The behavior of particles subject to a nonlinear drag force in a medium (nonlinear Brownian motion) is usually treated by means of the Fokker-Planck equation proposed by Klimontovich [Phys. Usp. 37, 737 (1994)]. However, the derivation of such an equation appears to be questionable since it involves unproved assumptions and does not take into account any relation between the assumptions and approximations leading (in kinetic theory) to a velocity-dependent friction coefficient and those necessary to obtain an appropriate kinetic equation. In spite of this, as far as we know, the limits of validity of the said equation have not yet been investigated. Such circumstances pose the general problem: what is (if any) the appropriate (approximate) differential kinetic equation which is really compatible with the Langevin equation for nonlinear Brownian motions? To adequately solve this problem, previous papers dealing with the behavior of heavy particles in light gases are reexamined. The study leads to the final conclusions that (1) the heavy-particle speed dependence of the friction coefficient (and of the fluctuating force), in the Langevin equation, makes it very difficult to derive an appropriate Fokker-Planck equation for nonlinear Brownian motion, particularly in the most general three-dimensional case, and (2) the searched appropriate (approximate) differential kinetic equation for the heavy particles really exists and is the fourth-order equation following from the Boltzmann equation for a quasi-Rayleigh gas. Then it is shown that, on the contrary, the equation for the heavy-particle mean velocity relaxation following from the Fokker-Planck equation proposed by Klimontovich conflicts with that directly derived from the Langevin equation, a circumstance which now proves the inadequacy of the Klimontovich approach. Subsequently, some relaxation processes of particular interest are studied. So both the correct time-dependent mean velocity and the mean distance traveled by an anisotropic swarm of heavy particles are obtained, and the results are discussed assuming an interaction force between heavy- and gas-particles proportional to inverse powers of their mutual distance. Moreover, the correct relaxation equation for the heavy-particle mean-square speed is also achieved, and the difference with the corresponding result following from the Klimontovich theory is pointed out. An approximate method of solution of the said correct relaxation equation is also proposed. Finally, the limits of validity of the present theory are discussed.

This work presents a quantum graph neural network architecture in which each edge of the input graph carries a $d$-level quantum system and the model acts on the associated line graph. The set of circuits generated by the model is dense in the unitary group of the total Hilbert space, subject to an irrationality condition on the angular parameters. The model constructively separates pairs of 1-WL-indistinguishable graphs whose line graphs are $k$-WL-distinguishable, making it a quantum model that lies beyond the 1-WL ceiling. The function class realizable at dimension $d$ admits a strictly positive residual loss floor, which yields a criterion for adaptive expansion of the local dimension. Numerical validation confirms the impossibility of separation by node-based architectures for $\mathrm{SRG}(16,6,2,2)$, establishes the 3-WL distinguishability of its line graphs, and demonstrates direct separation by the edge-qudit architecture between the Petersen graph and the pentagonal prism. Taken together, these results establish the structural expressivity of edge qudits as an advantage over existing node-based architectures.

The melting of stripe-forming systems involves changes in connectivity that are not fully captured by conventional structural and orientational descriptors. We investigate a two-dimensional model with competing interactions whose low-temperature phase consists of one-particle-thick stripes. Molecular dynamics simulations along seven heating isochores are analyzed using thermodynamic, orientational, dynamical, and graph-based observables. Heating produces a multistage reconstruction in which the loss of stripe alignment and the reorganization of filament connectivity occur over distinct temperature ranges. Density controls whether the disordered filaments fragment into finite polymerlike clusters or remain joined in a dynamically fluid, system-spanning network. The topological observables distinguish these outcomes, which are not resolved by the thermodynamic and orientational responses alone. Thus, the same ordered stripe microphase can melt into topologically distinct fluids selected by density.

