Entropy production (EP) is a central quantity in nonequilibrium physics as it monitors energy dissipation, irreversibility, and free energy differences during thermodynamic transformations. Estimating EP, however, is challenging both theoretically and experimentally due to limited access to the system dynamics. For overdamped Langevin dynamics and Markov jump processes it was recently proposed that, from thermodynamic uncertainty relations (TURs), short-time cumulant currents can be used to estimate EP without knowledge of the dynamics. Yet, estimation of EP in underdamped Langevin systems remains an active challenge. To address this, we derive a modified TUR that relates the statistics of two specific currents—one cumulant current and one stochastic current—to a system's EP. These two distinct but related currents are used to constrain EP in the modified TUR. One highlight is that there always exists a family of currents such that the uncertainty relations saturate, even for long-time averages and in nonsteady-state scenarios. Another is that our method only requires limited knowledge of the dynamics—specifically, the damping coefficient to mass ratio and the diffusion constant. This uncertainty relation allows estimating EP for both overdamped and underdamped Langevin dynamics. We validate the method numerically, through applications to several underdamped systems, to underscore the flexibility in obtaining EP in nonequilibrium Langevin systems.

Stochastic resonance is a phenomenon possessed by some nonlinear oscillators, in which a weak signal is boosted by noise. Both the monostable and bistable Duffing oscillators can exhibit this property. However, stochastic resonance has a strong frequency-dependence, as only a band of frequencies may be boosted by noise. Adaptive oscillators are a subset of nonlinear oscillators that can learn the features of an external force. Here, an adaptive state is added to a Duffing oscillator. This adaptive state enables the Duffing adaptive oscillator to outperform stochastic resonance over a wide range of frequencies by learning a resonance condition.

Stochastic perturbations can steer the response of a nonlinear system to specific regions of the system state space, including equilibrium points, periodic solutions, and aperiodic (including chaotic) solutions. This study constructs a discrete partial control scheme to confine the response trajectories of a forced Duffing oscillator, showing transient chaos, within a chaotic attractor, thereby avoiding undesired behavior. The system is analyzed in the presence of white noise, using a minimally bounded control input. The primary motivation is to evaluate the viability of applying bounded control inputs at discrete time instants over intervals differing from the forcing period and to examine the required control input magnitude for response confinement. Results demonstrate that the system can be effectively controlled with relatively small magnitude control inputs applied at discrete time instants. A careful selection of time steps is crucial to ensure that the response trajectory remains within the desired region. The scheme's formulation and the associated algorithm are independent of the system dimension.

We consider evolutionary games in which the agent selected for update compares strategies with $q$ neighbors, rather than a single neighbor as in standard evolutionary game theory. If all $q$ sampled neighbors agree, then agents update their strategy using a smoothed best-response rule. Following ideas from social impact theory, we also study mathematical generalizations of the dynamics to noninteger $q>0$. Looking at fixed point stability and fixation times for $2\times 2$ games with all-to-all interactions, we find that the flow changes significantly as a function of $q$. Further, we develop the pair approximation for the $q$-deformed dynamics on uncorrelated graphs. We also study games with more than two strategies, such as the rock-paper-scissors game where we show that changing $q$ leads to the emergence of new types of flow within the simplex.

Eukaryotic cells perform chemotaxis by determining the direction of chemical gradients based on stochastic sensing of concentrations at the cell surface. To examine the efficiency of this process, previous studies have investigated the limit of estimation accuracy for gradients. However, most studies have treated a circular cell shape, and the few considering elongated shapes assume the elongated direction as fixed. This leaves the question of how adaptive regulation of cell shape affects the estimation limit. Dynamics of cell shape during gradient sensing is biologically ubiquitous and can influence the estimation by altering the way the concentration is measured, and cells may strategically regulate their shape to improve estimation accuracy. To address this gap, we investigate the estimation limits in dynamic situations where elongated cells change their orientation adaptively depending on the sensed signal. We approach this problem by analyzing the stationary solution of the Bayesian nonlinear filtering equation. By applying diffusion approximation to the ligand-receptor binding process and the Laplace method for the posterior expectation under a high signal-to-noise ratio regime, we obtain an analytical expression for the estimation limit. This expression indicates that estimation accuracy can be improved by aligning the elongated direction perpendicular to the estimated direction, which is also confirmed by numerical simulations. Our analysis provides a basis for clarifying the interplay between estimation and control in gradient sensing and sheds light on how cells optimize their shape to enhance chemotactic efficiency.

