We discuss the order statistics of the particle positions of a gas of $N$ identical independent particles performing Brownian motion in one dimension in a potential that asymptotically behaves like $V\left(x\right)\sim x^{\gamma }$ for $x\to +\infty$, with a positive power $\gamma >0$. We show that in the stationary state, the order statistics that describe how the leaders are reshuffled are universal and independent of $\gamma$. What depends on $\gamma$ is the timescale of the leaders' reshuffling, which scales as a power of the logarithm of the population size: $t\sim {(lnN)}^{\frac{2(1-\gamma )}{\gamma }}\tau$, where $\tau$ is of order one. We derive the probability that the particle which has the $k$-th largest value of $x$ at some time $t_1$ will have the $j$-th largest value at time $t_2=t_1+t$ in the form of an explicit expression for the generating function for the reshuffling probabilities for all $k\ge 1$ and $j\ge 1$. The generating function, expressed in scaled time $\tau$, is independent of $\gamma$. In particular, we show that the average percentage overlap coefficient of leader lists takes the universal, $\gamma$-independent form $\mathrm{erfc}\left(\sqrt{\tau }\right)$ for long lists.

How should one define thermodynamic quantities (internal energy, work, heat, etc.) for quantum systems coupled to their environments strongly? We examine three definitions, inspired by the literature, of a quantum system's internal energy under strong-coupling conditions. Each internal-energy definition implies a definition of work and a definition of heat. Our study focuses on quenches, common processes in which the Hamiltonian changes abruptly. In these processes, the first law of thermodynamics holds for each set of definitions by construction. However, we prove that only two sets obey the second law. We illustrate our findings using a simple spin model. Our results guide studies of thermodynamic quantities in strongly coupled quantum systems.

The fast-ignition paradigm for inertial confinement fusion allows for extremely high gains but requires fuel to be heated very quickly to outpace hotspot disassembly and energy losses. This demands lasers with high power and intensity, posing engineering challenges that have called into question the fundamental practicality of fast ignition. Magnetized liner inertial fusion (MagLIF) circumvents these problems through its large-aspect-ratio cylindrical geometry and strong axial magnetic fields that allow for ignition at lower areal densities. Furthermore, MagLIF's large aspect ratio and higher yields relax other constraints on energy deposition and repetition rate, while its axial magnetic fields can be used to collimate ignitor electrons and thereby increase allowed standoff distance and save on ignitor energy. This tremendous overall relaxation of the engineering constraints that have historically limited the practicality of fast ignition suggests that the paradigm may be considerably more viable in a MagLIF context.

Understanding the atomic-scale dynamics of liquids in the terahertz (THz) regime remains challenging because liquids sustain phonon-like collective excitations governed by complex interactions. To clarify the role of hydrogen bonding in such dynamics, we performed inelastic neutron scattering on hydrogen sulfide $\left({\mathrm{H}}_2\mathrm{S}\right)$ and its deuterated form $\left({\mathrm{D}}_2\mathrm{S}\right)$, molecular analogues of water without a hydrogen-bonded network. By combining coherent and incoherent measurements, we show that liquid ${\mathrm{H}}_2\mathrm{S}$ exhibits a clear fast-sound mode, with a propagation velocity about twice the adiabatic sound speed, as in water. In contrast to hydrogen-bonded liquids, which display a secondary, nearly momentum-independent mode at 6–7 meV, no corresponding feature is observed in liquid ${\mathrm{D}}_2\mathrm{S}$. This absence highlights the central role of hydrogen bonding in enabling additional low-energy excitations. Overall, while fast sound is not exclusive to hydrogen-bonded systems, the detailed vibrational spectrum is strongly shaped by the local bonding environment.

The random clusters introduced by Fortuin and Kasteleyn (FK) and analyzed by Coniglio and Klein (CK) for Ising and related models have led first Swendsen and Wang and then Wolff to formulate remarkably efficient Markov chain Monte Carlo sampling schemes that weaken the critical slowing down. In frustrated models, however, no standard approach has yet been identified to achieve comparable gains in configurational sampling at low frustration—let alone to efficiently sample the strongly frustrated regime. To understand why formulating appropriate cluster criteria for frustrated models has thus far been elusive, we here study minimal short-range attractive and long-range repulsive as well as spin-glass models on Bethe lattices. Using a generalization of the CK approach and the cavity-field method, the appropriateness and limitations of FK-CK-type clusters are identified. We find that a standard, constructive cluster scheme is then inoperable, and that the frustration range over which generalized FK-CK clusters are even definable is finite. These results demonstrate the futility of seeking constructive cluster schemes for frustrated systems but leaves open the possibility that alternate approaches could be devised.

