Understanding limit-cycle oscillators' collective dynamics and phase transitions is crucial in science and engineering. We study a system of identical nonisochronous Stuart-Landau oscillators with asymmetric and symmetric couplings on a ring network. Due to the interplay of the two couplings, oscillators can have various modes of phase relationships: phase-lockings, and phase-drifts. Phase-locked states give rise to different symmetric collective states, whereas phase-drift states lead to asymmetric collective states. We derive the common frequency and amplitude solutions of oscillators in phase-locking states. By defining an order parameter for the system and computing quantities such as effective frequencies, amplitudes, and phase dispersions, we obtain a phase diagram of the system and identify distinct collective dynamical regimes: (i) Complete phase locking, resulting in full synchronization. (ii) Phase drifts among oscillators, leading to quasiperiodic traveling waves. (iii) Uniform phase distributions among oscillators, giving rise to various splay states. (iv) Oscillation death with different steady states. Continuous and discontinuous phase transitions operate across different collective dynamical regimes. The phase transition operating across complete synchronization and traveling wave states is continuous, while the phase transition operating across traveling wave and splay states is discontinuous and is associated with hysteresis. Furthermore, the oscillators impart an explosive oscillation death, even if all the oscillators have the same frequency, along the routes across the splay and complete synchronization states. Finally, we highlight the implications for biological and physical applications.

We examine learning dynamics in deep recurrent networks, focusing on the behavior near the boundary in the depth-width plane separating under- from overparametrized networks, known as the interpolation transition. The training data are Bach chorales in four-part harmony, and the learning is by stochastic gradient descent with a cross-entropy loss function. We find critical slowing down of the learning, approaching the transition from the overparametrized side: For a given network depth, learning times to reach small training loss values appear to diverge proportional to $1/(w-w_c)$ as the width $w$ approaches a (loss-dependent) critical value $w_c$. We identify the zero-loss limit of this value with the interpolation transition. We also study aging (the slowing down of fluctuations as the time since the beginning of learning increases). Taking a system that has been learning for a time ${\tau }_w$, we measure the subsequent mean-square fluctuations of the weight values at times $\tau >{\tau }_w$. In the underparametrized phase, we find that they are well-described by a single function of $\tau /{\tau }_w$. While this scaling holds approximately at short times at the transition and in the overparametrized phase, it breaks down at longer times when the training loss gets close to the lower limit imposed by the stochastic gradient descent dynamics. Both this kind of aging and the critical slowing down are also found in certain spin glass models, suggesting that those models contain the most essential features of the learning dynamics.

While hard-sphere models form the foundation of theoretical condensed matter physics, real systems often exhibit some degree of softness. We present a theoretical and numerical study of a class of nearly hard-sphere systems, generalized Hertzian hyperspheres, where particles interact via a finite-range repulsive potential that allows slight overlaps. Well-studied examples of this class include particles with harmonic repulsions, Hertzian spheres, and Hertzian disks. We derive closed-form expressions for thermodynamic properties, coexistence pressures, and scaling laws governing structure and dynamics. The theory predicts how quantities scale with temperature, density, spatial dimension, and potential softness. These theoretical predictions are tested through numerical simulations in dimensions ranging from one to eight.

Stick-slip on preexisting faults has long been associated with velocity-dependent and time-dependent friction. The description, however, has not adequately addressed position-dependent friction, an aspect that is relevant to spatially complex fault zones in nature. Here we show how the interaction between irregular surfaces results in heterogeneous Coulomb friction along the interface. We then conduct dynamic simulations of a spring-slider model and find that heterogeneity in Coulomb friction alone is capable of reproducing a wide range of stick-slip phenomena that may be associated with complex fault slip behaviors, from fault creep and low frequency earthquakes to ordinary earthquakes and slow slip events.

We consider a coupled atom-photon system described by the Tavis-Cummings dimer (two coupled cavities) in an open environment, to investigate the quantum signature of dissipative chaos. The appropriate classical limit of this model allows us to obtain a phase diagram identifying different dynamical phases, especially the onset of chaos. Both classically and quantum mechanically, we demonstrate the emergence of a steady state in the chaotic regime and analyze its properties. The interplay between quantum fluctuation and chaos leads to enhanced mixing dynamics and dephasing, resulting in the formation of an incoherent photonic fluid. The steady state exhibits an intriguing phenomenon of subsystem thermalization even outside the chaotic regime; however, the effective temperature increases with the degree of chaos. Moreover, the statistical properties of the steady state show a close connection with the random matrix theory, which, in contrast, is absent in the Liouvillian spectrum for this system. Finally, we discuss the experimental relevance of our findings, which can be tested in cavity and circuit quantum electrodynamics (QED) setups.

Physical systems with nonreciprocal or dissipative forces evolve according to a generalization of Liouville's equation that accounts for the expansion and contraction of phase space volume. Here we connect geometric descriptions of these non-Hamiltonian dynamics to a recently established classical density matrix theory. In this theory, the evolution of a “maximally mixed” classical density matrix is related to the well-known phase space contraction rate that, when ensemble averaged, is the rate of entropy exchange with the surroundings. We extend the definition of mixed states to include statistical and mechanical components, describing both the deformations of local phase space regions and the evolution of ensembles within them. As a result, the equation of motion for this mixed state represents the rate of contraction for an ensemble of dissipative trajectories. Recognizing this density matrix as a covariance matrix, its contraction rate is another measure of entropy flow characterizing nonequilibrium steady states.