I derive the impedance response of an ideal electrolyte containing an arbitrary number of mobile ionic species between blocking planar electrodes, described by the Poisson-Nernst-Planck equations. By transforming the linearized equations to a charge-salt description, the response is written in terms of a multi-component diffusion-migration matrix and its eigenvalues and eigenvectors. When all diffusivities are equal, the charge mode decouples from the neutral concentration subspace and the classical binary-electrolyte result is recovered. In contrast, unequal diffusivities couple charge relaxation to one or more neutral composition modes. For a ternary electrolyte with two cations and one anion, this coupling produces additional diffusive features and broadens the crossover between resistive and capacitive regimes. The results found in this paper provide a minimal continuum explanation for why mixed electrolytes can display impedance spectra that cannot be interpreted as a simple binary electrolyte with an averaged diffusion coefficient.

The dynamic behavior of the three-phase contact line (TPCL) during droplet evaporation plays a critical role in governing particle assembly processes, enabling spatially controlled deposition of nanomaterials at micro/nanoscales. This phenomenon utilizes evaporation-driven capillary and possible Marangoni flows arising from surface tension gradients to direct particle migration, resulting in pattern formation through TPCL pinning-depinning dynamics and solvent gradient effects. In this study, we report the formation of unique nonconcentric semicircular arrays of nanoparticle deposition on hydrophilic silicon substrates. This patterning is achieved by manipulating TPCL dynamics through gravity-induced asymmetric droplet deformation. A droplet placed on an inclined substrate adopts an asymmetric shape due to the competing effects of surface tension and gravity, producing spatially heterogeneous TPCL motion characterized by distinct pinning and depinning regimes. These events drive directional deposition of suspended particles, facilitating the self-organization of semicircular stripes. The physical features of the stripe patterns, such as density and spacing, are precisely tunable by adjusting the effective parallel gravitational force ($F_{g\parallel }$), which we control via substrate inclination ($\alpha$) and droplet volume ($V$). A universal linear relationship between stripe density and $F_{g\parallel }$ is found, independent of the mode of force modulation. However, subtle changes in the structural morphology of the surface pattern have been observed, resulting in periodic semicircular patterns with varying radii of curvature, while the final structure is significantly influenced by the sample preparation-related memory effect. An assembly mechanism and an empirical relation based on the experimental results are provided to correlate stripe density with $\alpha$ and $V$, capturing the interplay between gravitational forces and TPCL kinetics.

We analyze the finite-size corrections to the crosscap overlap in the two-dimensional classical Ising model along its self-dual critical line. Using a fermionic formulation, we express the lattice crosscap overlap in terms of Bogoliubov angles and develop a contour-integral approach by analytically continuing the lattice momentum to the complex plane. This leads to a remarkably simple expression for the crosscap overlap, which demonstrates that the finite-size corrections decay exponentially with system size. We further derive an exact analytical formula for the corresponding decay constant and show that it is determined by the complex singularity structure of the Bogoliubov angle.

Universal scaling in phase separation is typically assumed to be isotropic in systems with conserved dynamics. Here we show that boundary forcing alone can break this dynamical scaling symmetry, leading to anisotropic coarsening, with different effective global growth laws observed parallel and perpendicular to the boundary. We consider a ternary mixture with two conserved components and a passive species undergoing surface evaporation, which provides a simple setting to investigate this effect. In this case, evaporation leads to a progressive mass loss and to the formation of macroscopic concentration gradients, which, in turn, drive anisotropic coarsening, with different effective growth laws observed parallel and perpendicular to the boundary. At the same time, bulk regions appear to retain the standard Model B scaling, suggesting that the observed anisotropy is mainly induced by boundary fluxes rather than by changes in the intrinsic dynamics. Our results indicate that boundary conditions can play an important role in breaking scaling symmetry and may offer a way to influence coarsening behavior in nonequilibrium phase separation.

