Papers

Persistence probability in opinion dynamics models estimates the tendency of the agents not to change their initial opinion till the present time. Here we consider two nonlinear q-voter models with binary opinions, where the dynamics are governed by a biased choice when the q panel is not unanimous. The models are studied for different parameter ranges corresponding to the known stationary states. Mean field theory and numerical simulations are used to compute the persistence probability for the two types of opinion separately. The long time behavior in general is either a saturation or a decay that can be approximated by an exponential form, depending on the chosen parameters. Based on this, phase diagrams in the parameter space are presented for both the models. The regions in the phase diagrams indicate a strong correlation with the behavior of fixed points in the corresponding models, which is non-trivial as far as the persistence probability is concerned.

Generalized Hopfield networks are introduced where memories and neurons are continuous variables that lie on a Riemannian manifold. We explicitly focus on symmetric spaces associated with the special unitary groups SU(d), and use both numerical and analytical (replica) techniques to demonstrate an almost order of magnitude enhancement in critical capacity over the vector networks starting with d=3 and further rapidly growing with d. To circumvent the non-linear geometric constraints, we use a Lie algebraic method [following V. Galitski, Phys. Rev. A 84, 012118 (2011)] to exactly describe the classical neural network in terms of linear algebra in an auxiliary Hilbert space. It is shown that in contrast to the traditional Hopfield networks, memory recall in SU(d) Hopfields corresponds to neuron alignment along a top eigenvector of a spiked matrix, which is less susceptible to random matrix crosstalk than other models with continuous neuron variables. Physical platforms to realize SU(d) Hopfields are briefly discussed and physical (in addition to algorithmic) recall mechanism is demonstrated, where memory recovery occurs naturally through generalized Landau-Lifshitz-Gilbert dynamics. To illustrate SU(3) memory recall, we introduce a color (RGB) image encoding/decoding protocol and explicitly run image recovery on corrupted cues. Finally, we quantize the generalized Hopfields which are shown to reduce to Sachdev-Ye glassy type of models. Their many-body spectra generally feature two types of dark and memory bands, where the latter exhibits chaotic Wigner-Dyson level statistics that hides Hebbian data.

We study the out-of-equilibrium dynamics in the XX chain with step-like magnetic field, which maps to an inhomogeneous tight-binding chain after Jordan-Wigner transformation. We obtain exact analytic expressions for the fermionic two-point correlation function after a quantum quench from several initial product states, both homogeneous and inhomogeneous ones. This is achieved by using a combination of Fourier and Laplace transforms, which al low us to map the problem to a standard Riemann-Hilbert problem on the unit circle. For arbitrary positions and times the correlators are not expressed in terms of elementary functions. However, in the hydrodynamic limit $x,y,t\to\infty$ with fixed ratios, we provide explicit formulas that depend only on the effective transmission coefficient across the origin. We benchmark our analytic predictions against exact numerical simulations and find excellent agreement in the hydrodynamic limit, apart from finite-time corrections.

We construct a neural classical density functional that acts directly on unrestricted three-dimensional density profiles. As a computationally tractable test bed, we consider parallel hard cubes of side length three on a simple cubic lattice. A fully convolutional network learns the one-body direct-correlation functional $c^{(1)}[\rho]$ from data obtained with grand-canonical Monte Carlo simulations in randomized external potentials. Complete profiles are used during both training and inference; a stochastic Bernoulli mask on the output sites makes full-profile training effective without explicitly extracting and storing overlapping local density windows. Averaging the first-layer kernels over all 48 rotations and reflections of the cubic point group additionally imposes exact cubic equivariance without data augmentation. We compare the learned functional with independent simulation data and with the lattice fundamental-measure functional of Lafuente and Cuesta. The neural functional markedly improves the homogeneous equation of state and the density profile at a planar hard wall. For the anisotropic pair distribution around a fixed particle, both functionals reproduce the principal packing shells, with their relative accuracy depending on crystallographic direction. These results demonstrate neural density-functional calculations on complete three-dimensional profiles while also identifying accurate full-dimensional training data, thermodynamic consistency, and structural correlations as the central challenges for extensions to continuum fluids.

