Papers

The quantum-thermal escape rate from a metastable well in the absence of dissipation is customarily written as a Boltzmann average of the Hill-Wheeler flux over a continuum of energies. However, this continuum treatment diverges exponentially at low temperature. The divergence originates in a mismatch between a quantum partition function and a classical flux integral; retaining the discrete nature of the quasibound spectrum removes it exactly, and the resulting semiclassical Arrhenius law is obtained in closed form on both sides of the crossover temperature. Evaluating a uniform Kemble transmission probability on Bohr-Sommerfeld levels further recovers the standard WKB decay rate of the lowest resonance at zero temperature. Benchmarked against the resonances of cubic and quintic potentials, obtained by complex scaling, this uniform semiclassical result stays within $9\%$ of the exact rate over eleven orders of magnitude.

We investigate electrons subject to a magnetic field that may tunnel along a square lattice and additionally to two electronic leads. Using non-equilibrium Greens functions we evaluate the transmission and from that particle and energy currents and zero-frequency noise of the setup. We identify sweet spots for which the transmission reaches unity over broad frequency windows – carried by topologically protected boundary modes that are quite robust against local fluctuations of on-site energies. These transmission plateaus can be exploited for the implementation robust current standards, clocks and thermoelectric devices.

We uncover an emergent phase separation in a system of inertial dumbbells put into persistent rotation by a constant applied torque, breaking chiral symmetry, while interacting with each other through repulsive hard-core interactions. We numerically determine the phase diagram of the model, showing that there exists a binodal and a spinodal region where the system separates into a dense liquid and an ordered phase, which we identify as hexatic. An analysis of the system pressure unveils that the mechanism underlying the phase separation is heavily controlled by inertia. Locally injected kinetic energy is steadily stored into the system in a way that depends on density and that, at large inertia, results in a hot dilute and cool dense state, leading to a generically unstable homogeneous phase. The interplay between dumbbells' arrangement into clusters and chiral forces produces a spontaneous rigid rotation of the clusters around their centers of mass.

We investigate the emergence of rigid polycrystalline structures in atomistic ionic particle systems as low-energy configurations. The interaction between particles of opposite charge is modeled by hard spheres that interact when they are tangential. The interaction between particles of same charge is modeled as a hard repulsion that forces a minimal distance between them. The atomistic energy is frame invariant, and no underlying reference lattice is assumed on the ionic configurations. The asymptotic behavior of configurations with finite surface energy scaling is identified by means of $\Gamma$-convergence. The related continuum theory is described by piecewise constant fields that encode the local orientation of the configuration. The limiting energy is local and concentrates at grain boundaries, which correspond to the boundaries of the regions where the underlying configuration has a constant orientation. The limiting energy density is anisotropic and depends on the relative misorientation of the two grains, their translation misfit, and the normal to their interface. Furthermore, we perform a fine analysis of surface energies for solid-solid and solid-vacuum phase transitions and determine energetically favorable orientation mismatches. This relies on a structure result for our grain boundaries, which shows that, due to the rigid setup, interpolating layers near the grain interface are energetically not favorable.

The Lorenz system is the canonical low-order model of convective instability, yet its dissipative and driving terms have never been checked against, nor constructed from, an explicit thermodynamic bookkeeping. We derive a modification that satisfies the local-detailed-balance condition for macroscopic relaxation toward nonequilibrium steady states, thereby identifying the thermodynamic force, entropy-production rate and frenesy of the resulting flow. The resulting model produces a transition from a quiescent fixed point to a robust, large-amplitude relaxation oscillation, closely analogous to recharge-discharge oscillator paradigms used for the El Nino-Southern Oscillation. The system alternates between a long, nearly reversible recharge phase and a brief, violently frenetic discharge burst, during which essentially all of the cycle's activity and entropy production is concentrated.

