Papers

We determine exact finite-volume spin form factors and two-point correlations for a massive Kramer–Wannier interface in the scaling Ising model, separating ordered and disordered phases. The interface admits two complementary lattice realizations: a self-dual inhomogeneous Ising chain whose continuum limit has a sign-changing Majorana mass profile, and, after exchanging Euclidean space and time, a mixed thermal average amenable to Toeplitz/Fredholm methods. Combining these descriptions, we obtain the exact Neveu-Schwarz-Ramond spin matrix elements and a factorized Fredholm determinant for the interface correlator. Defect TCSA and independent fermionic spin-chain calculations confirm the predicted spectrum and matrix elements.

Using Vicsek-like self-propulsion rule, we study kinetics of liquid-liquid phase separation in mixtures of passive and active particles. For evolutions following temperature quenches of homogeneous configurations to the immiscible region of the phase diagram, we identify a remarkably strong dependence of the domain growth exponent on the choice of final state point. The singular dependence is indicative of the possibility of even an exponentially fast growth for suitable choices of system parameters. This striking observation, supported by finite-size scaling and other advanced analyses, is despite the fact that the overall mixture density is quite high that risks congestion with the prospect of slowing down particle transport. From flocking in pure biological systems to robotic swarming, these results are of much practical relevance. Theoretical pictures suitable for interpreting such high growth rates are discussed. In this regard, we discuss coarsening in the velocity field as well that describes how the density field coarsening follows the latter.

The evolution of DNA sequences can be viewed as stochastic dynamics on a high-dimensional discrete space, but it is unclear when empirical transition biases reduce to an effective energy landscape versus retain irreducible non-equilibrium circulation. Human context-dependent mutation probabilities offer a direct test: every single-nucleotide substitution in a local context has a reverse substitution, so the logarithm of the forward-to-reverse probability ratio defines an antisymmetric field-the human mutation field. We show this field has a dominant gradient component and a smaller but reproducible curl component. Using seven-base human germline substitution probabilities, we infer an effective mutational landscape with a Siamese neural network constrained to predict only energy differences. This model predicts forward-to-reverse log-ratios for held-out mutations with a correlation of about 0.93, close to both an unconstrained predictive reference (0.948) and the empirical reversible ceiling from Hodge projection (about 0.96). Although trained only on mutation probabilities, the inferred landscape largely recovers short-word genomic composition and Chargaff reverse-complement symmetry for sequences up to length four. Deviations from equilibrium structure reveal a small but detectable nonequilibrium component: a residual irreversible circulation violating the Kolmogorov cycle condition for detailed balance, reproducible across African, Asian, and European populations, and strongest in CpG-linked cycles and CpG-transition edges, consistent with methylcytosine deamination. These results give a thermodynamic decomposition of the human mutation field: most mutation bias is organized by a local equilibrium-like energy landscape aligned with genome composition, while the residual circulation points to specific directional mutational mechanisms.

Active systems navigate complex environments through non-equilibrium fluctuations, rendering standard equilibrium transition-rate theories inadequate. Moreover, transition rates provide only partial information on how stochastic dynamics explore metastable states, whereas knowledge of the optimal paths offers deeper physical insight. Using an active Ornstein-Uhlenbeck particle, i.e., a particle driven by exponentially correlated noise, as a paradigmatic model, we establish an exact mapping of the non-Markovian optimal path onto a higher-dimensional Hamiltonian dynamical system so that, for arbitrary force fields, optimal paths can be computed systematically by solving the corresponding Hamilton equations. Tuning the conserved energy allows us to explore diverse dynamical regimes, ranging from classical instanton trajectories strictly confined to the zero-energy surface to finite-time optimal paths at non-zero energies, which can spiral around shoulders of the energy landscape, a distinct signature of the non-Markovian active dynamics.