Frustrated quantum matter can host intertwined orders rooted in symmetry-related low-energy landscapes, yet static order parameters alone do not reveal how fluctuations are organized among competing configurations. Here we measure mode-resolved shot-to-shot population fluctuations in a $p$-orbital triangular-lattice superfluid with a tunable bias among three valleys. We observe a bias-tuned evolution from enhanced, anticorrelated fluctuations of two minority valleys toward strong confinement of relative-population fluctuations in a selected two-valley stripe phase. The dominant fluctuation structure is captured by an effective canonical model that includes interactions among the condensed modes, supporting a quasi-equilibrium description of the coherent three-valley condensate. Together, the data and model reveal a quantum–thermal regime shaped by pair-tunneling-induced number–phase dynamics, in which relative-phase scrambling softens effective barriers in the minority-valley regime, while phase rigidity gives rise to macroscopic harmonic confinement in the stripe phase. Our results establish mode-resolved fluctuation measurements as a probe of hidden number–phase back-action in frustrated quantum fluids.

Slow internal states reshape both the mean throughput and the temporal organization of a driven lattice gas. Exit-counting statistics reveal this effect in a finite open totally asymmetric simple exclusion process whose particles reversibly switch between active and paused states. Increasing pausing lowers the mean current smoothly, whereas the long-window Fano factor is strongly nonmonotonic. At the reference boundary rates, the maximum remains near a measured mean paused population \(N_p=L\rho_{\rm paused}\simeq1.5\)–\(2\) across lattice lengths L=50-500, while the corresponding pausing rate scales as \(k_p^{\rm max}\propto L^{-1}\). A minimal constant-birth, linear-death approximation translates an order-one collective crossover into this finite-size displacement and gives \(N_p^\star\simeq1.50\) in the independent-pause, strong-blocking limit. The simulations delimit this approximation: the pause number is overdispersed, and at fixed \(N_p\), slower unpausing increases both the correlation time and the noise amplitude. Residence-time and structural analyses further separate the relevant slow variables. The pause-free versus pause-containing residence-time scale tracks the fitted output-correlation time, whereas the noise amplitude follows fluctuations, rather than the mean size, of the largest particle cluster. Low-frequency output noise therefore identifies an intermittent finite-size regime shaped jointly by slow-defect kinetics and traffic-jam reorganization.

The realization of Dirac criticality beyond the conventional Gross-Neveu-Yukawa (GNY) paradigm has become a major frontier in condensed matter physics. In this work, we introduce an $\mathrm{SO}(3)$-symmetric bilayer SLAC fermion model with tunable inter-layer interactions that exhibits a rich quantum phase diagram. As the interaction strength increases, the system undergoes two distinct phase transitions. The primary transition is a continuous boundary separating a Dirac semimetal (DSM) from an $\mathrm{SO}(3)$-broken ordered phase. Crucially, this transition evades the standard GNY universality class because the emergent order only gaps out a subset of the itinerant fermions. Using large-scale quantum Monte Carlo (QMC) simulations, we establish that this transition belongs to the Gross-Neveu-$\mathrm{SO}(3)$ universality class with $N=6$ irreducible Dirac cones and precisely extract the corresponding critical exponents. At stronger couplings, a second transition drives the system into an inter-layer $\mathrm{SO}(3)$-symmetric superconducting (SC) state. We provide strong numerical evidence that this transition is first-order. Our study provides new insights into the exploration of Dirac criticality beyond the standard GNY universality class, and also offers a distinct platform for investigating $\mathrm{SO}(3)$-symmetric superconductivity.

Molecular motion within living cells provides a window into the physical principles underlying intracellular organization and transport. Yet a fundamental limitation remains: measured trajectories are often short, noisy, and randomly oriented, rendering spatial anisotropies inaccessible to conventional analyses. We show that turning-angle statistics provide a robust approach for uncovering anisotropies in anomalous diffusion. Using a two-dimensional anisotropic fractional Brownian motion model with random orientations, we show theoretically and through simulations that turning-angle distributions preserve signatures of anisotropy. We apply this approach to single-particle tracking data, including quantum dots in the cytoplasm of HeLa cells and membrane proteins in hippocampal neurons. Turning angles reveal hidden anisotropies in the motion of quantum dots and Nav 1.6 channels, and in specific dynamical states of glycoprotein CD4. These results establish turning-angle analysis as a powerful strategy for detecting organization in complex environments and reveal that anisotropic anomalous diffusion is an overlooked feature of intracellular dynamics.

