The cover-time problem, i.e., the time to visit every site in a system, is one of the key issues of random walks with wide applications in natural, social, and engineered systems. Addressing the full distribution of cover times for random walk on complex structures has been a long-standing challenge and has attracted persistent efforts. Usually it is assumed that the random walk is noncompact, to facilitate theoretical treatments by neglecting the correlations between visits. The known results are essentially limited to noncompact and homogeneous systems, where different sites are on an equal footing and have identical or close mean first-passage times, such as random walks on a torus. In contrast, realistic random walks are prevailingly heterogeneous with diversified mean first-passage times. Does a universal distribution still exist? Here, by considering the most general situations of noncompact random walks, we uncover a generalized rescaling relation for the cover time, exploiting the diversified mean first-passage times that have not been accounted for before. This allows us to concretely establish a universal distribution of the rescaled cover times for heterogeneous noncompact random walks, which turns out to be the Gumbel universality class that is ubiquitous for a large family of extreme value statistics. Our analysis is based on the transfer matrix framework, which is generic in that, besides heterogeneity, it is also robust against biased protocols, directed links, and self-connecting loops. The finding is corroborated with extensive numerical simulations of diverse heterogeneous noncompact random walks on both model and realistic topological structures. Our technical ingredient may be exploited for other extreme value or ergodicity problems with nonidentical distributions.

We calculated virial coefficients up to the eighth order for hard dumbbells in two-, three-, and four-dimensional Euclidean spaces employing Mayer-sampling Monte Carlo simulations. We improved and extended available data in two dimensions, provide virial coefficients in ${\mathbb{R}}^4$ in dependence on their aspect ratio, and recalculated virial coefficients for three-dimensional dumbbells. Highly accurate, semianalytical values for the second virial coefficient of homonuclear, four-dimensional dumbbells are provided. We compare the influence of the aspect ratio and the dimensionality to the virial series for this concave geometry. Lower-order reduced virial coefficients ${\tilde{B}}_i=B_i/B_2^{i-1}$ depend in first approximation linearly from the inverse excess part of their mutual excluded volume.

With the finite-tape autonomous information ratchet modeled by He et al. [Phys. Rev. E 105, 054131 (2022)], we recast the information processing second law, giving a tighter bound on the work extracted, in terms of the marginal bit-ratchet distribution defined from the joint tape-ratchet distribution. The marginal distribution is further utilized to probe and elucidate the conditions that lead to the presence of equilibrium and nonequilibrium stationary states in general, which are related to the effects of correlation. Applying our analysis to two designs of this information ratchet, where correlations within manifest differently, we uncover the mathematical condition for equilibrium stationary states for information ratchets that harness correlation, to identify them for engine operation during the transient phase.

The coupled dynamics of two similar and disparate electrochemical cells oscillators are analyzed. For the similar case, the cells are intentionally operated at different system parameters such that they exhibit distinct oscillatory dynamics ranging from periodic to chaotic. It is observed that when such systems are subjected to an attenuated coupling, implemented bidirectionally, they undergo a mutual quenching of oscillations. The same holds true for the configuration wherein two entirely different electrochemical cells are coupled via bidirectional attenuated coupling. Therefore, the attenuated coupling protocol seems to be universally efficient in achieving oscillation suppression in coupled oscillators (similar or heterogeneous oscillators). The experimental observations were verified by numerical simulations using appropriate electrodissolution model systems. Our results indicate that quenching of oscillations via attenuated coupling is robust and therefore could be ubiquitous in coupled systems with a large spatial separation prone to transmission losses.

We consider the nonlinear dynamics of a diatomic polar molecule under a linearly polarized laser field. We assume a model in which the molecule dipole is coupled with a time-dependent electric field. This system presents a bound energy region where the atoms are bound, and a free-energy region where the atoms are dissociated. Due to the nonalignment between the dipole axis and the laser direction, and the time dependence of the external field, this system presents two and a half degrees of freedom, namely the vibrational degree, the rotation degree, and the time. To investigate the system dynamics, instead of using the Poincaré surface-of-section technique, we propose the use of the Lagrangian descriptor associated with the escape times. The Lagrangian descriptor is a quantity that reveals complex structures in the phase space, whereas the escape times are the time span in which a trajectory is initially in the bound region before escaping to the unbound region. The combination of these two quantities allows us to distinguish between real stability regions from other complex structures, including stickiness regions, and a different formation, which we call escape islands. With the help of these tools, we find that for high-field amplitudes the inclusion of rotation leads to an increase of the stability regions, which implies a decrease of the dissociation in comparison with the one-dimensional case.

