We present a theory that combines the framework of irreversible thermodynamics with modified integral theorems to model arbitrarily curved and deforming membranes immersed in bulk fluid solutions. We study the coupling between the mechanics and permeability of a viscous and elastically bendable membrane, and a multicomponent bulk fluid solution. An equation for the internal entropy production for irreversibilities at the membrane is derived, determining the generalized thermodynamic forces and fluxes, from which we identify the deviatoric stress as a novel driving force for permeability. A complete set of equations of motion, constitutive laws, and boundary conditions to model the lipid membrane and bulk fluid system are provided.

Large language models (LLMs) have been widely employed across various application domains, yet their black-box nature poses significant challenges to understanding how these models process input data internally to make predictions. In this paper, we introduce a precise and quantitative law that governs the learning of contextualized token embeddings through intermediate layers in pretrained LLMs for next-token prediction. Our findings reveal that each layer contributes equally to enhancing prediction accuracy, from the lowest to the highest layer—a universal phenomenon observed across a diverse array of open-source LLMs, irrespective of their architectures or pretraining data. We demonstrate that this law offers different perspectives and actionable insights to inform and guide practices in LLM development and applications, including model scaling, pretraining tasks, and interpretation.

A network model based on players' aspirations is proposed and analyzed theoretically and numerically within the framework of evolutionary game theory. In this model, players decide whether to cooperate or defect by comparing their payoffs from pairwise games with their neighbors, driven by a common aspiration level. The model also incorporates a degree of stochasticity, or decision noise, through an effective temperature in the Fermi function. The level of cooperation in the system is fundamentally influenced by two social attributes: satisfaction, defined as the fraction of players whose payoffs exceed the aspiration level, and the degree of noise in decision-making. Deterministic players tend to maintain their initial strategies for sufficiently low aspiration levels, while the presence of noise promotes a state of perfect coexistence, resulting in half of the agents cooperating. The transition between these two behaviors can be critical, often leading to abrupt changes in cooperation levels. When the aspiration level is high, all players become dissatisfied, regardless of the effective temperature. Intermediate levels of aspiration lead to diverse behaviors, including sudden transitions for deterministic agents and a nonmonotonic relationship between cooperation and noise intensity. The study also carefully examines the effects of the interaction structure, initial conditions, and the strategy update rule (asynchronous versus synchronous). Special attention is given to the prisoner's dilemma, where significant cooperation levels can be achieved in a structured environment, with moderate aspiration and low decision noise intensity, and following a synchronous strategy updating scheme.

Direct numerical simulation is utilized to resolve the drag and turbulent characteristics of supersonic fully developed channel turbulence in this paper. Distinct from spatially developing turbulence, a relaxation of the skin friction coefficient is discovered after blowing in fully developed turbulence. The blowing intensity affects the oscillation amplitude of the relaxation instead of frequency and tendency, while the blowing length lowers the skin friction in the blowing zone but decreases the global drag reduction rate. Additionally, the drag reduction effect of various blowing intervals is also examined. The research findings indicate that implementing the second blowing at the location where the drag initially declines to its valley during the relaxation process can yield optimal drag reduction results. This blowing interval contributes to a reduction in the velocity gradient and the global Reynolds shear stress, and an enhancement in the lifting effect on the vortices, consequently leading to a further reduction in turbulent skin friction.

We study a system of diffusing point particles in which any triplet of particles reacts and is removed from the system when the relative proximity of the constituent particles satisfies a predefined condition. Proximity-based reaction conditions of this kind are commonly used in particle-based simulations of chemical kinetics to mimic bimolecular reactions (those involving just two reactants) and have been extensively studied. The rate at which particles react within the system is determined by the reaction condition and particulate diffusion. In the bimolecular case, analytic relations exist between the reaction rate and the distance at which particles react allowing modellers to tune the rate of the reaction within their simulations by simply altering the reaction condition. However, generalizing proximity-based reaction conditions to trimolecular reactions (those involving three particles) is more complicated because it requires understanding the distribution of the closest diffusing particle to a point in the vicinity of a spatially dependent absorbing boundary condition. We find that in this case the evolution of the system is described by a nonlinear partial integrodifferential equation with no known analytic solution, which makes it difficult to relate the reaction rate to the reaction condition. To resolve this, we use singular perturbation theory to obtain a leading-order solution and show how to derive an approximate expression for the reaction rate. We then use finite-element methods to quantify the higher-order corrections to this solution and the reaction rate, which are difficult to obtain analytically. Leveraging the insights gathered from this analysis, we demonstrate how to correct for the errors that arise from adopting the approximate expression for the reaction rate, enabling for the construction of more accurate particle-based simulations than previously possible.