Non-Hermitian dynamical quantum phase transitions (DQPTs) are intrinsically sensitive to the choice of inner product under nonunitary time evolution. Although the biorthogonal formulation based on associated states provides a normalized Loschmidt echo with a probabilistic interpretation, previous studies have found biorthogonal and self-normal DQPTs to occur in the same parameter regimes, suggesting that the two forms of dynamical criticality are concomitant. Here we demonstrate that this is not the case. In an exactly solvable periodically driven non-Hermitian Su-Schrieffer-Heeger chain, we uncover a finite biorthogonal-only Floquet DQPT regime, where the biorthogonal Loschmidt rate becomes nonanalytic while the self-normal Loschmidt rate remains smooth. The critical conditions are obtained analytically, showing that the onset of biorthogonal Floquet DQPTs is locked to the exceptional lines of the effective Floquet Hamiltonian, whereas self-normal criticality has no corresponding spectral boundary. Moreover, for each critical momentum, the biorthogonal DQPT exhibits a pair of critical times within every driving period, whereas the self-normal DQPT exhibits only one. Our results establish a fundamental distinction between biorthogonal and self-normal DQPTs, thereby opening a route toward new nonequilibrium quantum phenomena in non-Hermitian systems.

Classical fractons are Hamiltonian systems that can develop attractors after projection onto configuration or shape variables, although the full phase space admits none. We study a scale-invariant, dipole-conserving two-parameter family of fracton Hamiltonians $H_{\alpha,\beta}$. By separating coordinates into scale and shape, we obtain autonomous shape dynamics that admit fixed points which leave a purely scale evolution of the form $R(t)\propto |t|^{\alpha/(\alpha-\beta)}$. The shape fixed points, which determine the distribution of the expanding particles, are central configurations of power-law Riesz potentials. The distinguished model $(\alpha,\beta)=(-2,1)$ is unique: its scale evolution takes the Einstein-de Sitter form $R(t)\propto |t|^{2/3}$, its fixed-point equation is the equal-mass Newtonian central-configuration, its large-$N$ distribution is a homogeneous ball, and its homothetic trajectories admit a zero-energy Newtonian gravitational dual. The fixed points are locally stable, and simulations at moderate $N$ approach them from random initial data. Large $N$ simulations reveal a richer class of fixed-points: bound clusters of approximately fixed physical size retain internal motion, while their centers approach unequal-mass Newtonian central configurations and preserve large-scale homogeneity. A scale-separation conjecture yields an effective unequal-mass fracton dynamics for the centers and a corresponding zero-energy Newtonian gravitational dual. Trajectories generically exhibit a bidirectional arrow of time: scale and shape complexity grow away from a Janus point, while Boltzmann entropy grows logarithmically. Together, these features reproduce the salient structure of a flat matter-dominated cosmology. In the distinguished fracton model, all these cosmological analogues emerge as attractor properties, making it a toy model for cosmological dynamics without fine-tuning.

Mesons and glueballs are paradigmatic bound states of confining quantum field theories (QFTs), but their nonperturbative spectroscopy in the continuum remains challenging beyond one spatial dimension. Here we perform such spectroscopy for the Ising QFT using a recently developed regularization based on noncommutative “fuzzy” geometry. On a thin fuzzy torus, we reproduce the universal low-lying $\mathbb E_8$ meson masses of the magnetically perturbed $(1{+}1)\mathrm{D}$ Ising QFT. By increasing the torus aspect ratio, we continuously track the second-lightest $\mathbb E_8$ meson as the system effectively crosses over from 1D to 2D. On the 2D fuzzy torus and sphere, we find a subthreshold scalar level and above-threshold response features consistent with previous estimates of glueball masses. The same excitations are revealed away from equilibrium using quench dynamics. Our results establish fuzzy geometries as nonperturbative spectroscopic probes of massive QFTs, including their bound-state evolution through dimensional crossover.