Light propagating in a resonant atomic vapor performs a Lévy flight random walk, in which rare long steps dominate the transport. In contrast to the ideal power-law step-length distributions usually assumed in the literature, the actual distribution in atomic vapors is characterized by a local Lévy index $\alpha \left(\ell \right)$ that varies smoothly with the step length $\ell$. Moreover, the walker alternates between two distinct step-length distributions, depending on whether an atomic collision occurs during each scattering event. In this work, we experimentally investigate which effective Lévy index is retrieved from diffuse transmission measurements in such a nonideal Lévy flight, and how the alternation between the two distributions influences the transmission. From measurements in a dense cesium vapor for different system sizes and atomic densities, we show that the measured Lévy index corresponds to the local exponent evaluated at the system size, ${\alpha }_T=\alpha (\ell =L)$. Surprisingly, the measured values are consistent with the step-length distribution associated with atomic collisions, that has heavier tails, even when the collision probability is small. Monte Carlo simulations of walkers alternating between two Lévy-like step-length distributions support the experimental findings. Our results extend the study of Lévy flights to a class of nonideal random walks, with length-dependent local exponents and alternating step-length distributions, that are likely to be relevant in other real physical systems.

Understanding persistence properties of a system with respect to its underlying microscopic dynamics provides important insights into the associated first-passage behavior. Here, we investigate the persistence properties during phase ordering in the two-dimensional ($d=2$) Ising model evolving under competing nonconserved spin-flip and conserved spin-exchange dynamics by means of Monte Carlo simulations at zero temperature. We examine three distinct persistence probabilities: (i) the total persistence probability, defined as the probability that a lattice site has never experienced a change in the sign of the spin residing there; (ii) the spin-flip persistence probability, which exclusively measures the fraction of sites that have never undergone a spin-flip event; and (iii) the composite persistence probability, defined as the fraction of sites that have experienced neither a spin-flip nor a spin-exchange event. In the asymptotic regime, both the total and spin-flip persistence probabilities exhibit identical power-law decay, irrespective of the relative occurrence probability of the spin-flip move $p_r$. The corresponding persistence exponent ${\theta }_i\approx 0.225$, is found to be consistent with the value reported for systems evolving purely under nonconserved dynamics. We further demonstrate that both persistence measures satisfy the scaling relation $d-d_f^i={\theta }_i/{\alpha }_i$, where $d_f^i$ is the fractal dimension of the corresponding persistence lattice and ${\alpha }_i\approx 1/2$ characterizes the asymptotic power-law growth of spatially correlated regions of nonpersistent spins. In contrast, although the composite persistence probability also exhibits asymptotic power-law decay, the corresponding persistence exponent ${\theta }_{\mathrm{c}}$ and fractal dimension $d_f^{\mathrm{c}}$ of the persistence lattice show strong dependence on $p_r$. Combined with the presence of a universal growth exponent ${\alpha }_{\mathrm{c}}\approx 1/2$, this leads to the breakdown of the scaling relation among the characteristic exponents for the composite persistence.

Echo-state networks (ESNs) are known to reproduce the trajectories and main properties of chaotic attractors. Here, we investigate the question of whether ESNs can reproduce rare events in a system with a stationary distribution with slowly decaying bounded tails. In particular, we consider the chaotic four-dimensional competitive Lotka-Volterra system, and quantify extremes using the generalized extreme value distribution. In stationary simulations, the ESN reproduces the stationary distributions and rare event statistics with high accuracy. In addition, the ESN also reproduces rare event statistics for nonequilibrium ensemble simulations for initial conditions relatively close to the underlying attractor. However, the performance of the ESN deteriorates as the ensemble spread increases and trajectories explore regions of phase space not represented in the training data.

Phase transitions constitute fundamental mechanisms underlying abrupt or qualitative changes in the collective dynamics of interacting units across a wide range of natural and engineered systems. In dynamical networks, such transitions lead to significant reorganization in the coordinated behavior of coupled elements. In adaptive dynamical networks, the connectivity evolves dynamically in response to the states of the nodes, resulting in a coevolution of the structure and dynamics. In this work, we report two distinct forms of heterogeneous nucleation that give rise to single-step and multistep phase transitions toward global synchronization in finite-size adaptive networks with connection delays. We demonstrate that the nature of the nucleation transition is governed by both the presence and magnitude of the delay, as well as the class of natural frequency distribution. Using a collective coordinate framework, we develop a mean-field description of cluster dynamics and derive an analytical upper bound condition for the existence of two-cluster states, which shows excellent agreement with numerical simulations. Furthermore, we extend the analysis to systems with distributed delays and obtain corresponding analytical conditions. Our results provide a theoretical framework for understanding synchronization transitions in adaptive networks with time-delayed interactions.