Using numerical simulations, it is shown that a random, athermal pack of soft frictional grains will store an arbitrary waveform that is applied as a small time-dependent shear while the system is slowly compressed. When the system is decompressed at a later time, an approximation of the input waveform is recalled in time-reversed order as shear stresses on the system boundaries. It is shown that this effect depends on friction between the grains, and is independent of some aspects of the friction model. By systematically increasing the complexity of the stored waveform, it is found that a pack of ${10}^4$ grains can recall any one of 128 different waveforms with 100% classification accuracy and 512 different waveforms with over 90% classification accuracy, as measured by a neural net trained only on the inputs. This type of waveform memory might be observable in other types of athermal random media that form internal contacts when compressed, such as crumpled sheets and nestlike fiber assemblies.

We numerically investigate hyperuniformity in two-dimensional jammed packings of frictionless bidisperse particles. Hyperuniformity is characterized by the suppression of density fluctuations at large length scales, and the structure factor vanishes at small wave numbers as $S\left(q\right)\propto q^{\alpha }$, where $\alpha >0$. Jammed packings are known to exhibit hyperuniformity over a wide-wave-number window, down to $q^{*}\sigma \approx 0.2$, where $\sigma$ is the particle diameter. In two dimensions, we find that the exponent $\alpha$ is approximately $0.6–0.7$, in contrast to the reported value $\alpha =1$ in three-dimensional systems. We then extend the analysis to a wide range of particle size ratios, from the monodisperse limit to highly disparate mixtures. To reduce unwanted spatial heterogeneities associated with polycrystalline domains at very large or very small size disparities, we employ a recently proposed method by Rissone et al. [Phys. Rev. Lett. 127, 038001 (2021)]. This method enables a more precise determination of $\alpha$. We find that $\alpha$ remains nearly constant over the entire range of size ratios, except in the monodisperse case, where crystallization occurs.

We show that nonequilibrium dynamics can play a constructive role in unsupervised machine learning by inducing the spontaneous emergence of latent-state cycles. We introduce a model in which visible and hidden variables interact through two independently parametrized transition matrices, defining a Markov chain whose steady state is intrinsically out of equilibrium. Likelihood maximization drives this system toward nonequilibrium steady states with finite entropy production, reduced self-transition probabilities, and persistent probability currents in the latent space. These cycles are not imposed by the architecture but arise from training, and models that develop them reproduce the empirical distribution of data classes more faithfully, with a clear correlation between agreement with the data and entropy production. Compared with equilibrium approaches such as restricted Boltzmann machines, our model breaks the detailed balance between the forward and backward conditional transitions and relies on a log-likelihood gradient that depends explicitly on the last two steps of the Markov chain. Hence, this exploration of the interface between nonequilibrium statistical physics and modern machine learning suggests that introducing irreversibility into latent-variable models can improve the fidelity of the generated data distribution.

We develop a variational neural-network framework to determine the most probable path (MPP) of a 3D active Brownian particle (ABP) by directly minimizing the Onsager-Machlup integral (OMI). To obtain the OMI, we use the Onsager-Machlup variational principle for active systems and construct the Rayleighian of the ABP by including its active power. This approach reveals geometric transitions of the MPP from in-plane I- and U-shaped paths to 3D helical paths as the final time and net displacement are varied. We also demonstrate that the initial and final boundary conditions have a significant impact on the MPPs. Our results show that neural optimization combined with the Onsager-Machlup variational principle provides an efficient and versatile framework for exploring optimal transition pathways in active and nonequilibrium systems.

We investigate phase transitions in the two-dimensional Ising model with competing nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions, where the NNN coupling generates a degenerate superantiferromagnetic (SAF) ground state, by combining machine-learning methods with Monte Carlo simulations and finite-size scaling (FSS). Going beyond earlier studies, we show that unsupervised learning can recover order-parameter and Binder cumulant analogs that obey FSS. Principal component analysis (PCA) reveals the two-component structure of the SAF order parameter, and k-means clustering is used to visualize qualitative differences between ordered and disordered configurations in the principal-component space. A fully connected neural network (FCNN), trained without prior knowledge of the order parameter or the critical temperature, learns the strength of SAF stripe ordering and uses it as the internal feature for phase classification. FSS of thermodynamic observables, the PCA-derived quantity, and the FCNN output yields consistent estimates of the critical temperature and correlation-length exponent. These results demonstrate that machine learning can decode ordering patterns, extract meaningful order-parameter analogs, and reproduce critical behavior in frustrated Ising systems.