We study the two-dimensional $(q+r)$-state ferromagnetic Potts model with $q$ interacting visible states and $r$ noninteracting invisible states on a square lattice. We concentrate on the case $q=2$ for various $r$ values. The density of states is obtained by the Wang-Landau Monte Carlo simulation method, by which the specific heat, the partition function zeros, and the probability distribution of the internal energy are calculated. There exists a second-order phase transition for $r<r_c$, while a first-order phase transition appears for $r>r_c$, where $r_c$ is the critical value. From all analyses, we conclude that $r_c\approx 29$ for $q=2$. Calculating the density of visible spins using the Metropolis algorithm, we find that the first-order phase transition is due to the abrupt density change of the visible spins. The critical temperatures are measured to compare with a conjecture.

We disclose a class of stable nonlinear traveling waves moving at specific constant velocities within symmetric two-dimensional quantum droplets. We present a comprehensive analysis of these traveling bubbles and identify three qualitatively distinct regions within the one-parameter family of solutions, classified by velocity: (i) well-separated phase singularities at low velocity, (ii) singularities within the same density dip at intermediate velocity, and (iii) rarefaction pulses without singularities at higher (subsonic) velocities. Then, we generalize the discussion to unstable cases, incorporating higher-order vortex-antivortex pairs and arrays of vortices that move cohesively with a common velocity within the fluid. In all cases, we provide analytic approximations that aid the understanding of the results in different regimes.

A variety of key processes can be mathematically described in terms of directed transport networks, where a scalar (e.g., a signal or a substance) travels through a network until it reaches the exit. Different actions can intervene during transport, causing signal evolution. Here we focus on dispersion due to network connectivity. A mathematical model is built, under the hypothesis of exponential edge travel times. Three network elements are considered: network topology, edge-specific mean travel times, and signal partition at bifurcations. An analytical solution is given, which accounts for possible decay processes. Applications to model transport of conservative scalars in braided rivers and in random and small-world networks show the crucial role played by network-induced dispersion, the dominant role of edge travel times over bifurcation splitting rules, and occurrence of quasi-Gaussian shapes as the signal travels through the network.

Predicting future interactions or novel links in networks is an indispensable tool across diverse domains, including genetic research, online social networks, and recommendation systems. Among the numerous techniques developed for link prediction, those leveraging the networks' community structure have proven highly effective. For example, the recently proposed MapSim predicts links based on a similarity measure derived from the code structure of the map equation, a community-detection objective function that operates on network flows. However, the standard map equation assumes complete observations and typically identifies many small modules in networks where the nodes connect through only a few links. This aspect can degrade MapSim's performance on sparse networks. To overcome this limitation, we propose to incorporate a global regularization method based on a Bayesian estimate of the transition rates along with three local regularization methods. The regularized versions of the map equation compensate for incomplete observations and mitigate spurious community fragmentation in sparse networks. The regularized methods outperform standard MapSim and several state-of-the-art embedding methods in highly sparse networks. This performance holds across multiple real-world networks with randomly removed links, simulating incomplete observations. Among the proposed regularization methods, the global approach provides the most reliable community detection and the highest link prediction performance across different network densities. The principled method requires no hyperparameter tuning and runs at least an order of magnitude faster than the embedding methods.

The study of interlayer similarity of multiplex networks helps us understand the intrinsic structure of complex systems, revealing how changes in one layer can propagate and affect others, thus enabling broad implications for transportation, social, and biological systems. Existing algorithms that measure similarities between network layers typically encode only partial information, which limits their effectiveness in capturing the full complexity inherent in multiplex networks. To address this limitation, we propose an interlayer similarity measuring approach named Embedding Aided Interlayer Similarity (EATSim). EATSim concurrently incorporates intralayer structural similarity and cross-layer anchor node alignment consistency, providing a more comprehensive framework for analyzing interconnected systems. Extensive experiments on both synthetic and real-world networks demonstrate that EATSim effectively captures the underlying geometric similarities between interconnected networks, significantly improving the accuracy of interlayer similarity measurement. Moreover, EATSim achieves state-of-the-art performance in two downstream applications: predicting network robustness and network reducibility, showing its great potential in enhancing the understanding and management of complex systems.

The programming of computations in living cells is achieved by manipulating information flows within genetic networks. Typically, gene expression is discretized into high and low levels, representing 0 and 1 logic values to encode a single bit of information. However, molecular signaling and computation in living systems operate dynamically, stochastically, and continuously, challenging this binary paradigm. While stochastic and probabilistic models of computation address these complexities, there is a lack of work unifying these concepts to implement computations tailored to these features of living matter. Here we design genetic networks for stochastic and probabilistic computing, developing the underlying theory. Moving beyond the digital framework, we propose random pulses and probabilistic-bits (p-bits) as better candidates for encoding and processing information genetic networks. Encoding information through the frequency of expression burst frequency offers robustness to noise, while p-bits enable unique circuit designs with features like invertibility. We illustrate these advantages by designing circuits and providing mathematical models and simulations to demonstrate their functionality. Our approach to stochastic and probabilistic computing not only advances our understanding of information processing in biological systems but also opens new possibilities for designing genetic circuits with enhanced capabilities.