Fragility and fictive temperature are two central measures of glass formation, but their connection is usually expressed indirectly through structural relaxation times or cooling-rate dependence. Here we formulate this connection in terms of configurational enthalpy fluctuations. The time evolution of the fictive temperature during cooling or aging is controlled by the enthalpy–enthalpy correlation function that also determines the frequency-dependent configurational heat capacity. Fragility enters through the strong temperature dependence of the structural relaxation time and therefore controls how rapidly the corresponding enthalpy-relaxation spectrum shifts to long times near the glass transition. This leads to a direct time-domain description of the onset and growth of the fictive-temperature lag, including the Debye limit, and connects calorimetric relaxation, fragility, and nonequilibrium structural arrest within a common framework.

We develop a dynamical mean-field theory for nonreciprocal associative networks that store an extensive number of dynamical attractors, from limit cycles to strange attractors. Using a path integral calculation under quenched disorder, we derive self-consistent dynamical mean-field equations for pattern overlaps, autocorrelations and response functions. Memory retrieval capacity is governed by the spectral structure of the coupling matrices encoding stored patterns. When their eigenvalues are coherently aligned, retarded self-interactions and quenched noise feed back destructively: at zero eigenphase (fixed point attractors) the classical equilibrium capacity bound is recovered, while for limit cycles retrieval collapses far below it. In contrast, for uniformly distributed eigenphases, retarded self-interactions and much of the quenched noise cancels, reducing the dynamics to an effective single-spin process and amplifying capacity substantially. We validate the theory against microscopic simulations for limit-cycle and chaotic attractors, identifying eigenvalue decoherence as the mechanism enabling enhanced storage of dynamical memories.

Critical universality does not determine the microscopic conditional structure of a probability measure. We study a bicolored cluster recursion constrained to remain critical at every generation, with no equilibrium spin measure or fixed coupling imposed. In both two and three dimensions, the resulting history-dependent sequence develops a common Ising/Fortuin–Kasteleyn (FK) compatibility structure: the second-shell dependence of a one-site conditional law is strongly suppressed, nearest-neighbor effective couplings move progressively toward one another near the critical Ising value, and cluster and interface observables organize around the corresponding FK geometry. An exact cluster-coloring factorization singles out $q=2$ as the point where the residual connectivity weight disappears from the two-color spin marginal. Thus equilibrium-compatible conditional structure can emerge along a trajectory that remains critical throughout.

Heterogeneity in contractile systems of molecular motors interacting with their tracks plays a key role in numerous cellular physiological and pathological processes. However, its effects cannot be captured by theoretical models assuming identical mechano-kinetic properties for all motors. We developed a stochastic framework to describe heterogeneous ensembles of myosin motors comprising two populations with distinct mechano-kinetic properties. Assuming that motors interact with the actin filament (their track) as independent force generators, we derived the probability distribution of the isometric force and characterised the statistics of finite-size force fluctuations. The proposed framework includes an estimation procedure that simultaneously infers the mechano-kinetic parameters and the size of the motor ensemble, eliminating the need to prescribe it a priori. Validation against synthetic and experimental data shows the model accurately captures force fluctuations and provides realistic estimates of the ensemble size. Furthermore, the framework enables a quantitative assessment of ensemble heterogeneity and the inference of an unknown motor species properties. This approach provides a quantitative tool for characterising heterogeneous actin-myosin systems, such as those arising from the co-presence of different protein isoforms in cardiac and skeletal muscle, or from the partial replacement of native proteins with mutation-derived or engineered variants.

We investigate the full counting statistics of a one-dimensional multicomponent impenetrable gas released from a bipartite state in the presence of a local defect. Spin–charge separation reduces the multivariate generating function to an exact Fredholm determinant whose kernel is built from time-evolved single-particle orbitals. Expressing these orbitals in terms of scattering data gives access to both transient dynamics and the nonequilibrium steady state. We derive the leading long-time asymptotics and identify persistent oscillations in densities and currents caused by multiple bound states at the junction. We argue that interactions between components lead to a linear low-temperature correction to the current, in contrast to the quadratic correction in the single-component case. At equilibrium, we obtain the M-Wright distribution of spin transfer with $t^{1/4}$ scaling.

