model lies in that it combines the profile correction method with the level-set approach, and thus, it effectively improves the deficiency of the classical Cahn-Hilliard equation in terms of volume nonconservation of each phase. Based on this modified Cahn-Hilliard equation, we further propose an accurate interface-capturing lattice Boltzmann model. After that, we perform a range of numerical simulations, including two stationary droplets immersed in the gas phase, single vortex, Rayleigh-Plateau fluid instability, and droplet deformation under a shear flow. These simulations illustrate that the proposed lattice Boltzmann model has superior performance in maintaining local volume conservation and accurately capturing interfaces. More importantly, compared to the lattice Boltzmann model derived from the classical Cahn-Hilliard equation, it not only achieves more precise volume conservation for each phase but also provides a more consistent representation of the droplet's interface morphology more consistently, especially in dealing with small droplet problems.

The calculation of the demagnetizing field is the performance bottleneck in micromagnetic simulations that solve the Landau-Lifshitz-Gilbert equation. Although treecode is already recognized as a competitive method, achieving an $O\left(N{log}_{8}N\right)$ algorithm for general domains remains highly challenging. In our work, we tackle two essential parts of the treecode calculation: constructing high-quality non-neighbor element patches and performing multipole expansions. For each of these crucial components, we provide robust and accurate solutions. Regarding the construction of non-neighbor elements, we have thoroughly harnessed the potential of the hierarchy geometry tree data structure. By leveraging this structure, we successfully build high-quality non-neighbor element patches for each element, striking an excellent balance between computational complexity and accuracy. In the multipole expansion part, we systematically analyze the errors arising from multipole expansions and propose a $p$-adaptive method based on this error analysis, further enhancing computational efficiency. Our numerical results highlight the strengths of the proposed algorithm: in general domains, we achieve the $O\left(N{log}_{8}N\right)$ complexity of the treecode algorithm. Moreover, the $p$-adaptive method effectively improves the simulation efficiency of mainstream issues such as magnetic skyrmions, showing great promise for becoming a quality candidate in computational micromagnetics.

The signal dynamics in MRI are influenced by the geometry of the dephasing domain, as well as by diffusion and susceptibility effects. Analytical solutions exist only for simple models, and there is a clear need for improved simulation methods. The purpose of this work is to develop and validate a physics-informed neural network (PINN) framework for predicting MR signal dynamics from arbitrary local Larmor field distributions. We implemented a PINN-based simulation framework to solve the Bloch-Torrey equation including diffusion and susceptibility terms, with the complex Bloch-Torrey operator formulated with reflecting boundary conditions and an initially isotropic magnetization. The spatiotemporal magnetization distribution was represented as the output of a neural network, with spatial coordinates and time as inputs. We demonstrated that predicting local and total magnetization is feasible using the PINN-based framework for parameter regimes with intermediate oscillation frequencies, though adapted neural networks incorporating dual encoders or Fourier features were necessary, as conventional multilayer perceptrons failed to learn higher-frequency solutions. Most PINN algorithms performed similarly, except for the gradient-enhanced PINN algorithm, which failed to converge to a correct solution. Evidence of significant spectral bias was found and was partially alleviated by the use of Fourier features. We show that simulating the signal dynamics of MRI is possible using a PINN framework, demonstrating high accuracy and greater flexibility than conventional methods.

We numerically investigate the transport behavior of multiple deformable particles in time-oscillating potentials. For a fixed potential asymmetry, the transport direction is determined by the competition between two nonequilibrium driving mechanisms: the self-propulsion speed and the oscillation frequency of the potential. Particle deformability can either enhance or impede transport depending on which driving force dominates. Both rotational noise and particle density exhibit nonmonotonic influences, including velocity reversals. By carefully tuning system parameters, multiple reversals of the average particle velocity can be achieved, providing a potential mechanism for selective particle separation. Compared to single-particle systems, collective interactions give rise to richer dynamics and stronger transport rectification. These findings deepen the theoretical understanding of active soft matter in time-dependent potentials and may guide the design of experimental strategies for controlling and separating deformable particles in complex environments.

