We introduce a neural-network framework for parametrizing fundamental measure theory (FMT) applied to hard-sphere and Lennard-Jones fluids. Lutsko's extended scalar FMT expresses the excess free-energy density through two parameters, $A$ and $B$, which control the relative weight of tensor invariants in the third-order contribution ${\mathrm{\Phi }}_3$; different choices recover established functionals (Rosenfeld, White Bear, and the recent optimized values of Gül et al.). We treat $A$ and $B$ as learnable functions of the packing fraction $\eta =(\pi /6)\rho {\sigma }^3$, predicted by a small neural network and optimized end to end through a JAX-based density functional solver. Four training strategies are examined, namely matching the Carnahan-Starling equation of state, minimizing chemical-potential and compressibility deviations, optimizing wall contact densities, and a combined multiobjective loss. All four converge to parameters near the Percus-Yevick line defined by the constraint combination $C=8A+2B-9\approx 0$, with subpercent agreement with Carnahan-Starling thermodynamics. Extension to Lennard-Jones fluids via Weeks-Chandler-Andersen perturbation theory, using the temperature-dependent Barker-Henderson effective diameter to define an effective packing fraction, yields vapor-liquid coexistence with a critical temperature $T_c^{*}=k_B{T}_c/\epsilon \approx 1.28$, consistent with mean-field DFT benchmarks. The learned parameters remain close to the fixed Lutsko baseline ($A=1, B=0$) throughout, confirming that bulk phase equilibria alone cannot drive nontrivial density dependence. A three-phase training strategy—bulk equation-of-state fitting, test-particle sum-rule optimization, and wall-contact density fine tuning via numerical gradients—breaks this degeneracy, yielding contact densities within 1–2% of molecular dynamics data across all benchmark packing fractions and a 5.5-fold reduction in final wall-contact loss relative to direct wall fine tuning, at the cost of 5–8% bulk thermodynamic errors that quantify a genuine bulk-interface tradeoff. These results provide a starting point for adaptive density functionals trained on inhomogeneous simulation data.

At low temperatures, glasses exhibit distinctive properties compared to crystalline solids. A notable example is the phonon echo, a phenomenon that motivated the two-level-system (TLS) model. This model has successfully explained many universal anomalies in glasses. Here, we extend the TLS framework to a multilevel system and show that phonon echoes persist when nonlinear energy structures and disorder are included. By incorporating virtual phonon exchange, we introduce many-body interactions between these multilevel systems, leading to nonequidistant energy spacings that enhance the echo signal. Interestingly, we find that phonon echo coherence within the TLS model is suppressed under many-body interactions, originating from the fixed natural frequency inherent to each TLS unit. In stark contrast, the multilevel-system model exhibits remarkable robustness against such interactions, arising from anharmonicity-induced frequency adaptability and multiple parallel transition channels. Even if one transition falls out of resonance, the remaining channels can still preserve quantum coherence collectively. Meanwhile, finite-temperature thermal fluctuations cause dephasing, resulting in a decay of echo amplitude over time. Analytical and numerical results, examined perturbatively in both semiclassical and quantum regimes, show consistent agreement. Our work validates this multilevel extension of the TLS framework and underscores the role of many-body interactions in low-temperature glassy dynamics.

An antagonistic random sequential adsorption (RSA) model is investigated to describe competitive adsorption on an adsorbent surface by particles of two types, $a$ and $b$. The degree of competition is characterized by the probability $p_a(p_b=1-p_a)$ that an arriving particle is of type $a$. Particles of both types are modeled as discorectangles with aspect ratio $\varepsilon =l/d$, where $l$ and $d$ denote the particle length and thickness, respectively. Each particle is surrounded by a shell of thickness $\delta$, which induces repulsive interactions between particles of opposite types. This repulsion leads to the spontaneous formation of clusters consisting of particles of a single type, separated from clusters of the opposite type by cavities of thickness approximately $2\delta$. The mechanisms governing the formation of the adsorption layer and the resulting jamming coverage are analyzed. In addition, the dependence of the shell percolation threshold, ${\delta }_c$, on the probability $p_a$ is examined for different values of the shell thickness $\delta$ and the aspect ratio $\varepsilon$.