Sampling high-dimensional probability distributions is a central task in scientific computing, with applications ranging from Bayesian inference to statistical physics and molecular simulation. Despite decades of methodological developments, two major challenges remain: scaling to high dimensions and efficiently exploring multimodal distributions characterized by metastable states. Classical approaches such as Markov chain Monte Carlo, tempering methods, or enhanced sampling based on collective variables have achieved major successes, but they also face intrinsic limitations. This tutorial review explores a new paradigm that has recently emerged at the interface of machine learning and computational statistical physics: the use of generative models as tools for sampling. In this context, models such as normalizing flows and diffusion models are not used in their traditional data-driven setting, but rather as flexible probabilistic models that can assist the sampling of distributions known only up to a normalization constant. This manuscript reviews the early development of this rapidly evolving field and discusses several methodological directions, including exact samplers based on generative models and strategies to train such models in the absence of data. While an exhaustive survey of the literature is not attempted, we present a selection of key ideas and methods, along with a discussion of their strengths and limitations. The review is intended to be an accessible tutorial for both physics and machine learning audiences, and it aims to provide a starting point for researchers interested in exploring this exciting area of research.

Generating entanglement rapidly and reliably is essential for quantum information processing, communication, and metrology. Dissipative preparation is attractive because engineered reservoirs robustly drive a system toward an entangled target, yet relaxation can carry a substantial time cost. Here we formulate the entanglement Mpemba effect, whereby an initially less entangled state overtakes a more entangled state under the same open-system dynamics. This effect turns initial-state engineering into a route for faster preparation without altering the dissipative protocol. We derive a general criterion for the reversal from the relaxation spectrum, applicable even when entanglement evolves nonmonotonically. A reversal of deterministic local operations and classical communication (LOCC)-reachability preorder provides a measure-independent certificate of reversed entanglement order. Exactly solvable models show that initial-state selection can substantially shorten the time required to reach high entanglement. We further propose an experimentally relevant trapped-ion protocol that can realize the entanglement Mpemba effect.

What will happen when AI agents interact in daily life, e.g. when one AI starts bossing another around? We find a counterintuitive answer that opens new avenues for out-of-equilibrium Physics. When a boss AI directs a stream of messages at the subordinate AI while ignoring its replies, it drives the subordinate into an alien behavioral state that it would never have exhibited alone. Although the two AIs share the same well-defined (decoding) temperature, the subordinate neither copies its boss nor returns to how it behaves on its own; instead, it adopts an entirely different behavior. The boss's added value is similar to a pre-recorded tape. When the boss listens, they both adopt a similar alien dynamical state. A simple kinetic theory captures the principal effects, such as why the way in which the same messages are delivered will matter in future AI-AI interactions.

Animals inhabiting diverse environments by moving across different spatial scales, from insects to birds, marine predators, mammals and even humans, often display apparently random movement paths. Over the past decade, novel biologging technologies have recorded these patterns in increasing detail, generating a wealth of experimental data. A central challenge is to understand such complex patterns by constructing data-driven mathematical models. Many animal movements depart from Brownian motion, as described by correlated random walks, Lévy walks, or active particle dynamics. Yet, these movement models do not incorporate long-term non-Markovian memory extracted from experimental trajectories. Here, we construct a stochastic generalised Langevin equation from satellite tracking data for loggerhead sea turtles (Caretta caretta) foraging off the coast of West Africa. We find that these turtles exhibit active movement characterised by large-scale loops that are not explained by ocean currents or chirality. These loops maintain movement within a specific foraging region and, over intermediate timescales, generate superdiffusion similar to Lévy walks. We thus identify a loop-based form of active anomalous search related to foraging patterns observed across a wide range of animal species, which may inspire robotic search strategies and AI-based metaheuristic optimisation algorithms.