Dense granular flows are commonly investigated using the soft-sphere Discrete Element Method (DEM), whereas large-scale applications generally require continuum models. Granular temperature, defined as the variance of particle velocity fluctuations, is a relevant variable to collapse rheological scaling. However, experimental forcing approaches used to control temperature also affects other aspects of the granular state. An alternative is to use a thermostat algorithm to control the fluctuation energy directly. Although thermostat algorithms are well established in Molecular Dynamics (MD), their behaviour in dissipative DEM systems has received comparatively little attention. We show that conventional Langevin and Nosé-Hoover thermostats exhibit complementary limitations: Langevin control requires strong coupling to offset collisional dissipation, thereby damping particle dynamics, whereas Nosé-Hoover does not independently disrupt the correlations and segregation generated by repeated inelastic collisions, leading to non-ergodic and numerically unstable states. To address these limitations, we introduce two pairwise hybrid formulations combining deterministic temperature regulation with stochastic decorrelation. Both enforce the prescribed temperature, while variation of the stochastic decorrelation timescale modifies velocity statistics and associated mesoscopic properties such as diffusivity. Finally, application to pressure-controlled simple shear shows that increasing granular temperature at fixed inertial number reduces the apparent friction, consistent with previously reported trends. The framework therefore provides controlled reference states for comparing differently forced granular systems and identifying the variables required for temperature-dependent constitutive models.

Soft materials linked by reversible covalent or supramolecular bonds form a diverse class of assemblies with promising applications from nanoscience to medicine. Experiments typically probe bulk phase behavior and rheology, but it remains difficult to measure how microscopic bonding kinetics and the mechanics of the constituent elements give rise to bulk properties. Coarse-grained molecular dynamics (MD) simulations can bridge these scales, but most simulation approaches do not control individual bond kinetics, and those that do were mostly developed for bespoke applications that do not readily generalize. Here we present pySNAP (Simulation and Network Assembly Pipelines), a modular open source Python platform that integrates tunable dynamic bonding with a workflow, template, and analysis setup, so that users can study a wide range of systems with only small changes to input files. The platform is built on the GPU-accelerated HOOMD-blue MD engine and integrates DyBond, a GPU-accelerated plugin that forms and breaks bonds consistent with an equilibrium distribution and supports bonding between multiple types of partner species. Around this core, the snap_simulate package compiles a directory of parameter files into a HOOMD-blue simulation, and the snap_workflow package orchestrates the resulting parameter sweeps across workstations and high-performance computing schedulers. We describe the theory behind simulated dynamic bonding and how to use the package, from setting up a parameter sweep to analyzing its results, and demonstrate the framework on a diverse range of dynamically bonded systems, showing that it accommodates distinct interaction mechanisms, geometries, and physical scenarios within a unified workflow, while enabling both reproduction of existing models and rapid construction of more complex composite systems.

Boltzmann counting possesses an exact scale invariance from which the specific entropy emerges without asymptotic approximation. Under thermodynamic replication, this invariance exposes the Gibbs excess at the multinomial level. Its resolution reveals a state-dependent scale structure whose additive refinement generates a continuum reference measure, giving a combinatorial origin to the reference structure of continuous statistical mechanics. The resulting entropy is relative to this measure, and for an ideal gas the scales acquire operational meaning through reversible work.

We prove that, for almost every uniformly random configuration of $N$ particles in a two-dimensional periodic square domain, there exists a unique nearest stealthy hyperuniform configuration, up to floppy-mode displacements, when a finite set of $m$ low-$k$ Fourier modes is constrained to vanish exactly. Under an explicit full-rank Jacobian assumption, the displacement component orthogonal to the tangent space of the finite-mode hyperuniform manifold is uniquely determined, while residual tangent-space degrees of freedom correspond to first-order floppy modes that preserve the constrained low-$k$ density fluctuations. We restrict attention to the regime of constrained-mode fraction $\chi=m/(dN)$ in which such configurations are known to remain disordered rather than crystallize. To corroborate this framework, we implement a gradient-based generator network that iteratively displaces particles to minimize a combined loss functional of hyperuniformity, short-range repulsion, and smoothness, and whose dynamics is expected to approximate the theory's minimal orthogonal projection onto the hyperuniform manifold. The network drives generic random configurations toward disordered hyperuniform representatives, progressively suppressing long-wavelength density fluctuations as measured by the structure factor and number-variance exponent\textcolor{black}{, over a reciprocal-space window that extends well beyond the explicitly constrained modes and is bounded, as we show, by the structure-factor sum rule. Local bond-orientational analysis and the absence of Bragg peaks confirm that the resulting states remain disordered}. This integrated analytical and computational approach provides a rigorous geometric foundation for understanding stealthy hyperuniformity and for generating its disordered representatives from generic random configurations.