We extend the $N$ branching Brownian motions model of population invasion to higher-order asexual reproduction. Increasing reproduction order leads to qualitative changes: Invasion fronts generically cease to exist beyond binary reproduction, and in the binary case itself, their speed becomes diffusion independent. Ternary reproduction shows critical behavior, with collapse into a strongly localized “invasion bullet” in the supercritical regime, diffusive spreading in the subcritical regime, and a continuous family of fronts at criticality. These results suggest that the dominance of division and binary reproduction in nature reflects fundamental constraints on invasion dynamics.

The non-Markovian Fokker-Planck equation (FPE) provides an effective description of anomalous diffusion within the classical continuous time random walk framework. However, its nonlocal memory kernel leads to strong non-Markovian behaviors, making the non-Markovian FPE difficult to solve. Conventional numerical approaches must rely on intricate discretized implementations of fractional-calculus differentiation to approximate the fractional operators, leading to substantial computational and storage costs. To overcome these challenges and obtain the probability density function (PDF) across the entire temporal regime, we propose a subordination-based neural network (SNN) that leverages an equivalent subordinated integral representation of non-Markovian dynamics. This representation transforms the original non-Markovian FPE into a Markovian FPE formulated in an operational time where all memory effects are encapsulated by the subordination mechanism, enabling the solution via a deep neural network and direct transfer to FPEs with different memory kernels that share the same auxiliary Markovian FPE, without retraining. We validate the method for non-Markovian FPEs under nonlinear external forces, general boundary conditions, and with arbitrary memory kernels. With four representative examples, the SNN maintains high accuracy in both the central and tail regions of the PDFs, as well as in additional statistical quantities, e.g., survival probability and mean velocity, that are relevant to practical applications. The SNN approach provides a promising route for future extensions to higher-dimensional systems.

We study a probabilistic cellular automaton obtained as a mixture of the additive elementary rules 60 and 102. We prove that for any finite periodic lattice and for mixing parameter $\lambda =1/2$, the system almost surely reaches the absorbing all-zero configuration in finitely many steps. In addition, Monte Carlo simulations indicate as well the presence of a zero-density stationary state in a finite interval around $\lambda =1/2$. Despite this absorbing behavior, both mean-field and block-approximation schemes predict a stationary state with nonzero density. This failure highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata.

The statistics of correlations are central quantities characterizing the collective dynamics of recurrent neural networks. We derive exact expressions for the statistics of correlations of nonlinear recurrent networks in the limit of a large number $N$ of neurons, including systematic $1/N$ corrections, in the regime of Gaussian quenched disorder. Our approach uses a path-integral representation of the network's stochastic dynamics, which reduces the description to a few collective variables and enables efficient computation. This generalizes previous results on linear networks to include a wide family of nonlinear activation functions, which enter as interaction terms in the path integral. These interactions can resolve the instability of the linear theory and yield a strictly positive participation dimension. We present explicit results for power-law activations, revealing scaling behavior controlled by the network coupling. In addition, we introduce a class of activation functions based on Padé approximants and provide analytic predictions for their correlation statistics. Numerical simulations confirm our theoretical results with excellent agreement. We also compare with previous works that have studied the complementary case with annealed disorder, and based on this we propose a conjectural self-consistent equation for the more general case of colored noise.