Computing analytically the $n$-point density correlations in systems of interacting particles is a long-standing problem of statistical physics with a broad range of applications, from the interpretation of scattering experiments in simple liquids, to the understanding of their collective dynamics. For Brownian particles, i.e., with overdamped Langevin dynamics, the microscopic density obeys a stochastic evolution equation, known as the Dean-Kawasaki equation. In spite of the importance of this equation, its complexity makes it very difficult to analyze the statistics of the microscopic density beyond simple Gaussian approximations. In this work, resorting to a path-integral description of the stochastic dynamics and relying on a saddle-point analysis in the limit of high density and weak interactions between the particles, we go beyond the usual linearization of the Dean-Kawasaki equation, and we compute exactly the three- and four-point density correlation functions. This result opens the way to using the Dean-Kawasaki equation beyond the simple Gaussian treatments, and it could find applications to understand many fluctuation-related effects in soft and active matter systems.

Percolating networks of nanoparticles (PNNs) exhibit a range of brain-like properties and are well modeled by continuum percolation with tunneling. All of the PNNs studied to date for their potential use in neuromorphic computing have been two-dimensional (2D), formed by depositing particles onto flat insulating substrates. Since the brain itself is three-dimensional (3D), it is natural to ask whether improvements in the brain-like characteristics of the networks and in their usefulness for neuromorphic computing could be achieved by building 3D percolating-tunneling systems. Using realistic simulations, we investigated whether the properties would be significantly different if PNNs were fabricated in 3D, such that the network is no longer able to be represented as a planar graph. We found that the properties of 3D PNNs, including degree distributions and small-world characteristics, are surprisingly similar to those of 2D PNNs. We outline a possible method for fabrication of a 3D PNN and discuss the practical difficulties of achieving a truly percolating network. A straightforward analysis of power dissipation shows that 3D PNNs are likely to be considerably more difficult to cool than 2D PNNs. These results suggest that moving to 3D neuromorphic systems may not bring an improvement in computational performance.

Microbial motion is typically analyzed by simplified models in which trajectories exhibit straight runs (perhaps with added Gaussian noise) followed by random, discrete tumbling events. We present the results of a statistical analysis of the angular dynamics for four different swimming microbes: tumbling and smooth-swimming strains of Bacillus subtilis and two Eukaryotic algae, Tetraselmis suecica and Euglena gracilis. We show that the angular statistics closely resemble a Voigt profile, the convolution of a Gaussian (Lévy index $\alpha =2$) and Lorentzian (Lévy index $\alpha =1$) distribution. This distribution is ubiquitous for all four microbes. Rather than modeling tumbling as a discrete process, we model tumbling dynamics as a continuous process: Lévy flights in the orientational dynamics using a Lorentzian noise model. This model is analytically solvable. Each individual microbe trajectory has both stochastic behavior (noise) and varying drift behavior, such as helices of different sizes and frequencies and circular arcs with different radii. We model the distribution of different drift behavior via an ensemble theory. The drift behavior (e.g., circular arcs) comes from physical observations of the swimming behavior and explains many of the qualitative features of the experimental data that cannot be explained by a pure noise model. From this theory, we estimate the strength of Lorentzian noise, the physical rotational diffusion constant, and some relevant parameters relating to the distributions of deterministic drift behavior. This analysis shows that in some cases Gaussian noise is not the dominant process responsible for the angular statistics following a Voigt profile.