Several important dynamical systems are in ${\mathbb{R}}^2$, defined by the pair of differential equations $(x^{\prime },y^{\prime })=(f(x,y),g(x,y))$. A question of fundamental importance is how such systems might behave quantum mechanically. In developing quantum theory, Dirac and others realized that classical Hamiltonian systems can be mapped to their quantum counterparts via canonical quantization. The resulting quantum dynamics is always physical, characterized by completely positive and trace-preserving evolutions in the Schrödinger picture. However, whether non-Hamiltonian systems can be quantized systematically while respecting the same physical requirements has remained a long-standing problem. Here, we resolve this question when $f(x,y)$ and $g(x,y)$ are arbitrary polynomials. By leveraging open-systems theory, we prove constructively that every polynomial system admits a physical generator of time evolution in the form of a Lindbladian. We call our method cascade quantization, and demonstrate its power by analyzing several paradigmatic examples of nonlinear dynamics such as bifurcations, noise-activated spiking, and Liénard systems. In effect, our method can quantize any classical system whose $f(x,y)$ and $g(x,y)$ are analytic with arbitrary precision. More importantly, cascade quantization is exact. This means restrictive system properties usually assumed in the literature to facilitate quantization, such as weak nonlinearity, rotational symmetry, or semiclassical dynamics, can all be dispensed with by cascade quantization. We also highlight the advantages of cascade quantization over existing proposals, by weighing it against examples from the variational paradigm using Lagrangians, as well as nonvariational approaches.

Predicting extreme events is a challenging task due to their occasional appearance at irregular time intervals and with sudden large amplitudes. In particular, accurate forecasting of both the amplitude and timing of occurrence is difficult. We make an attempt to address the challenges using reservoir computing machine learning based on partial or complete information of the system variables in a few paradigmatic dynamical systems, namely, the forced Liénard system, the coupled FitzHugh-Nagumo model, and a model of hidden attractor. The efficacy of the machine learning approach has been tested using numerically generated data using models and a real-time experiment. The machine is able to successfully predict the transition points to extreme events via two well-known nonlinear processes, namely, Pomeau-Manneville intermittency and crisis-induced intermittency (interior crisis) against a system parameter by reconstruction of the bifurcation diagram by training the machine with simulated data. The machine is able to retrace the attractors of the systems quite efficiently and also reproduce the distributions of events and interevent intervals, thereby preserving the statistical properties of extreme events in the model systems. Prediction of time evolution is still limited to the length of a few Lyapunov times before it diverges from the original state as expected due to the complex nature of the extreme event dynamics. The amplitude of extreme events in the time series is preserved or predicted almost accurately. A prediction horizon metric is used for measuring the success level of the machine and deciding the minimal number of inputs necessary for efficient learning and prediction of the forced Liénard system and the hidden attractor model. For the coupled FitzHugh-Nagumo model, we use the mean square error measure to find an appropriate choice of a pair of inputs that is necessary for successful prediction. Using experimental data from an analog forced Liénard circuit, so far the machine could learn the dynamics and the long-term statistics of extreme events originating via Pomeau-Manneville intermittency.

One of the defining features of complex networks is the connectivity properties that we observe emerging from local interactions. Recently, hypergraphs have emerged as a versatile tool to model networks with nondyadic, higher-order interactions. Nevertheless, the connectivity properties of real-world hypergraphs remain largely understudied. In this work we introduce path size as a measure to characterize higher-order connectivity and quantify the relevance of nondyadic ties for efficient shortest paths in a diverse set of empirical networks with and without temporal information. By comparing our results with simple randomized null models, our analysis presents a nuanced picture, suggesting that nondyadic ties are often central and are vital for system connectivity, while dyadic edges remain essential to connect more peripheral nodes, an effect which is particularly pronounced for time-varying systems. Our work contributes to a better understanding of the structural organization of systems with higher-order interactions.