We performed numerical simulations of two-dimensional magnetohydrodynamic (2D MHD) turbulence on an ${8192}^2$ grid with mean magnetic fields $B_0=0,1$, 3, 6, and 10. The energy spectra and fluxes of Elsässer variables are in better agreement with Kolmogorov-like phenomenology ($k^{-5/3}$) than Iroshnikov-Kraichnan phenomenology ($k^{-3/2}$). Our numerical study shows that $k_{\parallel }\propto k_{\perp }$ for all $B_0$, which is contrary to the predictions of “critical balance.” Additionally, the averaged alignment angle between the velocity and magnetic fluctuations remains constant across all length scales, contrary to the predictions of dynamic alignment. These results indicate that local alignment and antialignment events coexist but statistically balance each other, leading to an overall absence of scale-dependent alignment. Thus, our results demonstrate violations of both critical-balance and dynamic alignment in 2D MHD turbulence. We explore regimes up to $B_0/\delta b\approx 10$, over which $k^{-5/3}$ spectral scaling is observed, relevant to solar-wind turbulence near 1 AU and beyond.

Active polymeric systems exhibit a rich spectrum of nonequilibrium phenomena arising from stochastic forces that explicitly break detailed balance. Despite the rapid growth of experimental and numerical studies, analytical progress remains limited. To date, theoretical understanding relies largely on variants of the active Rouse model, whose formal solutions, though exact, are often obscured by summations over Rouse modes and therefore provide limited direct physical insight. In this work, we develop a transparent scaling theory that captures the tagged-monomer mean-squared displacement (MSD) in active polymers through a compounding formula: The MSD of a monomer in the chain is expressed as that of an isolated active particle, modulated by a connectivity factor encoding tension propagation along the polymer backbone. This approach isolates the role of activity from that of polymer connectivity and reveals the emergent dynamical regimes in a physically intuitive manner. We test the scaling predictions against exact calculations for a broad class of generalized active polymer models driven by diverse noise statistics. The agreement demonstrates the robustness of the scaling framework across microscopic details. Our results provide a simple and extensible theoretical structure that can be applied to complex and analytically intractable active polymer systems, thereby offering a unifying perspective on nonequilibrium polymer dynamics.

A ring polymer in a confining space may exhibit at least two phases, namely, an expanded (or solvent-rich phase) if its concentration is small, or a collapsed (or polymer-rich phase) when it is concentrated and compressed. These phases are discussed in [P.-G. de Gennes (Cornell, Ithaca, New York, 1979)] and have been modeled, traditionally, in the mean field using Flory-Huggins theory [P. J. Flory, J. Chem. Phys. 10, 51 (1942); M. Huggins, J. Am. Chem. Soc. 64, 2716 (1942)]. In three dimensions, the ring polymer may also be knotted, or linked, and have its conformational degrees of freedom constrained by its topology. In a lattice model of confined knotted ring polymers, there are indications that the thermodynamic properties of the ring polymer (for example, the osmotic pressure [F. Gassoumov and E. J. Janse van Rensburg, J. Phys. A: Math. Theor. 52, 025004 (2019); E. J. Janse van Rensburg, Phys. Rev. E 100, 012501 (2019)]) are a function of its topology. In this paper, we explore a lattice knot model of a confined ring polymer as a function of its chemical potential. We show that a well-defined phase transition occurs between solvent-rich and polymer-rich phases when the lattice knot exhibits either the unknot topology or any other fixed knot type. Furthermore, we observe small yet significant variations in the free energy near the critical point when comparing trefoil knots with other nontrivial knot types. These findings indicate that the thermodynamic properties of confined ring polymers depend on their topological entanglement characteristics (namely, their knot type).

We introduce a generalized comb geometry based on a continuous-time random walk that unifies longitudinal advection, transverse trapping, and reversible reactions with temporal memory. By deriving macroscopic evolution equations from this microscopic stochastic process, we establish a direct link between waiting-time statistics and three distinct memory functions. Our framework reveals that temporal memory in longitudinal motion drives long-time superdiffusion, whereas memory in transverse exploration leads to subdiffusion. Furthermore, reaction memory governs the exchange between free and reactive populations, leading to algebraic growth of spatial non-Gaussianity in irreversible regimes and a constant non-Gaussian signature in the transverse direction. We find that ergodicity breaking is determined by the slower of two relaxation processes: branch exploration or reaction unbinding. This work bridges single-particle tracking with continuum modeling, uncovering the competition and synergy among distinct memory mechanisms to provide exact analytical benchmarks for reactive transport in disordered media.