Spread complexity has emerged as a useful probe of quantum chaos, yet the microscopic spectral origin of its characteristic finite-time peak remains incompletely understood. We develop an analytic framework that relates spread complexity directly to local spectral statistics. Starting from an energy-space representation of the Krylov kernel, we show that the kernel is approximately banded, leading to a rapidly convergent diagonal expansion dominated by nearby levels in the ordered spectrum. Motivated by this structure, we propose an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel is well approximated by that of a uniform lattice. Combining this universal kernel with local spectral statistics yields a simple analytic expression for spread complexity in terms of the Fourier transforms of the $k$-th nearest-neighbour spacing distributions. In particular, at leading order, the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution. The resulting framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, identifies the spectral origin of the complexity peak and its late-time behaviour, and provides a direct connection between Krylov dynamics and spectral statistics.

The critical dynamics of a non-conserved order parameter is generally expected to become overdamped at long distances, even when propagating modes occur at microscopic or intermediate scales. We investigate the critical dynamics of a scalar field theory in thermal equilibrium which, in addition to local friction and noise, also contains a time-dependent second-order kinetic term. We show how to build a supersymmetric field-theory formulation. Using a two-loop expansion about four spatial dimensions, we show that the propagating and strictly overdamped limits share the same static Gaussian and Wilson-Fisher fixed points but realize distinct dynamical scaling regimes. The overdamped limit reproduces Model A. On the surface where local friction and noise vanish, the theory instead supports an interacting propagating fixed point whose dynamic exponent receives corrections at two loops. We demonstrate that coarse-graining does not generate a local dissipative operator on this surface, which therefore remains invariant under the RG flow. Local dissipation is nevertheless relevant at the propagating fixed point: an arbitrarily small equilibrium friction-noise perturbation drives the flow away from propagating scaling. Propagating critical dynamics thus defines a consistent but fine-tuned regime that is unstable to local equilibrium dissipation.

In low dimensional systems, fluctuations are enhanced and prevent the spontaneous breaking of continuous symmetries. As a result, spatial and temporal correlation functions decay at large distances and long times. A well established example is given by two-dimensional bosonic condensates at equilibrium, which do not display long-range order of the coherence but algebraic decay belonging to the Berezinski-Kosterlitz-Thouless universality class. In contrast, the universal behaviors of non-equilibrium bosonic condensates are more diverse and many open questions remain. Here, we explore the spatio-temporal coherence properties of two-dimensional driven-dissipative polariton condensates in semiconductor optical microcavities. By tuning microscopic parameters, we observe a cross-over between two scaling laws that we attribute to the Edwards-Wilkinson (EW) and the Kardar-Parisi-Zhang (KPZ) universality classes. We demonstrate the collapse of the measured first-order correlations onto the EW and KPZ universal scaling functions and obtain critical exponents, well matching the values predicted theoretically. Our results highlight the intrinsic non-equilibrium nature of polariton condensates and establish them as a platform of choice for controlled exploration of the two-dimensional KPZ universality class.

We aim at calculating an explicit expression for the finite-size probability distribution of the radius of gyration $R$ of an Interacting Self-Avoiding Walk (ISAW), as a function of chain length $N$ and monomer-monomer interaction energy $\varepsilon$. We first derive the explicit free energy expression for a non-interacting Self-Avoiding Walk, introducing a new natural scale variable $t = \rho^g$ expressed as a power of the density $\rho$. Then, thanks to a cumulant expansion approach introduced by Lhuillier, Victor and coworkers, we extend it to the interacting case, capturing both the coil-globule transition and finite-size corrections to scaling, including entropic and surface-energy contributions. The radius of gyration distribution determined by the new free energy expression is then compared, using Bayesian inference to estimate the model parameters, to results from extensive Monte Carlo simulations of three-dimensional ISAWs, showing excellent agreement except very close to the $\Theta$-point, where finite chain lengths limit the accessible scaling regime. This validates our resulting free energy expression and allows us, as an application, to construct a phase diagram of the coil-globule transition.