The triangular Ising antiferromagnet, with its residual entropy density and critical correlations at absolute zero, is the archetype of a two-dimensional spin liquid. Its ground state is partitioned into topological sectors connected by global spin flip events that wrap the lattice boundaries, and become statistically irrelevant in the thermodynamic limit where magnetic fluctuations are restricted to the dominant sector. Thus far, these properties have been mainly investigated from a theoretical perspective, and one might wonder to what extent they can be transposed to real materials. Here, we present experimental observations obtained in an artificial, macroscopic realisation of the seminal triangular Ising antiferromagnet that consists of a lattice of millimeter-sized NdFeB cylinders put into motion by a mechanical shaker. Specifically, we demonstrate that the very-low-energy physics and the true ground state of this model can be reached experimentally. Besides, we are able to probe, in space and time, the magnetic properties within manually preselected pockets of the ground-state manifold that emulate, to a good approximation, the behaviour in distinct topological sectors. Our approach opens new avenues for naked-eye visualisation and hand manipulation of many-body phenomena associated to frustrated magnetism and models of statistical physics.

In a classical paper [K. Husimi, Prog. Theor. Phys. 9, 381 (1953)], Husimi showed that, for a single harmonic oscillator with a time-dependent angular frequency and initial conditions sampled from an equilibrium distribution, the averaged energy cannot decrease after a cyclic operation. This irreversibility is quantified by Husimi's adiabaticity parameter constituted with adiabatic invariants. In this work, we generalize Husimi's framework to a many-body system of one-dimensional harmonic oscillators coupled on an arbitrary connected network, with all spring constants sharing a common time dependence. For a system attached to a fixed wall, the dynamics can be decomposed into independent normal modes by transforming to mass-weighted coordinates and diagonalizing the resulting positive definite matrix, with each mode characterized by its own Husimi's adiabaticity parameter. By defining the generalized Husimi's adiabaticity parameter as their arithmetic mean, we derive the exact time evolution of the averaged energy and establish its non-decrease under cyclic operations. Numerical simulations confirm these results for both a uniform nearest-neighbor chain and a heterogeneous network.

Frictional relaxation after a sudden velocity change underlies the phenomenological laws that describe the stability of steady sliding and of seismic faults. To connect this relaxation with measurable statistics of microscopic contacts, we introduce a minimal model incorporating logarithmic aging of contact forces and a power-law contact-size distribution reflecting fractal interface roughness, and derive the relaxation function analytically. Depending on the power-law exponent $\alpha$, the intermediate-time response exhibits a plateau, logarithmic decay, or power-law decay, in contrast to the exponential decay assumed in the phenomenological laws. The theory provides experimentally testable links between microscopic contact statistics and macroscopic frictional relaxation.

The mean squared displacement (MSD) is a cornerstone in the analysis of diffusion processes in complex media. When the system is heterogeneous and, in particular, when single-particle trajectories are short, it is essential to extract maximal information from each measured trajectory. This is typically done by time-averaging squared increments and examining the scaling of the time-averaged MSD in log-log space. However, classical regression methods perform poorly in this setting because time-averaging introduces correlations aggravated by those inherent to anomalous diffusion. We tackle these limitations by applying a generalized least squares framework, which substantially reduces variance and bias in diffusion parameter estimates, especially for short (around 100 points) and ultra-short (around 10 points) trajectories. The method is fully automated and requires no supervision. Furthermore, it enables prediction of estimation error probability density, which is asymptotically Gaussian, for both classical and enhanced approaches. Leveraging this prediction, we introduce a specialized deconvolution algorithm that reconstructs the underlying particle ensemble structure from experimental data.