This paper shows that the configurational temperature of liquid-state theory, $T_{\mathrm{conf}}$, defines an energy scale, which can be used for adjusting model parameters of active Ornstein-Uhlenbeck particle (AOUP) models in order to achieve approximately invariant structure and dynamics upon a density change. The required parameter changes are calculated from the variation of a single configuration's $T_{\mathrm{conf}}$ for a uniform scaling of all particle coordinates. The resulting equations are justified theoretically for models involving a potential-energy function with hidden scale invariance. The validity of the procedure is illustrated by computer simulations of the Kob-Andersen binary Lennard-Jones AOUP model, showing the existence of lines of approximate invariance of the reduced-unit radial distribution function and time-dependent mean-square displacement.

This paper proposes using the configurational temperature $T_{\mathrm{conf}}$ for quantifying how far an active-matter system is from thermal equilibrium. We measure this “distance” by the ratio of the systemic temperature $T_{\mathrm{s}}$ to $T_{\mathrm{conf}}$, where $T_{\mathrm{s}}$ is the canonical-ensemble temperature for which the average potential energy is equal to that of the active-matter system. $T_{\mathrm{conf}}$ is “local” in the sense that it is the average of a function, which depends only on how the potential energy varies in the vicinity of a given configuration. In contrast, $T_{\mathrm{s}}$ is a global quantity. The quantity $T_{\mathrm{s}}/T_{\mathrm{conf}}$ is straightforward to evaluate in a computer simulation; equilibrium simulations in conjunction with a single steady-state active-matter configuration are enough to determine $T_{\mathrm{s}}/T_{\mathrm{conf}}$. We validate the suggestion that $T_{\mathrm{s}}/T_{\mathrm{conf}}$ quantifies the deviation from thermal equilibrium by data for the radial distribution function of the 3D Kob-Andersen and 2D Yukawa active-matter models with active Ornstein-Uhlenbeck and active Brownian Particle dynamics. Moreover, we show that $T_{\mathrm{s}}/T_{\mathrm{conf}}$, structure, and dynamics of the homogeneous phase are all approximately invariant along the motility-induced phase separation boundary in the phase diagram of the 2D Yukawa model. The measure $T_{\mathrm{s}}/T_{\mathrm{conf}}$ is not limited to active matter and can be used for quantifying how far any system involving a potential-energy function, e.g., a driven Hamiltonian system, is from thermal equilibrium.

It has been shown that a nonvibrating magnetic granular system, when fed by an alternating magnetic field, behaves with most of the distinctive physical features of active matter systems. In this work, we focus on the simplest granular system composed of a single magnetized spherical particle allocated in a quasi-one-dimensional circular channel that receives energy from a magnetic field reservoir and transduces it into a running and tumbling motion. The theoretical analysis, based on the run-and-tumble model for a circle of radius $R$, forecasts the existence of a dynamical phase transition between an erratic motion (disordered phase) when the characteristic persistence length of the run-and-tumble motion, ${\ell }_c<R/2$, to a persistent motion (ordered phase) when ${\ell }_c>R/2$. It is found that the limiting behaviors of these phases correspond to Brownian motion on the circle and a simple uniform circular motion, respectively. Furthermore, it is qualitatively shown that the smaller the magnetization of a particle, the larger the persistence length. It is so at least within the experimental limit of validity of our experiments. Our results show a very good agreement between theory and experiment.

We elucidate the theoretical relationships among fundamental physical concepts that are involved in the diffuse interface modeling for an isothermal single-component liquid-vapor system, which cover both the equation of state (EOS) and the surface tension force. As an example, a flat surface at equilibrium is discussed both theoretically and numerically by using two different approaches. Particularly, the force structure in the transition region is clearly presented, which demonstrates that the capillary contributions due to the density gradients can suppress the mechanical instability of the thermodynamic pressure and lead to constant hydrodynamic pressure (and chemical potential). Then, by comparing with the van der Waals (vdW) EOS for a flat interface at equilibrium, it is shown that applying the double-well approximation can give qualitative predictions for relatively high density ratio (${\rho }_l/{\rho }_g=7.784$) and satisfactory results for relatively low density ratio (${\rho }_l/{\rho }_g=1.774$). The main cause for this observation is attributed to the nonlinear variation of the generalized coefficient function in the double-well formulation at different density ratios. In addition, for the latter case, we simulate a droplet impact on a hydrophilic wall by using a recently proposed well-balanced discrete unified gas kinetic scheme (WB-DUGKS), which justifies the applicability of the double-well approximation to complex interfacial dynamics in the low-density-ratio limit. Furthermore, the reason for the inconsistency between the coefficients of the mean-field force expressions in the existing literature is explained.