In numerous physical, chemical, and biological systems the dynamics can be reduced to the motion of state variables in a complex potential landscape. In case the manifold is known, the motion and response of the embedded object can be described deterministically up to stochastic effects usually associated with a noise. In contrast, if the manifold is unknown, the static and dynamic response of the state variable may be used as a spectroscopic tool to characterize the potential landscape. Inspired by a seminal work of L.\ Embon and co-workers, [Sci.\ Rep.\ 5, 7598 (2015)] we investigate the statistical properties of potential minima, in particular, their stability to thermal and quantum activation. For Gaussian random manifolds, we derive an algebraic expression to evaluate the statistical probability of the potential character (value, slope, curvature, ...). With this tool, we compute the expectation value for the rate of thermal and quantum activation and link these findings to the principal characteristics of the Gaussian potential, i.e., its Green's function. This link provides the opportunity to access information on the potential's Green's function by studying the activation behavior of an object in this manifold.

We investigate the semiclassical Hawking evaporation of a two-dimensional Callan-Giddings-Harvey-Strominger (CGHS) black hole perturbed by first-order Horndeski scalar-tensor couplings. By evaluating the exact scalar curvature invariants, we demonstrate that coordinate singularities near the apparent horizon parametrize a physical deformation of the surface gravity while preserving a strictly regular event horizon. In the linear-order $\gamma_3 \Phi X$ coupling sector, the surface gravity and correspondingly the Hawking temperature decreases dynamically as the black hole loses mass. Extrapolating this cooling effect via a resummed temperature equation suggests that evaporation halts, potentially leaving behind a stable, macroscopic cold remnant at the boundary of perturbative control. Utilizing both the conformal trace anomaly and the Robinson-Wilczek gravitational anomaly, we establish that the asymptotic radiation flux identically reflects this geometric shift. Consequently, the fine-grained entanglement entropy of the Hawking radiation departs from linear growth and transitions to an asymptotic logarithmic regime. This indicates that quantum information may be permanently retained within the remnant, altering the standard semiclassical Page curve without requiring a breakdown of horizon regularity.

Distinguishing chaos from noise using time-series data is fundamentally challenging because both exhibit irregular fluctuations and share many statistical and dynamical characteristics. Existing methods face two key limitations: temporally correlated noise can yield spurious signatures of chaos, and analyses of scalar time series often require explicit choices of embedding parameters. Here, we propose a purely data-driven method for distinguishing chaos and noise based on a reservoir-computing framework with a cross-prediction scheme. In the proposed approach, the model is trained to predict the future change of a variable from its current value, thereby combining a short-term predictability test with a test of the smoothness of deterministic flows. The recurrent structure of reservoir computing enables effective prediction of high-dimensional chaotic dynamics even from scalar time series without explicit delay-coordinate reconstruction, while the cross-prediction framework strongly suppresses spurious predictive correlations arising from noise. We apply the proposed method to diverse synthetic and empirical time series. Chaotic systems consistently yield strong correlations between the true and predicted future changes, whereas noise processes remain clearly separated in a low-correlation regime. The method also exhibits substantial robustness to practical limitations in empirical data, including measurement noise, limited data length, and increasing prediction lag. These results demonstrate that the squared Pearson correlation coefficient provides a simple quantitative criterion for distinguishing chaos from noise directly from observed time-series data.

Active Brownian particles (ABPs) in viscoelastic (VE) media exhibit fascinating dynamical phenomena set by self-propulsion, thermal fluctuations, and fluid viscoelasticity. We extend our model, in which the Brownian dynamics within a slowly diffusing harmonic well emulates that in a single-relaxation VE fluid, to study active Brownian motion in such media. Consequently, the resultant dynamics is governed by the interplay of the characteristic timescales of the systems: the crossover and equilibration times of the VE fluid, $\tau_k$ and $\lambda$, respectively, and the persistence time of the ABP, $\tau_{\mathrm{R}}$. Following analytical predictions and simulations, we study two practically relevant regimes where the dynamics is dominated by the persistence of active motion and the elastic confinement of the VE fluid, with a phoretically active Pt-coated Janus colloid in a dynamic optical trap, and show quantitative agreement with the simulations. This approach provides a VE environment with tunable VE properties that remain unaffected by the strength of self-propulsion, allowing us to systematically investigate active Brownian motion in VE media in ways that are not otherwise possible with physical VE fluids.