In complex systems, accurately predicting the critical points of first-order phase transitions, such as explosive death, is crucial for early warning and risk management. This study constructs a parameter-aware next-generation reservoir computing (PNGRC) framework to achieve accurate prediction of critical transition. The framework embeds control parameters into the nonlinear vector autoregressive model to generate high-order nonlinear feature vectors that jointly encode system states and parameter information. The forward and backward trajectories under parameter variations are captured in the training stage, and the bidirectional parameter-aware model is constructed to effectively identify the bistable structure and transition paths within hysteresis regions. We evaluate the PNGRC framework on three representative coupled oscillator systems, highlighting its capability to identify first-order critical points across complex dynamical regimes. The results show that the PNGRC framework not only accurately reconstructs system trajectories under training parameters, but also generalizes well to unseen conditions, effectively capturing the period-doubling bifurcation structure and hysteresis phenomena. Compared to traditional parameter-aware reservoir computing, the PNGRC framework substantially reduces data requirements and training time. This work provides an efficient data-driven paradigm for predicting first-order phase transitions.

Active Brownian particles (ABPs) serve as a minimal model of active matter systems. When ABPs are sufficiently persistent, they undergo a liquid-gas phase separation and, in the presence of obstacles, accumulate around them, forming a wetting layer. Here, we perform simulations of ABPs in a quasi-one-dimensional domain in the presence of a wall, studying the dynamics of the polarization field. Over the course of time, we observe a transition from a homogeneous (where all particles are aligned) to a heterogeneous (where particles align only at the interface) polarization regime. We propose coarse-grained equations for the density and polarization fields based on microscopic and phenomenological arguments that correctly account for the observed phenomena.

We propose a new framework for analyzing vehicular traffic jams based on the space-time geometry of congestion. Using the simple but instructive model elementary cellular automaton rule 184 (ECA 184), we identify jammed regions as connected clusters in space-time and show that their statistical properties follow scaling laws characteristic of a percolation transition. Key traffic observables—including total delay, relaxation time, and jam lifetimes—exhibit consistent scaling behavior. We introduce an auxiliary quantity, termed “elementary jams,” that serves as the basis for computing these observables and interpreting jam propagation. The simplicity of ECA 184 allows for efficient analysis and reveals structure that we believe generalizes to more complex traffic models. We outline how this percolation-based approach can be applied to models such as the Nagel-Schreckenberg model.

Collisional Brownian engines have recently gained attention as alternatives to conventional nanoscale engines. However, a comprehensive optimization of their performance, which could serve as a benchmark for future engine designs, is still lacking. In this work, we build upon this by deriving and analyzing the optimal control strategy for a collisional Brownian engine. By maximizing the average output work, we show that the optimal strategy consists of linear force segments separated by impulsive deltalike kicks that instantaneously reverse the particle's velocity. This structure enforces constant velocity within each stroke, enabling fully analytical expressions for optimal output power, efficiency, and entropy production. We demonstrate that the optimal strategy significantly outperforms standard ones (such as constant, linear, or periodic drivings), achieving higher performance while keeping entropy production under control. Remarkably, when evaluated using realistic experimental parameters, the efficiency approaches near unity at the power optimum, with entropy production remaining well controlled. To analyze a more realistic scenario, we examine the impact of smoothing the deltalike forces by introducing a finite duration and find that, although this reduces efficiency and increases entropy production, the optimal strategy still delivers high power output in a robust manner.

Generic systems are associated with a mixed classical phase space. The question of the properties of the eigenstates for these systems remains less known, although it plays a key role for understanding several important quantum phenomena such as thermalization, scarring, tunneling, and (de-)localization. In this work, by employing the kicked top model, we perform a detailed investigation of the dynamical signatures of the mixed eigenstates via the out-of-time-order correlator (OTOC). We show how the types of eigenstates get reflected in the short- and long-time behaviors of the OTOC and conjecture that the dynamics of the OTOC can be used as an indicator of the mixed eigenstates. Our findings further confirm the usefulness of the OTOC for studying quantum complex systems and also provide more insights into the characteristics of the mixed eigenstates.