We investigate the de Almeida-Thouless (AT) stability of the replica symmetric (RS) solution for a spread-spectrum watermarking model incorporating Bayes-estimation-based image restoration. In our previously proposed model [1], which simultaneously performs message decoding and image restoration under Gaussian channel noise, a discrepancy between the theoretical predictions of the RS solution and Markov chain Monte Carlo (MCMC) simulations was observed in specific parameter regimes. By analyzing the blind case where both the original message and image are unknown, we explicitly derive the conditions for the AT stability of the RS solution by clarifying the structure of the replicon eigenspace for the hybrid system. Our results reveal that AT stability is broken precisely in the parameter region where the RS solution deviates from MCMC simulations, thereby successfully explaining the mechanism underlying this discrepancy in terms of replica symmetry breaking (RSB). [1] M. Kawamura, T. Uezu, K. Hayashi, and M. Okada, “Statistical mechanical evaluation of a spread-spectrum watermarking model with image restoration,” Phys. Rev. E 99, 062132 (2019).

Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.

Nanothermodynamics describes the process where large systems subdivide into equilibrium distributions of small subsystems. A key ingredient is Hill's subdivision potential (E) that ensures adherence to the 1st and 2nd laws of thermodynamics in systems of any size. In this review and reassessment, it is emphasized that nanothermodynamics gives new insight into many measurements, theories, and simulations. Measurements establishing the need for nanothermodynamics show thermodynamic heterogeneity from multiple effective temperatures (T_i) inside most types of materials. One theoretical result that requires E=0 is the stable solution of Ising's original model for finite chains of interacting spins, a solution Ising could not have found 40 years before Hill's work. Another result is a novel solution to Gibbs' paradox that makes the entropy of the semiclassical ideal gas exactly extensive. Molecular dynamics simulations reveal how a standard fluctuation relation is modified when local degrees of freedom fluctuate faster than their coupling to the heat bath, consistent with the measured thermodynamic heterogeneity. Simulations of a Creutz-like model, comprised of Ising spins coupled to an explicit heat bath of Einstein oscillators, are used to study the 2nd law. It is found that maximizing the total entropy (S_t) requires an intrinsically irreversible step, providing a counterexample to the usual claim that statistical mechanics emerges from reversible dynamics. Furthermore, fluctuations of this model are best described by Einstein's reversal of Boltzmann's relation and the 2nd-law, not by recent fluctuation theorems.

Metastability in open quantum systems is usually inferred from spectral separation in the Liouvillian, which governs unconditional, ensemble-averaged dynamics. We show that this diagnosis is incomplete at the level of individual quantum trajectories: conditioned realizations of the same unconditional dynamics can bypass, transiently access, or operationally preserve a metastable memory, depending on the monitored channel and the observed record. We demonstrate these mechanisms in a driven-dissipative nonlinear oscillator realizing a quantum associative memory, by comparing spectra, target fidelities, and phase-space distributions. The non-Hermitian dynamics obtained by post-selecting on the absence of detected events supports metastable retrieval, but with a distinct long-time fate: normalization selects the least-decaying mode of the non-Hermitian spectrum rather than the addressed memory branch. In contrast, stochastic jump trajectories can preserve retrieval over extended times when the post-measurement update remains compatible with the coherent memory structure. Thus trajectory-level metastability is not determined by the averaged generator alone; it requires compatibility between the monitored channel, the measurement record, and the metastable manifold.

We develop a comprehensive many-body theory for systems of particles obeying quantum unified statistics, or quons. After exploring the properties of the Fock space of this system, we formulate a systematic S-matrix expansion and a generalized Wick's theorem. A consistent Green-function formalism is constructed at both zero and finite temperatures, accompanied by a generalized Wick's theorem appropriate for infinite-statistics operator algebras. Within this framework, we establish diagrammatic rules for interacting quon systems. Employing the random phase approximation, we derive the dielectric function and reveal the emergence of anomalous plasmon modes that have no direct counterpart in conventional Bose or Fermi systems. We further analyze the ground-state energy, energy-loss function, generalized Thomas-Fermi screening wave vectors, and Friedel oscillations, elucidating how infinite statistics qualitatively modifies collective behavior and screening properties.