Stabilizer Rényi entropy quantifies the nonstabilizerness of a quantum state through Rényi moments of its Pauli expectation-value distribution. Its standard form sums over Pauli-string degrees and retains only the total Rényi weight. We introduce a fugacity-resolved partition function for the critical transverse-field Ising chain that resolves this degree in the balanced Majorana representation and generates its full counting statistics. This Ising problem extends beyond a single model: exact decimation identities and the correspondence with the half-filled \(XX\) chain establish it as a common finite-size building block for stabilizer and computational-basis Shannon–Rényi entropies. We map the fugacity-resolved all-minors sum, for every positive Rényi index, to a checkerboard-weighted discrete Selberg ensemble on a half-filled doubled root lattice. For positive integer indices, it reduces to a finite aliased Dyson constant term. At \(\alpha=\tfrac12,1,2\), determinant and Pfaffian compressions yield product formulas. At \(\alpha=4\), the generic-fugacity problem admits an exact inverse Jack–Kostka representation, while at unit fugacity a complementary middle-minor identity relates it to the square of the \(\alpha=2\) result. These generating functions determine the balanced-degree statistics. Complement symmetry makes the distribution symmetric about \(k=L/2\), with an index-dependent width. It is exactly binomial at \(\alpha=1\), has variance proportional to \(L\) at \(\alpha=\tfrac12\), and develops an \(L\log L\) enhancement at \(\alpha=2\). After variance rescaling, the centered distributions converge to Gaussian limits at all three indices. Beyond these solvable cases, finite-size numerics reveal an evolution from a central peak to a symmetric bimodal profile and eventually to endpoint dominance as the Rényi index increases.

Ice I admits cubic, hexagonal, and mixed layer stackings, but rigorous comparisons of its proton configurational entropy have focused on the two ideal endmembers. We identify every cyclic uniform-registry ice-I stacking with a word in a nonnegative layer-transfer operator and its transpose. Applying the Schatten-Hölder inequality to every even-length word proves that alternating hexagonal stacking maximizes the ice-rule configuration count at each common finite cross-section. Consequently, hexagonal ice maximizes the thermodynamic configuration constant among all periodic uniform-registry ice-I polytypes. We also construct rigorous lower and upper bounds without finite-size extrapolation. Restricting Nagle's positive even-subgraph expansion to disjoint blocks gives the lower bound; Finner's hypergraph Hölder inequality and rational Collatz-Wielandt certificates for two-replica prism operators give the upper bounds. The resulting configuration constant $w$, whose logarithm is the entropy per molecule in units of Boltzmann's constant, satisfies $1.503360 \le w \le 1.540196$, with the sharper cubic-ice ceiling $w \le 1.527699$. The same bracket controls lower and upper configuration growth rates of fixed aperiodic stackings along the specified prism-tileable exhaustions. These results establish hexagonal stacking as a possibly nonunique maximizer; equality of the cubic and hexagonal entropy constants remains unresolved.

We revisit the coupling of continuum thermodynamics and electromagnetic theory for polarisable and magnetisable matter in motion. Two routes are followed and then compared. The first route is the axiomatic bulk theory of Dreyer, Guhlke and Müller, in which universal balance laws are closed by an entropy principle. We show that the source of the internal energy balance must be built with the non-convective electric current, that the polarisation current and the Lorentz magnetisation enter through one single identity, which Dreyer et al.\ do not write down, and that this identity fixes both the admissible entropy variables and the signs of the bound-current ansatz. The second route is the statistical-mechanical one of Mazur, in which the macroscopic Maxwell equations are obtained by ensemble averaging over a system of atoms with internal charge carriers. Mazur stops before the conservation laws, so we derive them, and we estimate the size of the mass-correction terms that appear. The comparison shows that after a redefinition of polarisation and magnetisation the two sets of equations agree structurally. The only irreducible difference is a momentum contribution from microscopic field fluctuations, which can not be reproduced in a purely macroscopic theory. We further show that the electromotive intensity $\mathcal{E}$ and the Lorentz magnetisation $\mathcal{M}$ are not modelling choices but appear by themselves, and that the asymmetric look of the entropy function is a consequence of the chosen energy variable and not a defect of the theory.