The inflation of an inner radial (or spherical) cavity in an amorphous solid confined in a disk (or a sphere) served as a fruitful case model for studying the effects of plastic deformations on the mechanical response. It was shown that when the field associated with Eshelby quadrupolar charges is nonuniform, the displacement field is riddled with dipole charges that screen elasticity, reminiscent of Debye monopoles screening in electrostatics. In this paper we look deeper into the screening phenomenon, taking into account the consequences of irreversibility that are associated with the breaking of Chiral symmetry. We consider the equations for the displacement field with the presence of “odd dipole screening,” solve them analytically and compare with numerical simulations. Suggestions how to test the theory in experiments are provided.

The phenomenon of dispersive shock waves (DSWs) exerts a critical influence on nonlinear dynamics in various nonlinear fields, and simulating this complex physical process remains a significant challenge. In this paper, we dramatically enhance the ability of physics-informed neural networks (PINNs) to describe complex dispersive phenomena by integrating residual learning paradigms and introducing a dispersion factor into the existing PINN method. For iteration problems in the time domain, we propose a highly adaptable Deep Runge-Kutta method, which not only ensures high accuracy and flexibility but also facilitates the inference of numerically significant physical state variables. Furthermore, we formulate a physics-guided multistage neural network (PgMSNN) model using a multistage training strategy to address the evolution problem of the generalized Gardner equation with step initial conditions. Representative numerical experiments are conducted in selected regions to study forward problems, model stability, and parameter inversion issues. Comparisons with state-of-the-art numerical simulation neural networks highlight the efficacy of the PgMSNN model in addressing these challenges. This study not only improves the numerical simulation accuracy and stability of PINNs for DSW phenomena but also provides an efficient and flexible tool for the simulation and prediction of complex nonlinear dynamic systems. The successful application of the PgMSNN model highlights the significant potential of physics-guided strategies in deep learning for tackling complex physical problems.

We introduce a projected complex Langevin (CL) numerical sampling method—a fictitious Langevin dynamics scheme that uses numerical projection to sample a constrained stationary distribution with highly oscillatory character. Despite the complex-valued degrees of freedom and associated sign problem, the projected CL method succeeds as a natural extension of real-valued projected Langevin processes. In the newly proposed method, complex-valued Lagrange multipliers are determined to enforce constraints to machine precision at each iteration. To illustrate the efficacy of this approach, we adapt the projected CL method to sample coherent state quantum field theories describing interacting Bose gases, which are realized in modern cold-atom experiments. We apply projected CL to two scenarios with holomorphic constraints, namely the canonical and microcanonical ensembles, and we show that projected CL reproduces the correct thermodynamic observables. We further observe improved numerical stability and accuracy at larger time steps when compared to the previous state-of-the-art method for performing constrained CL sampling.

Focusing on the standard map of the kicked rotor, we investigate the effects of classical stickiness (orbits temporarily confined to a region of a chaotic phase space) on an open quantum system. The regions of stickiness that we identify emerge in the strongly chaotic map. By scanning the system's phase space with a leak, we show that stickiness reduces the degree of delocalization of the quantum states and the decay rate of generic initial states. We find an excellent correspondence between the classical dwell time and finite-time Lyapunov exponents with the quantum dwell time and Wehrl entropies of the quantum states. Knowledge of the structure of the classically chaotic trajectories can thus be used to determine the best placement of leaks to enhance or decrease the degree of delocalization of quantum states and to modify the quantum dynamics of open systems.

Marginal optima are minima or maxima of a function with many nearly flat directions. In settings with many competing optima, marginal ones tend to attract algorithms and physical dynamics. Often, the important family of marginal attractors is a vanishing minority compared with nonmarginal optima and other unstable stationary points. We introduce a generic technique for conditioning the statistics of stationary points in random landscapes on their marginality and apply it in three isotropic settings with qualitatively different structures: in the spherical spin-glasses, where the energy is Gaussian and its Hessian is a Gaussian orthogonal ensemble (GOE); in multispherical spin glasses, which are Gaussian but non-GOE; and in sums of squared spherical random functions, which are non-Gaussian. In these problems, we are able to fully characterize the distribution of marginal optima in the landscape, including when they are in the minority.

Totally asymmetric simple exclusion process (TASEP) sets the paradigm for one-dimensional driven single-file motion. We study a periodic TASEP with two road blocks or defects of different kinds, one point and another extended, across which particle flows are inhibited. We show how the interplay between particle number conservation and competition between the defects leads to inhomogeneous steady states with localized domain walls (LDW). The LDW locations jump discontinuously, indicating a discontinuous transition between these LDW states, as the system passes from being controlled by one defect to the other. When the defects are competing, instead of an LDW, a pair of delocalized domain walls appear, none of which can penetrate the extended defect. A minimum current principle can be used to identify the dominant defect that controls the domain wall formations. Our results should be important in diverse systems, ranging from protein synthesis by ribosomes in biological cells to urban traffic networks.