Framewise variance normalization removes fluctuation amplitude from an observation, but not from the dynamics that generate later patterns. The current normalized spectrum therefore need not determine its own future. We study this effect in three two-dimensional phase-separation models that change transport, add active driving, or suppress large-scale separation. For paired initial fields, their normalized spectra evolve identically in the linear regime. The restoring model provides an exact example: damping does not directly change how the normalized spectrum evolves when both equations are applied to the same field. It nevertheless changes the hidden amplitude, which alters the later nonlinear shape. An exact comparison separates the immediate effect of changing the equation from the effect of reaching a different state through earlier evolution. It also reveals strong cancellation between these effects in the mobility and active models. As a practical test, spectral histories improve classification by about 4-6 percentage points at several later observation windows, although the gain depends on quench depth and spectral construction. Temporal spectra can therefore reveal dynamical consequences of information absent from every individual normalized frame.

Massive jellyfish swarms observed at sea can extend over tens of kilometres and contain millions of individuals, yet the mechanisms governing their formation and large-scale dynamics remain poorly understood. Agent-based models provide a framework for describing this dynamics based on jellyfish responses to ocean currents and environmental cues, but become computationally prohibitive when extended to large populations and spatial scales relevant to ocean circulation. Here we derive a continuous kinetic theory from an active-particle model of jellyfish motion. The resulting Fokker-Planck framework incorporates transport by prescribed currents, stochastic reorientation, direct interactions and stimulated steering, allowing chemical signalling to be represented through a coupled field. We further derive a hydrodynamic closure for large swarms by exploiting the separation between fast orientational and slow spatial dynamics, yielding a reduced density equation suitable for implementation in ocean-current models. This framework provides a route from individual behavioural mechanisms to continuum descriptions of jellyfish populations and establishes a basis for constraining model parameters using observations and in-situ measurements. This approach offers a theoretical foundation for future numerical prediction of large jellyfish swarm formation and evolution in realistic ocean flows.

We consider an ensemble of particles performing a biased random walk on a two-dimensional lattice. Due to the adhesion between particles (nearest-neighbor Ising interactions), the probability of hopping depends on the number of neighbors and the strength of adhesive interaction. The asymmetry in the hopping directions leads to a macroscopic current of particles through the system. We theoretically compute the distribution of currents and verify the results in stochastic particle simulations. To make theoretical progress, one has to determine the probabilities of various particle configurations such as single particles, pairs, and different configurations of three-particle clusters. These probabilities are theoretically derived by formulating rate equations for the concentrations of various structures resembling Becker-Döring cluster equations. One intriguing finding is that the concentration of left-handed triplets is not equal to the concentration of right-handed triplets for non-zero asymmetry of hops and non-zero interaction between particles.

Non-reciprocal interactions are ubiquitous in active, biological, and disordered systems, generically driving them out of equilibrium. Here, we introduce a hierarchy of non-reciprocally interacting Ornstein-Uhlenbeck (NROU) models governed by a tunable non-reciprocity parameter $g$. At a special point $g=g^*$, the drift matrix becomes non-diagonalizable, realizing exceptional points (EP's) of different orders, where eigenvalues and eigenvectors simultaneously coalesce. The hierarchy encompasses non-reciprocally coupled dimers, their disordered counterparts, and a many-body chain exactly mapping onto the paradigmatic Hatano-Nelson model in the arena of non-Hermitian quantum systems. For the disordered model, we show that the distribution of the EP location $g^*$ across disorder realizations develops a universal edge singularity precisely at the clean-system EP, and is manifestly non-self-averaging. Across all models, we find that at the EP, the usual exponential relaxation of the autocorrelation and covariance functions is dressed by a polynomial-in-time prefactor whose degree is set by the order of the EP and whose detailed structure encodes the spatial architecture of the chain. At complete asymmetry, the many-body chain exhibits “pseudo-equilibrium”: its steady-state distribution factorizes into equilibrium-like single-particle measures despite a nonzero steady-state current. Moreover, the $N$-particle interacting system decomposes into $N/2$ independent complex OU processes. Finally, using the Harada-Sasa relation, we obtain a closed-form expression for the total steady-state heat dissipation and uncover a boundary refrigeration effect, in which the boundary particles switch from acting as a hot to a cold reservoir as the non-reciprocity is tuned.