This study investigates the impact of rarefaction and compressibility on the flow dynamics of a lid-rotating cavity, known as Vogel-Escudier (VE) flow, using the direct simulation Monte Carlo (DSMC) method in OpenFOAM. Simulations are performed for Knudsen numbers (${\mathrm{Kn}}_{\mathrm{ref}}$) from 0.025 to 0.5, and lid Mach numbers (${\mathrm{Ma}}_{\mathrm{lid}}$) ranging from 0.5 to 6, corresponding to Reynolds numbers in the range of $1.24\le \mathrm{Re}\le 298.5$. The results reveal that due to the lid rotation, the number density becomes inhomogeneous. This leads to an insensitivity of the temperature and velocity slip to Mach number under supersonic conditions. Substantial deviations from Navier-Stokes-Fourier (NSF) predictions are observed. This slip alters momentum transfer and thermal transport, with heat flux showing anti-Fourier behavior, where heat transfers counter to the temperature gradient, and the NSF equation significantly overpredicts the driving shear stress. The analysis highlights the emergence of continuum breakdown, marked by a gradient-based local Knudsen number (${\mathrm{Kn}}_{\mathrm{g}}$), which effectively delineates regions of continuum and rarefied flow.

Parametrized artificial neural networks (ANNs) can be very expressive ansatzes for variational algorithms, reaching state-of-the-art energies on many quantum many-body Hamiltonians. Nevertheless, the training of the ANN can be slow and stymied by the presence of local minima in the parameter landscape. One approach to mitigate this issue is to use parallel tempering methods, and in this work, we focus on the role played by the temperature distribution of the parallel tempering replicas. Using an adaptive method that adjusts the temperatures in order to equate the exchange probability between neighboring replicas, we show that this temperature optimization can significantly increase the success rate of the variational algorithm with negligible computational cost by eliminating bottlenecks in the replicas' random walk. We demonstrate this using two different neural networks, a restricted Boltzmann machine and a feedforward network, which we use to study a toy problem based on a permutation invariant Hamiltonian with a pernicious local minimum and the $J_1\text{−}{J}_2$ model on a rectangular lattice.

The rheological properties of tangentially propelled flexible polymers under linear shear flow are studied by computer simulations and are compared with analytical calculations. We find a significant impact of the coupled nonequilibrium active and shear forces on the polymer characteristics. The polar activity enhances shear-induced stretching along the flow direction, shrinkage in the transverse direction, and implies a strongly amplified shear-thinning behavior. The characteristic shear rate for the onset of these effects is determined by the activity. In the asymptotic limit of large activities, the shear-induced features become independent of activity and, for asymptotically large shear rates, shear dominates over activity with passive polymer behavior.

Using a wave-number-advection algorithm, we describe a method to add global profile curvature terms to the local gyrokinetic equations. This approach enables a high-precision global simulation capability without sacrificing the efficiency and spectral accuracy of local simulations. Preliminary numerical results suggest that there are two distinct types of global correction to the local result: (1) a trivial shift of the local result in radius and (2) a nontrivial decorrelation that is roughly equivalent to the well-known $\mathbf{E}\times \mathbf{B}$ shear effect. This interpretation provides an intuitive framework for understanding global gyrokinetic simulations.

We observe the formation of staircase patterns in the magnetic potential ($A$) in a weakly magnetized two-dimensional magnetohydrodynamic system driven by a forced, fluctuating vortex array. Layering occurs due to inhomogeneous mixing of $A$ by vortex cells. Magnetic Reynolds number ($R_m$)–dependent quenching of the turbulent diffusion of $A$ by weak magnetic fields increases the disparity between the (short) cell circulation time and the (long) time for intercell transport of magnetic potential. Thus, magnetic fields strengthen transport barriers between cells and reinforce the staircase, relative to its passive scalar counterpart. The analysis reveals a feedback mechanism, which promotes staircase formation. Magnetic staircases persist in both the flux expulsion ($R_m{v}_A^2/U_0^2<1$) and vortex disruption ($R_m{v}_A^2/U_0^2\ge 1$) limits. In the latter case, residual vortex cells homogenize $A$. Global layering morphology is shown to be well characterized by staircase curvature. Stochastic forcing of magnetic potential can support magnetic staircases against resistive decay.

We report numerical simulations of creep flow and yielding of particulate depletion gels under constant shear stress, combining data on different length and time scales. We characterize the statistics of mesoscale strand-breaking events in the gel, which are distributed homogeneously in space, corresponding to macroscopically ductile flow. At the microscale, a spatiotemporal analysis of structural and mechanical metrics connects properties of strands before and after they fail, indicating that strand breakage is statistically predictable. Using results from different scales, we discuss the interplay between creeping and aging dynamics, and we demonstrate a viscosity bifurcation.