Periodically driven classical many-body systems can host a rich zoo of prethermal dynamical phases. In this work, we extend the paradigm of classical prethermalization to aperiodically driven systems. We establish the existence of a long-lived prethermal regime in spin systems subjected to random multipolar drives. We demonstrate that the thermalization time scales as ${(1/T)}^{2n+2}$, where $n$ is the multipolar order and $T$ is the intrinsic time-scale associated with the drive. In the $n\to \infty$ limit, the drive becomes quasiperiodic and the thermalization time becomes exponentially long $[\sim exp(\beta /T\left)\right]$. We further establish the robustness of prethermalization by demonstrating that these thermalization time scaling laws hold for a wide range of initial state energy densities. Intriguingly, the thermalization process in these classical systems is parametrically slower than their quantum counterparts, thereby highlighting important differences between classical and quantum prethermalization. Finally, we propose a protocol to harness this classical prethermalization to realize time rondeau crystals.

By using the Krylov subspace technique to generate spin-coherent states in the kicked top model, a prototype model for studying quantum chaos, the accessible system size for studying the Husimi functions of eigenstates can be much larger than that reported in the literature and our previous study [Phys. Rev. E 108, 054217 (2023)]. In the fully chaotic kicked top, we show that the mean Wehrl entropy localization measure approaches the prediction given by the circular unitary ensemble, while the spectral statistics follows the circular orthogonal ensemble. For the mixed-type cases, we identify mixed eigenstates by the overlap of the Husimi function with regular and chaotic regions in classical compact phase space. Numerically, we show that the fraction of mixed eigenstates scales as $j^{-\zeta }$, a power-law decay as the system size $j$ increases, across nearly two orders of magnitude, expanding significantly beyond the previously limited range of system sizes. This provides supporting evidence for the principle of uniform semiclassical condensation of Husimi functions and the Berry-Robnik picture in the semiclassical limit.

In this paper, we study the temporal and spatiotemporal dynamics of a cancer growth model with the Allee effect. For the temporal model, we first prove its well-posedness and analyze its rich local bifurcations, such as the transcritical bifurcation, the saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. For the diffusion system, we first prove that cross-diffusion is a key mechanism for the formation of spatial patterns. Then, we study the Turing instability and derive the amplitude equation by using the weak nonlinear analysis method. Using the stability of the amplitude equation, we discuss the hexagonal patterns and stripe patterns and its combinations of several kinds of patterns under different parameter conditions. Finally, all possible cases of the system under different parameter sets are investigated, and a large number of numerical simulations are carried out to verify the analytical results. Further, we compare the stochastic model with Gaussian white noise to the deterministic model, and we find that the noise can improve the stability.

Coupled phase oscillators with both attractive and repulsive interactions offer a valuable framework for exploring collective dynamics in nonlinear dynamics, serving as an analog to the frustrated Ising model in statistical physics. In this work, we study a ring of phase oscillators with nearest-neighbor and next-nearest-neighbor interactions. The coupling strengths, denoted as $J_1$ and $J_2$, dictate the behavior of these interactions. For an infinite number of oscillators, the model allows for a continuum of equilibria. For identical oscillators, linear stability analysis shows the prevalence of multistability. To complement the linear stability analysis, we delve into the basin stabilities of these equilibria. We categorize them based on basin stability into in-phase equilibria with nearly zero phase difference between adjacent oscillators, antiphase equilibria with nearly $\pi$ phase difference, and heterogeneous equilibria characterized by the phase difference, concentrating on the most unstable modes of the uniform equilibrium. The resulting phase diagram in the $J_1\text{−}{J}_2$ plane mirrors that of the zero-temperature $J_1\text{−}{J}_2$ Ising model. For nonidentical phase oscillators, similar qualitative outcomes are observed.

We investigate the phenomenon of transition to synchronization in the Sakaguchi-Kuramoto model in the presence of higher-order interactions and global order parameter adaptation. The investigation is done by performing extensive numerical simulations and low-dimensional modeling of the system. Numerical simulations of the full system show both continuous (second-order) as well as discontinuous transitions. The discontinuous transitions can either be associated with explosive (first-order) or tiered synchronization states depending on the choice of parameters. To develop an in depth understanding of the transition scenario in the parameter space we derive a reduced order model (ROM) using the Ott-Antonsen ansatz, the results of which closely match with those of the numerical simulations of the full system. The simplicity and analytical accessibility of the ROM help to conveniently unfold the transition scenario in the system having complex dependence on the parameters. Simultaneous analysis of the full system and the ROM clearly identifies the regions of the parameter space exhibiting different types of transitions. It is observed that the second-order transition is connected with a supercritical pitchfork bifurcation (PB) of the ROM. On the other hand, the discontinuous tiered transition is associated with multiple saddle-node (SN) bifurcations along with a supercritical PB and the first-order transition involves a subcritical PB alongside a SN bifurcation. Finally, the stability analysis of the different synchronization states of the system is performed analytically.