We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is, the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the Euclidean case. We explain this anomaly by a careful rederivation of the Kramers-Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature $T$ and the fraction of antiferromagnetic bonds $p$. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first-order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.

We consider random walks evolving on two models of connected and undirected graphs and study the exact large deviations of a local dynamical observable. We prove, in the thermodynamic limit, that this observable undergoes a first-order dynamical phase transition (DPT). This is interpreted as a “coexistence” of paths in the fluctuations that visit the highly connected bulk of the graph (delocalization) and paths that visit the boundary (localization). The methods we used also allow us to characterize analytically the scaling function that describes the finite-size crossover between the localized and delocalized regimes. Remarkably, we also show that the DPT is robust with respect to a change in the graph topology, which only plays a role in the crossover regime. All results support the view that a first-order DPT may also appear in random walks on infinite-size random graphs.

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and it also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of a long time, even in the absence of noise. Noise destroys the correlations and increases diffusion at small friction. Diffusion thus exhibits a nonmonotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained using the well-established relationship between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high-friction limit but not so in the intermediate regime. We explain the deviation in the low- to intermediate-friction regime using the modified random walk theory. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. To quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

Mean-field theory links the physiological properties of individual neurons to the emergent dynamics of neural population activity. These models provide an essential tool for studying brain function at different scales; however, for their application to neural populations on large scale, they need to account for differences between distinct neuron types. The Izhikevich single neuron model can account for a broad range of different neuron types and spiking patterns, thus rendering it an optimal candidate for a mean-field theoretic treatment of brain dynamics in heterogeneous networks. Here we derive the mean-field equations for networks of all-to-all coupled Izhikevich neurons with heterogeneous spiking thresholds. Using methods from bifurcation theory, we examine the conditions under which the mean-field theory accurately predicts the dynamics of the Izhikevich neuron network. To this end, we focus on three important features of the Izhikevich model that are subject here to simplifying assumptions: (i) spike-frequency adaptation, (ii) the spike reset conditions, and (iii) the distribution of single-cell spike thresholds across neurons. Our results indicate that, while the mean-field model is not an exact model of the Izhikevich network dynamics, it faithfully captures its different dynamic regimes and phase transitions. We thus present a mean-field model that can represent different neuron types and spiking dynamics. The model comprises biophysical state variables and parameters, incorporates realistic spike resetting conditions, and accounts for heterogeneity in neural spiking thresholds. These features allow for a broad applicability of the model as well as for a direct comparison to experimental data.

Recurrent neural networks are widely used for modeling spatiotemporal sequences in both nature language processing and neural population dynamics. However, understanding the temporal credit assignment is hard. Here, we propose that each individual connection in the recurrent computation is modeled by a spike and slab distribution, rather than a precise weight value. We then derive the mean-field algorithm to train the network at the ensemble level. The method is then applied to classify handwritten digits when pixels are read in sequence, and to the multisensory integration task that is a fundamental cognitive function of animals. Our model reveals important connections that determine the overall performance of the network. The model also shows how spatiotemporal information is processed through the hyperparameters of the distribution, and moreover reveals distinct types of emergent neural selectivity. To provide a mechanistic analysis of the ensemble learning, we first derive an analytic solution of the learning at the infinitely large network limit. We then carry out a low-dimensional projection of both neural and synaptic dynamics, analyze symmetry breaking in the parameter space, and finally demonstrate the role of stochastic plasticity in the recurrent computation. Therefore, our study sheds light on mechanisms of how weight uncertainty impacts the temporal credit assignment in recurrent neural networks from the ensemble perspective.