We investigate the sandpile model on complex networks by developing a branching-process framework that explicitly incorporates dissipation during avalanche propagation. Unlike classical branching descriptions, which assume conservative transport and locally treelike independence, the present approach introduces grain-loss effects directly into the offspring distribution, yielding generalized generating functions for dissipative avalanche dynamics. In the dissipative regime, avalanche-size distributions acquire exponential cutoffs while preserving topology-dependent scaling behavior. Numerical simulations confirm the theoretical predictions on sparse random networks and reveal systematic deviations in highly structured topologies. In particular, using Holme-Kim clustered scale-free networks, we show that the avalanche exponent is relatively insensitive to weak-to-moderate clustering but decreases substantially in the strongly clustered regime, especially as the triangle-closing probability approaches unity. This decrease is accompanied by an enhanced probability of large cascades, showing that sufficiently abundant short cycles generate strong correlations that challenge the classical independent-branch approximation. Surprisingly, trees also exhibit substantial deviations from a power law because low edge density and the abundance of leaves constrain avalanche propagation. These results show that dissipation, clustering, and sparse connectivity fundamentally reshape the avalanche size distribution of the sandpile model on networks and establish quantitative limits for branching-process descriptions of avalanche dynamics.

In this work, we integrate theoretical modeling, molecular simulation, and empirical analysis to identify and characterize evolutionary hysteresis. We first show how epistatic interactions create bistable fitness landscapes and structural hysteresis in a two-locus Wright-Fisher model, revealing two distinct hysteresis regimes under cyclic and noisy selection. Notably, an epistatically constrained population achieves maximal average fitness at an intermediate level of environmental stochasticity. We then extend this framework to more complex systems, demonstrating robust hysteresis loops in both a disordered multilocus model and in biophysically realistic simulations of protein structural flexibility. Finally, we present empirical evidence of evolutionary hysteresis. By analyzing two decades of metagenomic time-series data from freshwater C. Nanopelagicaceae experiencing strong seasonal temperature cycles, we find that approximately 63% of seasonally oscillating alleles exhibit nonzero loop area versus temperature, and a majority retain substantial area under any constant phase shift of temperature, inconsistent with a simple evolutionary lag. Together, our results establish hysteresis as a general, measurable feature of evolution with a potential for probing features of complex fitness landscapes.

We report the experimental observation of highly nonlinear coherent structures in a linear magnetized plasma characterized by a strong background density gradient and significant $\mathbf{E}\times \mathbf{B}$ velocity shear under high ion-neutral collisionality. These structures, identified as drift acoustic waves, exhibit large normalized density fluctuations reaching amplitudes of up to $\sim 10\%$ and show periodic sawtoothlike waveforms. These observed waveforms are well described by cnoidal functions, corresponding to stationary nonlinear wave trains. Cnoidal waves are exact solutions of Korteweg–de Vries-type equations, alongside the more commonly studied soliton solutions. To the best of our knowledge, this work presents the first controlled experimental observation of cnoidal wave trains in a highly collisional magnetized plasma through systematic variation of profile gradients. These findings provide important new insights into the nonlinear evolution and saturation of drift acoustic waves in inhomogeneous, sheared, and collisional magnetized plasmas.

There is a growing recognition of diffusion in nonstatic media across a spectrum of scientific fields, from cosmology to biology. This paper focuses on more complex practical scenarios of center-of-mass (c.m.) motion of polymers, which is not only determined by the basic motion of time-changed Brownian motion but also regulated by the fluctuations of polymer size and the nonstatic medium. A diffusing diffusivity (DD) model describing the motion of the c.m. is established. By introducing comoving coordinates and the physical coordinate system, based on the DD model, the important statistical quantities of the c.m. are calculated, and the dynamic behavior of the c.m. is further systematically analyzed. Specifically, the motion characteristics of the c.m. in exponential and power-law media are investigated, leading to several interesting conclusions: in exponential media, the c.m. motion is primarily dominated by the medium; while in power-law media, classification is performed with $\alpha /2$ as the critical value, and the c.m. is dominated by the medium and time-changed Brownian motion, respectively. In particular, the kurtosis reveals that DD-induced non-Gaussianity persists in exponentially expanding media and in strongly expanding power-law media. Finally, we derive thoroughly the Feynman-Kac equation corresponding to the DD model under time-changed Brownian motion in nonstatic media and verify its correctness.