The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, $N$ Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For $t\lesssim \sqrt{N}$, after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At $t\sim \sqrt{N}$, the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading $1/N$ effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at $t\sim 2N$, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at $t\lesssim \sqrt{N}$ indicate lack of thermalization—a desired feature for better performance of quantum information devices.

We develop an operator-description for the linear stability in finite networks of Kuramoto oscillators on circulant graphs. This mathematical approach offers analytical predictions for the linear stability of $q$-states, which include phase synchronization $(q=0)$ and phase-locked states with different spatial frequencies $\left(\right|q|>0)$. This approach seamlessly incorporates the presence of time delays (represented by phase lags in the coupling). With this, we are able to determine the specific combination of connectivity and time delays (phase lags) that leads to any given $q$-state to be linearly stable. We apply our framework to a variety of networks, including k-ring graphs, distance-dependent graphs, and random circulant graphs. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual and finite oscillator networks.

We analyze quantum droplets formed in a two-dimensional symmetric mixture of Bose-Einstein condensed atoms. For sufficiently large atom numbers, these droplets exhibit a flattop density profile with sharp boundaries governed by surface tension. Within the bulk of the droplet, traveling matter waves—localized density dips—can propagate at constant velocity while maintaining their shape. Using numerical simulations and qualitative analysis, we investigate the rich phenomenology that arises when such excitations reach the boundary of a finite droplet. We show that they can emit a small outgoing droplet, excite internal modes of the host soliton, or, in the case of vortex-antivortex pairs, split into individual vortices propagating backward near the edge. Furthermore, we demonstrate that traveling waves can be dynamically generated near the boundary through the collision of distinct droplets, and we discuss their trajectories and interactions.

Hydrodynamic interactions play a crucial role in the formation of flagellated microswimmer clusters, yet they are still not clearly understood. In this article we try to elucidate the influence mechanism of the flagellum elasticity on the clustering-separation process of two flagellated microswimmers. We first systematically derive an active sphere-rod model from the undulating flagellum model. Based on this simplified model, we then construct autonomous two-dimensional dynamical systems to describe the clustering-separation process of two identical flagellated microswimmers in Newtonian fluid. The theoretical predictions without considering the flagellum elasticity are compared with results from smoothed dissipative particle dynamics simulations in which fluid-structure interactions are fully resolved. The flagellum elasticity is identified to be the key factor that alters the phase portrait and even triggers a clustering-separation transition. The elastic flagella are found to bend under the influence of nonuniform hydrodynamic forces induced by the nearby microswimmer. This bending leads to curved mean centerlines which generate rotational torques on the microswimmers thereby causing them to separate. With the inclusion of the bending effect, the theoretical model is also able to reproduce the clustering-separation transition, and predicts the appearance of a saddle point in a specific phase regime. However, fluid viscoelasticity has a much smaller impact on the phase portrait, it only considerably affects the temporal trajectory of the system. Our results highlight the critical roles that flagellum elasticity may play in the cluster formation of flagellated microswimmers.

We develop a comprehensive three-dimensional theory to describe the hydrodynamics of magnetic nanoparticles exposed to an external time-dependent magnetic field. This theory extends the mobility matrix formalism developed by J. Happel, to (quasi-) magnetostatic interactions and incorporates Brownian motion as a background actuation source. Motions of both ferromagnetic and superparamagnetic nanorods are calculated for a large variety of magnetic fields: constant or sinusoidally varying, as well as fully or partially rotating. The complete three-dimensional dynamics of such nanoparticles, mostly in analytical forms, is obtained, revealing transient behaviors and details of the alignment dynamics of the nanorod along the direction of the field or into its plane of rotation. It can also be used to measure hydrodynamical parameters such as the dynamic viscosity of the fluid. Applying these results to biological tissues and attaining the intracellular spacescale resolution is of significant interest to elucidate biological processes occurring therein. Extension to incorporate viscoelastic effects is a next stage that seems fully compatible with this approach.