We investigate the nonequilibrium evolution of density correlations in a thermal one-dimensional Bose gas following a trap quench. We focus on the Tonks–Girardeau limit and develop an analytical description based on a finite-temperature extension of Quantum Generalized Hydrodynamics. We benchmark our predictions against exact numerics for a quench from a double well to a harmonic trap and for a trap release, and assess their temperature range of validity. Our results capture the dynamical enhancement of correlations during the expansion stages, namely the crossover from thermally suppressed to power-law bulk correlations. For the trap release, the conformal distance associated with fixed bulk points decreases as $1/t$, so the correlations progressively probe a universal regime~$\sim1/x^2$ that is independent of temperature. We thus show that, in the hydrodynamic long-time regime, the bulk density correlations approach their zero-temperature equilibrium form independently of the initial temperature.

We study the first-passage time statistics of an active Ornstein–Uhlenbeck particle on a finite one-dimensional interval with an absorbing boundary at one end and a reflecting boundary at the other, subject to stochastic resetting. In the weak-activity regime, we develop a perturbative solution of the Fokker–Planck equation and use a renewal framework to obtain analytical expressions for the first-passage-time distribution and its low-order moments. In the absence of resetting, activity can either increase or decrease the mean first-passage time, depending on the relation between the persistence time and the diffusive time to absorption. When resetting is introduced, we determine both the onset of beneficial resetting and the finite range of resetting rates for which resetting lowers the mean first-passage time of the active system. We further show that the resetting transition depends on the velocity resetting protocol. Finally, by comparing the active particle with resetting to the passive Brownian particle without resetting, we construct a phase diagram that identifies the regions of parameter space in which activity and resetting together reduce the mean first-passage time.

The construction of a reliable renormalization group (RG) flow for discrete gravity models that preserves their underlying symmetry group, typically $U(N)$ or $O(N)$, remains an open problem. For random matrix models, which are the focus of this paper, this symmetry is intrinsically tied to the interactions encoding the random geometry of two-dimensional quantum Euclidean spacetime. We develop a novel approach based on the introduction of a partial matrix-valued intermediate field. In the large-$N$ limit, measure concentration strongly suppresses fluctuations of its singular values, allowing it to play the role of a self-consistent background field. This provides the basis for a relational RG in which the notion of scale is dynamically induced by the effective Gaussian measure in the basis where the intermediate field is diagonal. Our construction preserves the symmetry of the original model and admits a well-defined continuum limit. We show that the resulting infrared theory is described by a three-dimensional non-local Euclidean field theory with a non-trivial Wilson-Fisher-like fixed point and a single relevant direction. Remarkably, the associated critical exponent exactly matches the standard double-scaling exponent. We finally discuss extensions of the background-field approach to other discrete gravity models, such as random tensor models, and to different symmetry groups, as well as connections with more formal RG frameworks and information geometry.