The diffusion of ideas and language represents a typical example of collective dynamics in complex systems, conventionally described by S-shaped models such as the logistic function and its generalizations. However, the role of subexponential growth—a slower-than-exponential pattern known in epidemiology—has been largely overlooked in social diffusion processes. Here, we present a piecewise power-law model to characterize complex growth curves with a few parameters. We systematically analyzed a large-scale dataset of approximately one billion Japanese blog articles linked to Wikipedia vocabulary, and observed consistent patterns in web search trend data (English, Spanish, and Japanese). Our analysis of 2963 items, selected for reliable estimation (e.g., sufficient duration/peak, monotonic growth), reveals that 1625 (55%) diffusion patterns without abrupt level shifts were adequately described by one or two segments. For single-segment curves, we found that (i) the mode of the shape parameter $\alpha$ was near 0.5, indicating prevalent subexponential growth; (ii) the peak diffusion scale is primarily determined by the growth rate $R$, with minor contributions from $\alpha$ or the duration $T$; and (iii) $\alpha$ showed a tendency to vary with the nature of the topic, being smaller for niche/local topics and larger for widely shared ones. Furthermore, a microbehavioral model of outward (stranger) versus inward (community) contact suggests that $\alpha$ can be interpreted as an index of the preference for outward-oriented communication. These findings suggest that subexponential growth is a common pattern of social diffusion, and our model provides a practical framework for consistently describing, comparing, and interpreting complex and diverse growth curves.

Granular creep is the slow, sub-yield movement of constituents in a granular packing due to the disordered nature of its grain-scale interactions. Despite the ubiquity of creep in disordered materials, it is still not understood how to best predict the creep-to-failure regime based on the forces and interactions among constituents. To address this gap, we perform experiments to explore creep and failure in quasi-two-dimensional piles of photoelastic disks, allowing the quantification of both grain movements and grain-scale contact force networks. Through controlled external disturbances, we investigate the emergence and evolution of grain rearrangements, force networks, and voids to illuminate signatures of creep and failure. Surprisingly, the force chain structure remains dynamic even in the absence of observable particle motion. We find that shifts in force chains indicate larger, avalanche-scale disruptions. We connect these force signatures with the geometry of the voids in the pile. Overall, our experiments and analyses deepen our mechanical and geometric understanding of the creep-to-failure transition in granular systems.

We use stellar dynamics as a testbed for statistical closure theory. We focus on the process of “Vector Resonant Relaxation,” a long-range, nonlinear, and correlated relaxation mechanism that drives the reorientation of stellar orbital planes around a supermassive black hole. This process provides a natural setting to evaluate the predictive power of generic statistical closure schemes for dynamical correlation functions, in the fully nonlinear and nonperturbative regime. We develop a numerical scheme that explicitly implements the seminal “Martin-Siggia-Rose” formalism at one-loop order via an iterative fixed-point approach, thereby improving upon the bare order from the “Direct Interaction Approximation.” Using this framework, we quantitatively validate the ability of the formalism to predict (i) the two-point two-time correlation function, (ii) the renormalized three-point interaction vertex, and (iii) the three-point three-time correlation function. These predictions are compared to direct measurements from numerical simulations. We conclude by discussing the limitations of this approach and presenting possible future venues.

We study the Susceptible-Infectious-Susceptible model on arbitrary networks. The well-established pair approximation treats neighboring pairs of nodes exactly while making a mean-field approximation for the rest of the network. We improve the method by expanding the state space dynamically, giving nodes a memory of when they last became susceptible. The resulting approximation is simple to implement and appears to be highly accurate, both in locating the epidemic threshold and in computing the quasistationary fraction of infected individuals above the threshold, for both finite graphs and infinite random graphs.

Surfactant enrichment is central to foam-based separation processes, yet the composition of the foam liquid is widely assumed to remain uniform due to efficient dissipation of local concentration gradients. Using contrast-matched small-angle neutron scattering, we directly quantify micelle concentrations within the liquid channels of a steady-state foam over time and height, and distinguish them from surfactant adsorbed at the gas-liquid interface. Despite nearly constant macroscopic structure, the micellar concentration in Plateau borders increases steadily with both foam age and height, reaching more than two orders of magnitude above the initial solution concentration. This study provides the first direct measurement of persistent surfactant accumulation within the foam liquid. The observed gradients are fully accounted for by upward interfacial transport driven by bubble aging, while downward drainage and other equilibration mechanisms play a negligible role. These results overturn the common assumption of rapid surfactant homogenization within foam liquids and imply that micellar gradients must be considered when analyzing separation processes, foam stability, and, more broadly, any phenomenon involving exchange between interfaces and the foam liquid.