We develop a method to extract the shear viscosity of a lipid membrane from equilibrium molecular dynamics simulations. The method characterizes the rheology of general interfacial systems embedded in three-dimensional media; we term it molecular interfacial rheology. In our simulations the planar bilayer and surrounding water are confined between solid, parallel walls. Following Onsager's regression hypothesis, membrane and water fluctuations are assumed to relax according to the coupled continuum-mechanical equations governing the confined system—which predict that the membrane transverse velocity autocorrelation function (TVACF) decays exponentially, at a rate set by the membrane and water viscosities. The measured TVACF, however, exhibits damped oscillations followed by a slowly decaying tail. We reconcile these behaviors using the Mori–Zwanzig formalism, and extract the wavevector-dependent membrane viscosity from the time-integral of the TVACF. Results from theory and simulations agree over a decade of wavevectors, and extrapolating to long wavelengths yields shear viscosities ranging from 0.064 to 0.18 pN*us/nm across two representative single-component, fluid-phase bilayers. Our results are corroborated by nonequilibrium simulations where a spatially varying in-plane body force is applied to lipid molecules, thus validating the framework of molecular interfacial rheology.

Establishing a robust and physically interpretable link between static structure and heterogeneous relaxation dynamics remains a fundamental challenge in glass physics. Here, we introduce a weighted pair-entropy descriptor based on the conventional two-body excess entropy. For this, we multiply the integrand used to calculate the excess entropy by a weight function that is directly related to the length scale of the pair correlation function. This multiplication does not affect the contribution of the short range order to the local excess entropy, but allows to take into account the structure present on intermediate distances, i.e., the medium-range order. For a canonical two-dimensional Lennard-Jones glass former, the resulting descriptor exhibits a strong correlation with particle-level dynamical propensity at long times (multiples of the alpha-relaxation time), with a maximum structure–dynamics correlation reaching about 0.9, substantially outperforming the predictive power of the conventional local pair excess entropy. These results demonstrate that incorporating a physically motivated structural length scale into entropy-based descriptors markedly enhances their predictive power while preserving physical interpretability. Our findings provide a simple and general framework for investigating structure–dynamics correlations in glass-forming systems.

We investigate reflected diffusion in a bounded domain subject to two independent, competing stopping mechanisms: an exponentially distributed bulk lifetime of rate $p$ and a surface reaction triggered when the boundary local time exceeds an independent exponential threshold of rate $q$. Denoting by $T$ the stopping time and by $L$ the acquired boundary local time at stopping, we derive their marginal distributions, joint Laplace transform, and complete hierarchy of mixed moments. These statistics are determined by the splitting probability $\phi$ that surface reaction occurs before bulk decay. In particular, we establish the identity $p\expect{T}+q\expect{L}=1$ and show that the cumulative risk $pT+qL$ is exponentially distributed with unit rate. We further obtain equivalent representations of $\phi$ in terms of the Robin-Laplacian and the generalized Steklov spectra. Explicit results for a three-dimensional ball reveal how competing rates $p,q$ control $\phi$ and the $(T,L)$ statistics. Monte Carlo simulations test the universal cumulative-risk law.

We generalize the Mori-Zwanzig projection operator formalism by introducing an oblique projector using biorthogonal variables to coarse-grain the dynamics of systems far from equilibrium, such as active or living matter. Improving on previously used orthogonal projectors, the oblique projector correctly decomposes the emergent interactions of the coarse-grained particles into systematic and stochastic contributions. We show that the formalism directly connects to established results in non-equilibrium linear response theory and adiabatic perturbation theory. Based on this formalism, we develop a coarse-graining algorithm which we apply to extract the emergent non-equilibrium chiral dynamics of a passive tracer in a chiral active bath. Our theory correctly predicts the odd transport properties and a confinement-induced shift of the chiral frequency of the tracer.