We study the imaginary-time evolution of a domain-wall state constrained to return to itself in a staggered free-fermion chain. The resulting space-time profile separates into frozen, liquid, and gas phases divided by sharp boundaries known as arctic curves. The spectral gap produces an incompressible half-filled gas phase bounded by an inner arctic curve, in addition to the outer frozen-liquid boundary. We determine both arctic curves, the thermodynamic return amplitude, and the complete equal-time correlation kernel. Correlations decay algebraically in the liquid regions and exponentially in the gas, while both arctic boundaries arise as caustics of free quasiparticle trajectories. The return amplitude and correlation kernel are respectively controlled by the determinant and inverse of the same block-Toeplitz operator. We obtain an exact matrix Wiener-Hopf factorization of this operator by reducing the problem to a scalar Riemann-Hilbert problem on an elliptic spectral curve. The factorization also yields the exact thermodynamic return amplitude: its logarithm consists of a quadratic term with explicit gap dependence and a bounded periodic theta-function correction. The resulting frozen-liquid-gas structure is a continuous-time free-fermion counterpart of that found in doubly periodic dimer models.

Entanglement and magic are resources that reveal complementary aspects of quantum many-body systems. Their interplay is captured by nonlocal magic, the magic that survives arbitrary local changes of basis. Yet their markedly different dynamical behavior leaves open how this irreducible component of magic spreads. Here we connect nonlocal magic to the capacity of entanglement, a tractable quantity measuring fluctuations of the entanglement Hamiltonian. This connection enables analytically controlled predictions across a wide range of many-body dynamics, from chaotic to integrable systems, which we investigate also using large-scale numerical simulations. In chaotic systems, nonlocal magic exhibits a transient buildup, with logarithmic growth in time followed by decay to a size-independent value. For integrable systems, we develop a quasiparticle picture in quantitative agreement with numerics, showing that the same initial growth instead leads to saturation at a value logarithmic in subsystem size. These contrasting behaviors have an operational consequence for entanglement embezzlement: the strongest scramblers embezzle only transiently, whereas free-fermionic dynamics can sustain universal embezzlement in the steady state.

We identify a mechanism in which soft nonunitary fluctuations of the pair condensate can suppress phase coherence in a two-dimensional Rashba superconductor proximate to easy-plane ferromagnetic order. This spin polarization couples linearly to the supercurrent through a Lifshitz invariant, acting as an emergent gauge field for the superconducting $\mathrm{U}(1)$ phase. Integrating out soft magnetic fluctuations renormalizes the effective superfluid stiffness and drives the system into a "smectic" superconducting state, characterized by a finite-momentum helical phase texture and a long-wavelength stiffness that vanishes in one direction. Consequently, vortices no longer have a logarithmically divergent energy cost, and the quasi-long-range superconducting order is destroyed at any finite temperature. Weak lattice anisotropy pins the ferromagnetic order, cutting off the smectic-like behaviour at long wavelengths and thereby restoring a finite effective stiffness. Nonetheless, near the onset of magnetic order, the enhanced magnetic susceptibility leads to a vortex-deconfined regime for arbitrarily weak magnetoelectric coupling. The results provide insight into how intertwined superconducting and easy-plane ferromagnetic orders can reshape phase coherence, with relevance to rhombohedral graphene proximitized by a transition-metal dichalcogenide substrate.

We introduce a bootstrap method for deriving bounds on disorder-averaged observables in classical statistical systems with quenched disorder. The three main ingredients of the bootstrap are the positivity of ensembles of probability measures, disorder-averaged equations of motion, and explicit expectation values of random coupling variables. Together, these ingredients lead to a linear programming problem over the space of disorder averages. By further employing the positivity of the two-replica Gram matrix, we also formulate a semidefinite programming problem that produces bounds on two-replica correlators. We demonstrate the method using two examples: the two-dimensional random site-diluted Ising model and the one-dimensional disordered contact process. For the latter example, the bootstrap bounds exhibit a region of multiple kinks that we relate to the Griffiths region, whose finite-size scaling analysis yields a typical correlation length exponent in good numerical agreement with the known value.

Inspired by recent ARPES studies on multilayer ($n\geq3$ layers) cuprate superconductors, we use the unrestricted Hartree-Fock approximation to explore the ground-state phase diagram of two coupled, inequivalently doped square-lattice Hubbard layers. In the decoupled-layer limit, the lightly hole-doped ground state is typically an incommensurate spin-stripe state. However, with sufficiently strong interlayer coupling, stripe order is destabilized relative to a commensurate Néel-ordered metal. The resulting state exhibits a reconstructed Fermi surface with hole pockets centered at $(\pm\pi/2,\pm\pi/2)$ that are similar in character to those seen in experiments. Our results illustrate the qualitatively new physics that can arise from interlayer coupling in multilayer cuprates.