A very basic aspect in the interaction of ions with plasmas is the effect of screening of the ion charge. In the case of classical plasmas this effect is usually described by linear theories. However, for dense quantum plasmas, deviations from the linear models arise and new effects appear. These nonlinear effects have been extensively studied in the case of solid targets where the screening may be very appropriately modeled using the Friedel sum rule for the phase shifts of the scattering waves, as a constraint to adjust the screening potential. This approach considers the screening media as a degenerate electron gas. In the present work the formulation is extended to the general case of plasmas with arbitrary degree of degeneracy. The study covers the whole range of plasma conditions going from dense and degenerate plasmas to nondegenerate ones. A form of the phase-shift sum rule for the case of moving ions in plasmas is obtained. The formulation provides several closed expressions and analytical approximations to characterize the screening effects. Analytical expressions for the linearized regime for the cases of the Fermi-Dirac and Maxwell-Boltzmann distributions are obtained. The calculations show a weakening of the screening and a disappearance of nonlinear effects with increasing plasma temperature, plasma density, or ion speed, and a convergence of the nonlinear results to the linear ones when these conditions are reached. The expressions derived here are of general applicability to a wide range of plasma conditions.

This study presents a temperature-conditioned deep generative approach for efficiently generating spin configurations of the two-dimensional (2D) Ising model. The method employs a convolutional generator network, conditioned on spin states, temperature embeddings, and random noise, to generate thermodynamically accurate spin configurations on a 2D square lattice. Physics-based constraints, derived from magnetization, energy, and nearest-neighbor correlations obtained via Monte Carlo simulations, are integrated as auxiliary loss terms during training to ensure physical consistency. The model accurately captures phase transition behaviors near the critical temperature and supports the scalable generation of synthetic configurations for further analysis. Investigations across lattice sizes $L=16$, 32, and 64 demonstrate close alignment with Monte Carlo baselines, with generated configurations achieving mean absolute errors as low as 0.0016 for magnetization, 0.0063 for energy, and 0.0016 for correlation in the $L=64$ case. Phase classification using standard machine learning models further validates the quality of the generated data, with ensemble classifiers yielding F1 scores above 0.99. These findings highlight the proposed framework as a scalable, computationally efficient solution for generating spin configurations in an Ising model.

The Lieb-Mattis theorem orders the lowest-energy states of total spin $s$ of a system of $P$ interacting fermions. We generalize these predictions to fermionic mixtures of $P$ particles with more than $N=2$ spinor components or species in the thermodynamic limit $P\to \infty$. The lowest-energy state inside each permutation symmetry sector $h$, arising in the $P$-fold tensor product decomposition, is well approximated by a $\mathrm{U}\left(N\right)$ coherent (quasiclassical variational) state, especially in the limit $P\to \infty$. In particular, the ground state of the system belongs the most symmetric (dominant Young tableau $h_0$) configuration. We exemplify our construction with the $N=3$ level Lipkin-Meshkov-Glick model, with a previous motivation on pairing correlations and $\mathrm{U}\left(N\right)$-invariant quantum Hall ferromagnets. In the limit $P\to \infty$, each lowest-energy state within each permutation symmetry sector $h$ undergoes a quantum phase transition for a critical value ${\lambda }_c\left(h\right)$ of the exchange coupling constant $\lambda$, depending on $h$. This generalizes standard quantum phase transitions and their phase diagrams corresponding to the ground state belonging to the most symmetric sector $h_0$.

This study investigates the influence of varying degrees of freedom (DOFs) on the swimming performance of self-propelled undulatory swimmers navigating a straight path in three flow configurations: an unbounded fluid, near a solid wall, and in a side-by-side arrangement. Vertical and rotational DOFs are frozen or actively controlled, resulting in four trajectory control scenarios. In unbounded flow, freezing motion DOFs generally underestimate the cost of transport while overestimating time-averaged amplitudes and Strouhal numbers, discrepancies that grow with increasing Reynolds number. Additionally, swimmers with two frozen DOFs exhibit phase shifts in lateral force and torque oscillations. Near solid boundaries, constrained swimmers experience more pronounced wall effects, whereas in side-by-side configurations, frozen-DOF swimmers display intensified channel effects. Analysis of the pressure distribution and wake topology reveals marked differences in pressure concentration and vortex street structures depending on DOFs. These findings underscore the critical role of motion DOFs in shaping swimming dynamics and emphasize the importance of appropriate trajectory control strategies for accurate modeling of fish locomotion.