For a one-dimensional path graph and a lifted path graph constructed from a duplication of each of its sites, we study how a reversible Markov chain can be perturbed and gradually driven into non-reversibility. The reversible Markov chain has a transition matrix that is diagonalizable and features real-valued eigenvalues and eigenvectors. The left and right eigenvectors form a biorthogonal system. We discuss in concrete examples how the transition matrix of a non-reversible Markov chain may be diagonalizable or non-diagonalizable, and it may have real eigenvalues and complex-conjugate pairs. For a number of steady states (flat, square-wave, wedge, V-shape), we compute eigenvalue spectra on both graphs and discuss the speedup that can be achieved through lifting. We develop a Green's matrix formalism, which we use to compute Kemeny times and mean first-passage times, and which provides valuable information and allows us to interpret the results for the characteristic times.

We investigate the slowing down of dynamics in a glass-forming mixture interacting via an inverse-power-law (IPL) potential using a combination of theory and large-scale molecular dynamics simulations. We measure the static pair-correlation function, configurational entropy, inherent-structure energy, and structural relaxation time. We employ a theoretical framework to calculate the structural relaxation time $\tau_{\alpha}$, which is found to be in very good agreement with the simulation results. The theory identifies a local structural order which defines the cooperativity of the relaxation and brings forth a fluctuation induced parameter $\psi ( T )$ and a crossover temperature $T_a$ that characterize the density and temperature dependence of the glassy dynamics. Furthermore, we determine a crossover temperature using independent dynamical and thermodynamic criteria and compare with the theoretically predicted crossover temperature $T_a$. Relaxation dynamics is shown to obey density-temperature scaling, similar to thermodynamic properties, in terms of a variable $\Gamma$ formed by an appropriate combination of density and temperature, characteristic of IPL interactions. Finally, we show that, when the excess thermodynamic and dynamic quantities obtained at different densities are plotted as functions of the reduced temperature $T/T_a$ (or $T_a/T$), the data collapse onto master curves with excellent agreement between theory and simulation. These scaling relations provide a unified description of the thermodynamics and dynamics in IPL systems, enabling the prediction of relaxation behavior over a wide range of densities from data at a single state point.

Overparameterized neural networks carry far more hidden units than a task nominally requires, raising the question of which neurons are essential and whether that distinction is legible in the representation itself, without labels or gradients. We cast neuron selection as the problem of coarse-graining the hidden layer by retaining a subset of its neurons, and score each putative selection by the mapping entropy (ME). This quantity measures the loss of discriminatory power inherent in discarding part of the network neurons, and the selection that minimises the ME is taken as particularly informative. This criterion is fully unsupervised, in that it depends only on hidden-activation statistics. In teacher-student networks, ME optimisation recovers the minimal teacher-consistent representation and retains extra units in proportion to the hidden layer's residual variability; in a non-linear Gaussian process task, it selects coherent functional-class mappings whose preferred class shifts across training. On this task and on translation-augmented MNIST, ME-selected subnetworks outperform random subsets of equal size, most clearly under strong compression - linking configurational distinguishability to predictive performance.

We study the operator entanglement of local operators in one-dimensional brickwork circuits whose two-site gate satisfies the braid relation; throughout this work, we call such a gate a Yang–Baxter gate. We establish upper bounds for several structured, overlapping classes of Yang–Baxter gates. We show that the operator Schmidt rank remains uniformly bounded in time for all qubit Yang–Baxter gates and, in arbitrary local dimension, for permutation gates obtained from non-degenerate Yang–Baxter maps. We also show that it grows at most polynomially for involutive dual-unitary Yang–Baxter gates and for arbitrary phase dressings of permutation gates obtained from non-degenerate Yang–Baxter maps. These results imply, respectively, constant and logarithmic upper bounds on the operator entanglement. Conversely, we construct a seven-state involutive Yang–Baxter gate without dual unitarity and a one-site operator whose exact operator Schmidt rank grows exponentially, although the corresponding operator entropies remain undetermined. Entanglement growth in the general Yang–Baxter case remains open. All proofs and selected examples were constructed by ChatGPT 5.6 Sol.