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of $N$ nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree $k=2$, the resulting networks consist of cycles. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with $lnN$, while the length of the longest cycle scales with $N$. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles $\langle S\rangle$ scales with $lnN$. Here we present exact analytical results for the distribution $P_N(S=s)$ of the number of cycles $s$ in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large $N$ limit. The moments and cumulants of $P_N(S=s)$ are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of $N$ objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

A multiplex is a collection of network layers, each representing a specific type of edges. This appears to be a genuine representation of many real-world systems. However, due to a variety of potential factors, such as limited budget and equipment, or physical impossibility, multiplex data can be difficult to observe directly. Often, only partial information on the layer structure of the system is available, whereas the remaining information is in the form of a single-layer network. In this work we face the problem of reconstructing the hidden multiplex structure of an aggregated network from partial information. We propose an algorithm that leverages the layerwise community structure that can be learned from partial observations to reconstruct the ground-truth topology of the unobserved part of the multiplex. The algorithm is characterized by a computational time that grows linearly with the network size. We perform a systematic study of reconstruction problems for both synthetic and real-world multiplex networks. We show that the ability of the proposed method to solve the reconstruction problem is affected by the heterogeneity of the individual layers and the similarity among the layers. On real-world networks, we observe that the accuracy of the reconstruction saturates quickly as the amount of available information increases. In genetic interaction and scientific collaboration multiplexes, for example, we find that $10\%$ of ground-truth information yields $70\%$ accuracy, while $30\%$ information allows for more than $90\%$ accuracy.

The movement of a group of biological individuals, such as fish schools, can evolve from disordered motions to synergistic movements or even ordered patterns. However, the physical origins behind such emergent phenomena of complex systems remain elusive. Here, we established a high-precision protocol for studying the collective behavior of biological groups in quasi-two-dimensional systems. Based on our video recording of $\sim 600\mathrm{h}$ of fish movements, we extracted a force map of the interactions between fish from their trajectories using the convolution neural network. Presumably, this force implies the fish's perception of the surrounding individuals, the environment, and their response to social information. Interestingly, the fish in our experiments were predominantly in a seemingly disordered swarm state, but their local interactions were clearly specific. Combining such local interactions with the inherent stochasticity of the fish movements, we reproduced the collective motions of the fish through simulations. We demonstrated that a delicate balance between the specific local force and the intrinsic stochasticity is essential for ordered movements. This study presents implications for self-organized systems that use basic physical characterization to produce higher-level sophistication.

A three-dimensional blunt-base bluff body in a uniform flow is subjected to long-time stochastic dynamics of switching between two opposite wake states. This dynamic is investigated experimentally within the Reynolds number range Re $\simeq {10}^4–{10}^5$. Long-time statistics coupled to a sensitivity analysis to the body attitude (defined as the pitch angle of the body with respect to the incoming flow) show that the wake switching rate decreases as Re increases. Equipping the body with passive roughness elements (turbulators) modifies the boundary layers before separation, seen as the inlet condition for the wake dynamic. Depending on their location and Re, the viscous sublayer length scale and the turbulent layer thickness can be modified independently. This sensitivity analysis to the inlet condition shows that a decrease of the viscous sublayer length scale at a given turbulent layer thickness leads to a decrease in the switching rate, whereas the modification of the turbulent layer thickness has almost no effect on the switching rate.

We first derive a set of equations describing general stationary configurations of relativistic force-free plasma, without assuming any geometric symmetries. We then demonstrate that electromagnetic interaction of merging neutron stars is necessarily dissipative due to the effect of electromagnetic draping—creation of dissipative regions near the star (in the single magnetized case) or at the magnetospheric boundary (in the double magnetized case). Our results indicate that even in the single magnetized case we expect that relativistic jets (or “tongues”) are produced, with correspondingly beamed emission pattern.

We present a data-driven approach to learning surrogate models for amplitude equations and illustrate its application to interfacial dynamics of phase field systems. In particular, we demonstrate learning effective partial differential equations describing the evolution of phase field interfaces from full phase field data. We illustrate this on a model phase field system, where analytical approximate equations for the dynamics of the phase field interface (a higher-order eikonal equation and its approximation, the Kardar-Parisi-Zhang equation) are known. For this system, we discuss data-driven approaches for the identification of equations that accurately describe the front interface dynamics. When the analytical approximate models mentioned above become inaccurate, as we move beyond the region of validity of the underlying assumptions, the data-driven equations outperform them. In these regimes, going beyond black box identification, we explore different approaches to learning data-driven corrections to the analytically approximate models, leading to effective gray box partial differential equations.