This study investigates electron kinetics in a direct current magnetron sputtering discharge, advocating for the use of the cumulative distribution function (CDF) as a superior alternative to conventional probability distribution functions (PDFs), such as the electron energy distribution function. A three-dimensional simulation model is employed to track electron dynamics under balanced and two unbalanced magnetic field configurations. The analysis contrasts two critical phases: a prebreakdown stage with a vacuum electric field (Plasma Off) and a sustained discharge stage incorporating a strong sheath potential (Plasma On-). Results from the PDF analysis confirm that plasma formation profoundly modifies the electron distributions, leading to a substantial increase in the low-energy electron population and a general rise in the effective electron temperature due to sheath acceleration. Among the tested topologies, the unbalanced typeI (UB1) configuration consistently generates the most energetic electrons. However, the CDF provides a more intuitive and quantitatively robust analysis, free from binning artifacts. The rightward shift of the electron energy CDF (EECDF) for the UB1 configuration unequivocally confirms its higher median and mean energies. This approach reveals nuanced dynamics obscured by PDFs, such as a decrease in mean energy for the UB1 case upon plasma formation, while it increased for the other configurations. Furthermore, the CDF allows for precise quantification of distribution skewness; for instance, the fraction of electrons with energy below the mean in the UB1 configuration increased from approximately 60% in the prebreakdown phase to 80% in the sustained discharge. This work establishes the CDF as a powerful tool for characterizing non-Maxwellian plasmas, offering unambiguous insights into electron properties critical for optimizing magnetic confinement and process control in plasma-based technologies.

We show that the network of interparticle contacts in collectively handled jammed frictional and polydisperse sphere assemblies exhibits protocol-dependent self-organization. Here, by collective handling, we refer to boundary-controlled sphere assemblies where the creation of a jammed state from a stress-free state and the transition between jammed states are controlled only from the boundaries like in experiments and practical applications. We use the discrete element method and follow protocols similar to previous experiments to create $2d$ jammed states and quantify the self-organization of the network of interparticle contacts using the $k$-core decomposition method in graph analysis. The degree of self-organization depends on the method of preparation or protocol followed and affects the state attained in the jamming phase diagram. We explain the micromechanical origin and detail how the protocol followed can be tuned to limit self-organization. However, in contrast to algorithmic protocols where either the dynamics is not considered or particle features like size or interaction details like friction are tuned to attain dense states, we find that only a subregion of the phase diagram is attainable in collectively handled spheres and, remarkably, the jamming phase boundary of frictionless spheres marks the inaccessible region. We conjecture that the frictional interaction-dependent dynamics-driven formation of the 4-core subnetwork underlies the existence of the inaccessible region and propose the existence of a collective dynamical jamming region in the phase diagram of granular materials.

We present an experimental setup designed to investigate the statistical properties of extreme events in random elastic bending waves induced by an electromagnetic shaker on a thin stainless steel plate. In this setup, the standard statistical criteria used to define extreme events, such as rogue waves in the sea, are not sufficiently restrictive. Therefore, we introduce a new, more restrictive criterion to quantify the occurrence of rare events, similar to those observed in wave tanks [G. Michel et al., J. Fluid Mech. 943, A26 (2022)]. Using this refined criterion, we explore correlations between the amplitude of extreme events and other wave characteristics, such as slopes, energy, and periods of the waves. We find that extreme events in our setup are correlated to the longest wavelength of the plate, which corresponds to the plate's mode. We also observe that the steepness and kinetic energy of these events reach their time-averaged value, as expected for these slow-varying modes of the plate. The study raises questions about the purely statistical characterization of statistically rare events and rogue waves.

Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature $T_{\mathrm{c}}$ behaves similarly to an order parameter, and this analogy has been successfully used to locate $T_{\mathrm{c}}$ and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at $T=0$ and $T=\infty$ only. We directly compare its output to the same network trained at multiple temperatures below and above $T_{\mathrm{c}}$ to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify $T_{\mathrm{c}}$ and $\nu$, while the extraction of $\gamma$ becomes more challenging.

Dynamics in glassy systems near the phase transition is characterized by particle jumps. Approaches to describing this dynamics are based on models in which the time and length scales defining the jumps are parameters to be determined. We instead propose a model-independent method to detect these jumps. Our method uses the theory of the inspection paradox to analyze particle trajectories and reveals the time and length scales defining a jump without free parameters. Given its simplicity and generality, our method can be applied to resolve hopping motion in a broad class of systems, including experimental ones.