We report improved global-minimum configurations for the classical Thomson problem of $N=60$, $61$, $92$, and $99$ repulsive Coulomb charges confined to a disk. By combining the quenched molecular dynamics (QMD) method with the fixed-border heuristic introduced by Amore and Zarate, we systematically obtain configurations with energies $E_{\mathrm{QMD}}(60)=2159.3584240930$, $E_{\mathrm{QMD}}(61)=2237.19264190$, $E_{\mathrm{QMD}}(92)=5358.35353314$, and $E_{\mathrm{QMD}}(99)=6254.83029083$, which improve upon the previously best-known values. For $N=60$, the Voronoi diagram of our configuration differs from the one reported earlier. The symmetries for $N=61$ $(C_{2})$ and $N=99$ $(D_{1})$ are confirmed and are consistent with the known symmetry patterns for these configurations. For $N=92$, whereas the previously reported Voronoi diagram has lower symmetry, our solution exhibits a clear $D_{1}$ (axial) symmetry of defects. The results are highly reproducible across multiple independent runs, providing strong evidence that these configurations are robust global-minimum candidates.

Group interactions are widespread, and higher-order extensions of familiar models display collective phenomena absent from their pairwise baselines, which are routinely offered as evidence of a distinct higher-order physics. We ask whether that claim survives the test that gives new a precise meaning in statistical physics: that of universality. Revisiting canonical higher-order models, we argue that what classifies collective behavior are the infrared ingredients that survive at long scales. Arity is not a universality label. Beyond universality, we examine two further questions: whether higher-order structure is a fact about the system or a choice of description, and what data can and cannot tell us about interaction order. Higher-order descriptions remain indispensable when they expose the organizing structure, provide a better mechanistic language, or improve prediction. We close with what should be measured before new phenomena can be claimed, and where higher-order structure already earns its place.

Active Brownian particles confined to rigid substrates are known to accumulate near rigid boundaries and, under suitable conditions, undergo motility-induced phase separation (MIPS). A particularly intriguing manifestation of this behavior is the formation of self-sustained vortices around circular obstacles, which act as localized nucleation sites for particle aggregation. While several dynamical properties of such vortices have been previously characterized, their behavior in the thermodynamic limit remains largely unexplored. Here, we investigate how the mass and spatial extent of a dry active-matter vortex scale with system size. Using numerical simulations of repulsive active Brownian Particles interacting with a fixed circular obstacle, we measure the vortex mass, mean radius, and maximum radius as functions of the global area fraction, obstacle size, and system size. We find two distinct scaling regimes. At low densities, the vortex remains localized and its characteristic properties saturate as the system size increases. Above a critical density, however, the vortex mass grows extensively with the total number of particles, while its spatial dimensions scale linearly with the system size, indicating the emergence of an obstacle-stabilized condensed state.

We investigate martingale properties of heat dissipated by generic non-equilibrium overdamped Langevin dynamics with multiple dimensions and transport coefficients that may depend explicitly on time and/or space. We find quantitative criteria that establish that heat dissipation may exhibit transient martingale (time-conserved), submartingale (time increasing), and supermartingale (time decreasing) properties in Langevin dynamics with drift coefficients with spatial non-linear dependencies. Furthermore, we reveal a tight link between the heat statistics at stopping (e.g. first-passage) times with that of a functional quantifying the phase-space contraction in the system dynamics. The theoretical results are used to predict the statistical properties of the heat dissipated by a spherical microscopic particle subject to gravity and double-layer forces and hydrodynamically interacting with a rigid wall, whose validity is confirmed by numerical simulations. These results extend the scope of stochastic energetics by revealing a non-trivial relation between heat fluctuations' statistical properties and the non-linear features of non-equilibrium processes.