We present an exact method for calculating the large deviation function describing rare fluctuations in the number of particles for product-kernel aggregation. Starting from the master equation, we derive an exact integral representation for the probability $P(M,N,t)$ of observing $N$ particles at time $t$ starting from $M$ monomers for any finite $M,N,t$. From this, we obtain an exact expression for the exponential moment $\langle p^N\rangle$ for integer $p$. Employing a replica conjecture—numerically validated by finite-$M$ scaling—we extend this result to real $p\ge 0$. The convex envelope of the large deviation function, obtained via a Legendre-Fenchel transform of the exponential moment, shows singular behavior. The singular structure allows us to construct the full phase diagram of product-kernel aggregation, which contains a tricritical point, separating continuous and discontinuous transitions. We also compute the asymptotic form of the large-deviation function for small $N/M$.

Bacteriophage-bacteria interactions are central to microbial ecology, influencing evolution, biogeochemical cycles, and pathogen behavior. Most theoretical models assume static environments and passive bacterial hosts, neglecting the joint effects of bacterial traits and environmental fluctuations on coexistence dynamics. This limitation hinders the prediction of microbial persistence in dynamic ecosystems such as soils and oceans. Using a minimal ordinary differential equation framework, we demonstrate that environmental fluctuations can suppress destructive oscillations through resonance, promoting coexistence where static models otherwise predict collapse. Counterintuitively, we find that lower bacterial growth rates are helpful in enhancing survival under high infection pressure, elucidating the observed postinfection growth reduction. Our studies highlight bacterial hosts as active builders of ecological dynamics and environmental variation as a potential stabilizing force. Our findings thus bridge a theory-experiment gap and provide a framework for predicting microbial responses to environmental stress, which might have potential implications for phage therapy, microbiome management, and climate-impacted community resilience as well.

The significance of small-scale forcing of particles on the carrier two-dimensional turbulent flow has been shown to influence the spectral scaling properties of the carrier fluid. We investigate possible consequences of such two-way coupling in a turbulent suspension of inertial particles through one- and two-point Eulerian and Lagrangian statistics. In particular, we find signatures of enhanced intermittency in the vorticity distributions. We characterize the changes in the small-scale geometry of the flow via the Okubo-Weiss parameter. Finally, we examine the scaling properties of the second-order vorticity structure functions and find a nontrivial form of scale invariance at finite mass loading. Motivated by these observations, we propose an effective multiscale forcing framework in which particle feedback is modeled as a spatially localized small-scale forcing. This dual-scale forcing captures the emergence of modified spectral scaling and provides a minimal Eulerian description of particle-laden turbulence that reproduces key statistical signatures of the system.

Dewetting of polymer thin films on nonwettable substrates culminates in a late-stage morphological transition in which a connected fibrillar network fragments into isolated droplets. Because this transformation resembles a depercolation process, it raises the question of whether the network-droplet transition represents a genuine critical phenomenon. We address this question by defining a connectivity-based order parameter and analyzing spatial correlations of the evolving morphology. Although the order parameter exhibits a sharp decrease suggestive of critical behavior, the correlation length of the polymer-rich phase remains finite, evolves through irregular fluctuations, and shows no divergence near the transition. We explain this by noting that the length scales at which the Plateau-Rayleigh instabilities act are much smaller than the large-scale correlations encapsulated in the structure's correlation length. In addition, nondimensionalized order-parameter curves measured at different observation scales fail to collapse onto a universal master curve. These results demonstrate that the breakup of the fibrillar network is governed by localized Plateau-Rayleigh rupture events rather than system-spanning cooperative dynamics. The late-stage transition in dewetting films therefore mimics a hydrodynamic singularity-driven pseudocritical crossover rather than a true critical transition.