It has recently been shown that networks possessing scale-free and fractal properties may exhibit a bifractal nature, in which local structures are described by two different fractal dimensions. In this study, we investigate random walks on such fractal scale-free networks (FSFNs) by examining the walk dimension $d_{\text{w}}$ and the spectral dimension $d_{\text{s}}$, to understand how the bifractality affects their dynamical properties. The walk dimension is found to be unaffected by the difference in local fractality of an FSFN and remains constant regardless of the starting node of a random walk, whereas the spectral dimension takes two values, $d_{\text{s}}^{\text{min}}$ and $d_{\text{s}}^{\text{max}}(>d_{\text{s}}^{\text{min}})$, depending on the starting node. The dimension $d_{\text{s}}^{\text{min}}$ characterizes the return probability of a random walker starting from an infinite-degree hub node in the thermodynamic limit, while $d_{\text{s}}^{\text{max}}$ describes that of a random walker starting from a finite-degree non-hub node infinitely distant from hub nodes and is equal to the global spectral dimension $D_{\text{s}}$. The existence of two local spectral dimensions is a direct consequence of the bifractality of the FSFN. Furthermore, analytical expressions of $d_{\text{w}}, d_{\text{s}}^{\text{min}}$, and $d_{\text{s}}^{\text{max}}$ are presented for FSFNs formed by the generator model and the giant components of critical scale-free random graphs, and are numerically confirmed.

We formulate a general, high-dimensional, partial integrodifferential equation (PIDE) kinetic theory describing the internal state (such as gene expression or protein levels) of cells in a stochastically evolving population. The resolution of our kinetic theory also allows one to track subpopulations associated with each generation. Both intrinsic noise of the cell's internal attribute and randomness in a cell's division times (demographic stochasticity) are fundamental to the development of our model. Using our framework, we are able to marginalize the high-dimensional PIDEs in a number of different ways to derive equations which can be PIDEs themselves) that describe the dynamics of marginalized or “macroscopic” quantities such as structured population densities, moments of generation-dependent cellular states, and moments of the total population. We also show how nonlinear “interaction” terms in lower-dimensional integrodifferential equations can arise from high-dimensional linear kinetic models that contain rate parameters of a cell (birth and death rates) that depend on variables associated with other cells, generating couplings in the dynamics. Our analysis provides a general, more complete mathematical framework that resolves the coevolution of cell populations and cell states. The approach may be tailored for studying, e.g., gene expression in developing tissues, or other more general particle systems which exhibit Brownian noise in individual attributes and population-level demographic noise.

Athermal (i.e., zero-temperature) underconstrained systems are typically floppy, but they can be rigidified by the application of external strain. Following our recently developed analytical theory for the athermal limit, here and in the companion paper, we extend this theory to underconstrained systems at finite temperatures. Close to the athermal transition point, we derive from first principles the partition sum for a broad class of underconstrained systems, from which we obtain analytic expressions for elastic material properties such as isotropic tension $t$ and shear modulus $G$ in terms of isotropic strain $\varepsilon$, shear strain $\gamma$, and temperature $T$. These expressions contain only three parameters, entropic rigidity ${\kappa }_S$, energetic rigidity ${\kappa }_E$, and a parameter $b_{\varepsilon }$ describing the interaction between isotropic and shear strain. We provide analytical expressions for these parameters based on the microscopic structure of the system. Our work unifies the physics of systems as diverse as polymer fibers and networks, membranes, and vertex models for biological tissues.

While renormalization groups are fundamental in physics, renormalization of complex networks remains vague in its conceptual definition and methodology. Here, we propose a novel strategy to renormalize complex networks. Rather than resorting to handling the bare structure of a network, we overlay it with a readily renormalizable physical model, which reflects real-world scenarios with a broad generality. From the renormalization of the overlying system, we extract a rigorous and simple renormalization group transformation of arbitrary networks. In this way, we obtain a transparent, model-dependent physical meaning of the network renormalization, which in our case is a scale transformation preserving the transition dynamics of low-density particles. We define the strength of a node in accordance with the physical model and trace the change of its distribution under our renormalization process. This analysis demonstrates that the strength distributions of scale-free networks remain scale-invariant, whereas those of homogeneous random networks do not.