Natural killer cells (NKCs) can distinguish the difference between normal and abnormal cells, and kill the latter. Nevertheless, regardless of the past intensive studies on the intracellular crosstalk via extracellular vesicles, the generic dynamical behaviors of the NKCs around its co-cultured cells in their attacking process remains an elusive intriguing issue. Here, using the binary mixtures of NKCs and normal fibroblast/cancer cells, we experimentally reveal the above unexplored issue. It is found that, NKCs initially assess and identify the type of co-cultured cells. For cancer cells, NKCs attach to them, recruit neighboring NKCs to form clusters, and exhibit anchored fluctuating motion around the cancer cells in the killing process, until the death of cancer cells. In response, CCs also aggregate into larger clusters than those of the NKC free case, until their death, to avoid NKC attacking. In contrast, when interacting with normal fibroblasts, NKCs continuously scout around the cells, without forming large clusters. They cause little effect on fibroblast motion.

While nucleation in typical active and driven fluids often appears equilibrium-like, striking departures emerge when large-scale fluctuations are strongly suppressed. Here, we investigate nucleation in nonequilibrium hyperuniform fluids by projecting the full density-field dynamics onto relevant collective variables. We demonstrate that nucleation is governed by a nonequilibrium quasipotential rather than the reversible work of formation. Surprisingly, because of the reduced hyperuniform fluctuations, the nucleation probability no longer separates into the usual surface and volume contributions. Furthermore, accounting for capillary waves reveals a clear breakdown of detailed balance driven by nonreciprocal dynamics. More broadly, our framework can be readily extended to identify nonequilibrium signatures in conventional active fluids.

In business, politics, and life, folk wisdom encourages people to aim for above-average results, but not to let the perfect be the enemy of the good. Here, we mathematically formalize and extend this folk wisdom. We model a time-limited search for strategies having uncertain rewards. At each time step, the searcher is either satisfied with their current reward or continues searching. We prove that the optimal satisfaction threshold is both finite and strictly larger than the mean of available rewards—matching folk wisdom. This result is robust to search costs, unless they are high enough to prohibit all search. We show that being too ambitious has a higher expected cost than being too cautious. We show that the optimal satisfaction threshold increases if the search time is longer, or if the reward distribution is rugged (i.e., has low autocorrelation) or left-skewed. The skewness result reveals counterintuitive contrasts between optimal ambition and optimal risk-taking. We show that using upward social comparison to assess the reward landscape substantially harms expected performance. We show how these insights can be applied qualitatively to real-world settings, using examples from entrepreneurship, economic policy, political campaigns, online dating, and college admissions. We discuss implications of several possible extensions of our model, including intelligent search, reward landscape uncertainty, and risk aversion.

Bacteria track chemical gradients using a biased random walk, a process called chemotaxis. Experiments suggest that bacteria also communicate during this process. Using a mathematical model, we find that sufficiently strong communication succeeds in keeping a population of bacteria together but slows down chemotaxis. However, if the secretion of the communication molecule is coupled to the detection of the external chemoattractant, chemotaxis can be faster than without communication. This result assumes that cells can detect and subtract the average of the chemoattractant over the population, a mechanism we term adaptive communication. Intriguingly, in this regime we predict that, even though blocking the communication receptors should slow down chemotaxis, overexpressing them should also slow it down, and partially blocking or underexpressing them may actually speed it up. Our work provides physical insights on how communication and chemotaxis are connected and may help explain why chemotaxing bacteria communicate.

The percolation phase transition in complex network systems attracts much attention and has numerous applications in various research fields. Finite-size effects smooth the transition and make it difficult to predict the critical point of appearance or disappearance of the giant connected component. To this end, we introduce a new method for computing and measuring susceptibility of arbitrary random undirected and directed networks. The importance of the characteristics is that a strong increase of the susceptibility is the early warning signal of approaching the transition point. Our method is based on the introduction of “observers,” which are randomly chosen nodes monitoring the local connectivity of a network. To demonstrate efficiency of the method, we derive explicit equations determining the susceptibility and study its critical behavior near continuous and mixed-order phase transitions in uncorrelated random undirected and directed networks, networks with dependency links, and $k$-cores of networks. The universality of the critical behavior is supported by the phenomenological Landau theory of phase transitions.