Next to the directed-percolation (DP) universality class, parity-conserving directed-percolation (pcDP; also called parity-conserving branching-annihilating random walks) is the second-most important model with an absorbing state transition. Its distinction from ordinary DP is that particle number is conserved modulo 2, which implies that there are two distinct sectors in systems with a finite initial number of particles: Realizations with even and odd particle numbers show different scaling behaviors, and systems in the odd sector cannot die. An intriguing feature of pcDP is that some of its critical exponents seem to be very simple rational numbers. The most prominent is the one describing the average number of particles (or active sites) in the even sector, which is asymptotically constant. In contrast, the dynamical critical exponent (which is the same in both sectors) seems not close to any simple rational. Finally, the order parameter exponent $\beta$ (which is also the same in both sectors) is, according to the most precise previous simulations, rather close to 1, but incompatible with it. We present high statistics simulations which clarify this situation, and which indicate several other intriguing properties of pcDP clusters. In particular, we find that all exponents which were close to rationals are even closer, and $\beta =1.000$ with the error in the next digit.

We study the phase-ordering dynamics of a quasiclassical Holstein model. At half-filling, the zero-temperature ground state is a commensurate charge-density-wave (CDW) with alternating occupied and empty sites. This quasiclassical formulation enables us to isolate the role of electrons in coarsening dynamics. Following a thermal quench, CDW domains grow through the diffusion and annihilation of kinks—topological defects separating the two symmetry-related CDW orders. While standard diffusive dynamics predicts domain sizes scaling as the square root of time, our large-scale simulations reveal a slower power-law growth with a temperature-dependent exponent. We trace this anomalous behavior to a cooperative kink hopping arising from Fermi-Dirac statistics of electrons and quasiconservation of electron numbers. The correlated-hopping of kinks in turn gives rise to an effective diffusion coefficient that depends on the kink density. These results identify a cooperative kink-hopping mechanism for anomalously slow coarsening in the quasiclassical Holstein model and suggest that similar correlated defect dynamics may also be relevant to phase-ordering processes in related electron-phonon systems.

Hierarchical crack patterns that arise during the drying of thin films of colloidal dispersions or polymer solutions on a solid substrate are of interest both from a fundamental standpoint and in the context of the creation of transparent electrodes for optoelectronics. This paper analyzes the morphology of such patterns based on image processing of real-world samples. Three approaches are used to generate artificial hierarchical networks: random homogeneous tessellation, recursive Voronoi tessellation, and a crack growth simulation model, each modified to reproduce the hierarchical structure. A comparison was made of the geometric characteristics (distribution of crack angles, edge lengths, cell areas, and circularity coefficient) and topological properties (distribution of the number of cell sides) of real and simulated networks. It was shown that the simulation model best reproduces the key features of real cracks, including the characteristic right angles of their connections.

We introduce a two-parameter ensemble of random discrete-time Markov models that simultaneously captures critical slowing down and broken detailed balance. Extending a previously studied heterogeneous Markov ensemble, we incorporate correlations between forward and backward transition rates through a single asymmetry parameter $\gamma$, while heterogeneity is controlled by $\epsilon$. Using results from random matrix theory, we identify a critical locus ${\epsilon }_c(\gamma ,N)$ at which relaxation times diverge and spectral universality breaks down, in Markov models with $N$ states. We characterize the behavior of entropy production, predictive information, and relaxation dynamics across the ensemble, showing that many observables depend strongly on heterogeneity but only weakly on asymmetry, except near the symmetric limit. Applying maximum-likelihood inference to human fMRI and EEG data, we find that both modalities operate near the predicted critical locus and occupy a similar region of the $\epsilon -\gamma$ plane, supporting a superuniversality of human brain dynamics. While ensemble averages are well captured by the null model, empirical data exhibit substantially enhanced variability, indicating subject-specific structure beyond random expectations. Our results unify criticality and nonequilibrium measures within a single framework and clarify their intertwined role in the analysis of complex biological dynamics.