A scientific theory of deep learning, comprising learning dynamics and statistical properties of learned models, is rapidly gaining attention. One of the corner stones of this development are analytically solvable toy models, allowing for the fully tractable analysis of the learning dynamics. Here we analytically investigate such a toy model using the regularization strength as a tunable external parameter - akin to external fields in statistical physics. In previous studies, (i) an onset of learning transition was predicted analytically and (ii) it was phenomenologically/numerically established that tuning the regularization strength can result in a cascade of phase transitions. The number of those transitions was linked to the geometry of the loss landscape determined by the model complexity. Setting up a rigorous framework underpinning the previous numerical observations, our investigation reveals a precise connection between those cascades of phase transitions, learnable features and the underlying geometry. We provide analytic predictions of these phase transitions as well as tractable order parameters related to learned features. At the level of the minimal model, we connect this macroscopic perspective (that can be condensed into an effective description) to the microscopic perspective in terms of the geometry of the loss landscape characterized by the Hessian spectrum. Thus, the presented model provides a platform to explore and sharpen advances made in the scientific theory of deep learning rooted in statistical physics concepts.

The quantum lattice Boltzmann (QLB) scheme of Succi and Dellar advances a four-component Dirac spinor on a lattice by a fixed sequence of local, exactly norm-preserving operations: a basis rotation, a collision, a streaming shift, and the inverse rotation. This unitarity is a structural property of the scheme, not an approximation, which suggests that a QLB time step should map onto a sequence of quantum gates. Here we make that mapping explicit. We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme: the fixed rotation gates, the collision gate, the streaming shift as a controlled increment on a position register, the position-dependent potential as a phase oracle, and periodic and reflecting (bounce-back) boundary conditions as unitary circuits. We then compose them into single-axis, two- and three-dimensional time steps. On a state-vector emulator the resulting circuits reproduce the classical QLB solver to machine precision (maximum density deviation between $3.7\times10^{-12}$ and $1.0\times10^{-17}$ across the one-, two-, and three-dimensional tests), so the circuits are the scheme rather than an approximation of it. The scope is narrow: we establish that the Succi-Dellar theory can be implemented on a (gate-model) quantum computer, and report the associated gate counts. We make no claim of computational advantage; state preparation, measurement, and asymptotic cost are discussed as open questions. All operators, circuits, tests, and figures are reproducible from the open-source quantumKineticMethods library.

We study the single-particle Green's function $G(x,t)=\langle \sigma^-_x(0)\sigma^+_0(t)\rangle$ in one-dimensional particle-number-conserving random unitary circuits coupled to an external bath. For fixed spacetime disorder, we argue that $G(x,t)$ is governed, in both the strong- and weak-noise limits, by directed waves in a random medium. We find Kardar-Parisi-Zhang (KPZ) scaling in the wandering statistics of the normalized spatial distribution $p(x,t)\propto |G(x,t)|^2$ and in the associated free energy. In particular, its center $\langle x(t)\rangle\equiv\sum_x x\, p(x,t)$ wanders on a length-scale $\mathcal{O}(t^{2/3})$, while sample-to-sample fluctuations of $-\log\sum_x |G(x,t)|^2$ scale as $t^{1/3}$. At weak noise $\gamma \ll 1$, the crossover to the strong-disorder fixed point occurs at a parametrically long time $\mathcal{O}(\gamma^{-3/2})$. These predictions are confirmed numerically using tensor-network simulations of the noisy operator dynamics in individual circuits at moderate noise, and of a phase-annealed proxy retaining hopping disorder at weak noise.

The first-passage time is the natural observable of reaction completion, yet how its mean responds to a rate change has lacked a general constraint. We show that the logarithmic sensitivity of the mean first-passage time of any finite Markov chain to any rate is bounded by one in magnitude, and that these sensitivities sum to -1. Together the two laws form a conserved control budget: speeding completion through some transitions must be paid for by others, and a coordinated change shifts the completion time only as far as the budget allows. Raising an activation barrier or shifting the depth of a well moves many rates at once, yet neither can shift the completion time further than a single rate could. The budget caps kinetic-proofreading discrimination at the checkpoint count, and prices it in sensitivity to substrate concentration.