Granular discharge through a single orifice is classically described by the Beverloo law, but this localized geometry becomes prone to clogging when the aperture size $D$ is only a few grain diameters $d$. Here, we study dry avalanches of monodisperse glass beads over an inclined perforated mesh as a minimal system for distributed granular filtration. By measuring the mass passing through the mesh as a function of the downslope position, we show that the deposited mass profiles are independent of the initial released mass and are described by a spatial leakage law characterized by the filtration length $\ell$. This length increases with the size ratio $\chi =d/D$, indicating that grain avalanches travel longer distances before being filtered as they approach the aperture width. In contrast to localized silolike discharge, which is suppressed beyond an aperture-size threshold, the inclined mesh remains operational over the explored range by providing a sequence of distributed passage opportunities. A comparison with localized discharge and a first-passage model identify the aperture-scale passage probability as the quantity controlling the macroscopic filtration length. These results connect Beverloo-type discharge, first-order sieving kinetics, and spatially distributed granular filtration.

Stochastic dynamics with state-dependent diffusion arise broadly in confined, anisotropic, and hydrodynamically coupled systems. The conventional Langevin equation contains a spurious drift associated with multiplicative noise. Since the restricted free energy is generally nonscalar, the covariance is not explicit. Here, we reformulate state-dependent Langevin dynamics using a scalar free energy and a diffusion metric defined by the inverse diffusion tensor. The conventional spurious drift is then expressed as a Christoffel contribution. Although our formulation is mathematically equivalent to the conventional one through the relation between the restricted and scalar free energies, it makes coordinate covariance explicit and enables the phenomenological construction of thermodynamic potentials directly from system symmetries. We demonstrate its consistency in representative examples of state-dependent diffusion arising from coordinate transformations, geometric confinement, and projection from curved to flat spaces.

For standard percolation, directionality is known to increase the complexity of percolation, requiring the distinct statistical treatment of in messages and out messages for determining the percolation order parameters. Here we show that when connectivity is ensured by paths of length $R\ge 2$, network directionality, impeding backtracking, can significantly reduce the complexity of percolation. To illustrate this finding, we introduce directed extended-range percolation (DERP), defined on directed networks with nonreciprocal edges and motivated by applications in quantum communication. In this framework, message transmission is enabled between trusted nodes separated by a directed path of length at most $R$. Using a message-passing approach, we show that, on locally treelike structures, directionality enables an exact determination of the percolation threshold and the anomalous critical indices with a number of distinct statistical messages smaller than the naive expectation, which would require twice as many messages as in the undirected version. This reduction arises because directionality suppresses backtracking paths, thereby considerably simplifying the resulting equations. On random directed networks we find that the critical behavior of DERP depends sensitively on degree correlations. These analytical predictions are corroborated by extensive Monte Carlo simulations, highlighting the profound impact of directionality and correlations on long-range connectivity in complex networks.

We investigate the optimization of nuclear fusion reactivity relative to the choice of energy probability distribution. A single Maxwell-Boltzmann (MB) distribution is found to be optimal in the region where reactivity is a concave function of the average available kinetic energy, specifically near the reactivity peak. Conversely, in regions where the reactivity curve is convex, optimization is achieved through a bimodal MB distribution. We then discuss our results in the context of three classes of generalized energy distributions (superstatistics), which are modeled as a continuum of MB distributions. As expected, in all cases examined, the reactivity gains provided by superstatistics are bounded by the proposed optimal distribution. Finally, we apply these findings to deuterium-deuterium and deuterium-tritium fusion processes using empirical parametrizations of their respective cross sections. For both reactions, employing optimal bimodal MB distributions at low average energies—of particular interest for magnetically confined plasmas—yields a reactivity enhancement of approximately a factor of 2.