The use of nonperturbative Wilsonian functional renormalization group (FRG) equations within an unavoidable approximation scheme produces physical results that depend on the choice of the coarse-graining regulator. This dependence of physical observables on the regulator can be minimized according to the principle of minimal sensitivity (PMS). For the 1PI formulation based on the Wetterich-Morris equation, we propose to investigate the optimal choice within a space of compactly supported polynomial regulators parametrized by a set of variables. We test this approach using the derivative expansion of the effective average action up to fourth order, applying it to the scalar field theory representing the three-dimensional Ising universality class and minimizing the dependence of the anomalous dimension on the regulator. At fourth order, we find appreciable corrections to the critical exponents that improve the predictions, compared to previous one parameter analyses. We also observe families of compact regulators that rapidly converge towards an optimal functional form as a small number of parameters is progressively introduced.

Fluctuation-response relations (FRRs) connect the linear response of a system to an external perturbation with properties of spontaneous fluctuations in the unperturbed system. We provide a simple derivation of FRRs for non-stationary dynamics following an almost standard way based on introducing a variable conjugated to perturbation (a dynamic conjugate). The derivation relies solely on the Markovianity of the underlying dynamics, with the only additional assumption that the transition PDF of the process is differentiable with respect to the strength of a constant, time- independent perturbation. The structure of this conjugate variable is however unusual: it is a two-time one, and reduces to a usual single-time conjugate in stationary situations. We show how several known results follow from this approach.

Studying the dynamical behavior of a system often depends on characterizing how it transitions between long-lived states. Because such transitions are rare, observing them usually requires specialized enhanced sampling techniques. Transition Path Sampling (TPS) is a well-established method for generating reactive trajectories, which is simple to implement and does not require the definition of a preconceived reaction coordinate. However, its efficiency is limited by its sequential nature and the resulting correlations between sampled paths. Previous work addressed this limitation by combining TPS with a sampling scheme based on conditioned Boltzmann Generators, a generative machine learning model capable of sampling a given target probability distribution. This approach produces uncorrelated transition paths but relies on an accurate reaction coordinate, which is rarely known in advance. Building on recent advances in committor learning, specifically on the Artificial Intelligence for Molecular Mechanism Discovery (AIMMD) method, in this work we introduce GenAIMMD, an iterative algorithm that actively and self-consistently learns the ideal reaction coordinate (the committor) and trains a conditioned Boltzmann Generator to sample from arbitrary bias windows along it. GenAIMMD thereby provides a correlation-free and fully parallelizable path sampling scheme that does not require prior knowledge of the system's transition mechanism. We apply GenAIMMD to a two-dimensional toy model and a higher-dimensional polymer system. In both cases, GenAIMMD succeeds in training the Boltzmann Generator and learning the committor. Benchmark results show a substantial increase in performance compared to standard TPS.

The formulation of statistical mechanics in terms of ensembles, originally proposed by Gibbs, has proved to be extremely effective in describing the equilibrium state of many-particle systems, in the most diverse contexts (critical phenomena, quantum mechanics, biophysics, just to mention a few). The reasons of this success are still debated, and the connection between dynamics and probability in large systems remains somehow elusive. Indeed, in order to derive the main results of equilibrium statistical mechanics, strong assumptions on the ergodicity of the dynamics are usually required. Nonetheless, empirical observations seem to suggest that the predictions of the theory hold true even when such assumptions are not verified. In this paper we reconsider the point of view put forward by Khinchin, stating that the only relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom in the system and the choice of extensive observables, irrespectively of the details of the microscopic dynamics. In particular, the presence of dynamical chaos is not required. To this aim, we discuss some analytical and numerical results on a couple of classical integrable systems, the harmonic chain and the Toda model, showing that many important features predicted by equilibrium statistical mechanics, as for instance Maxwell-Boltzmann distribution, are found even in absence of chaos.

We investigate energy fluctuations of solvated DNA in three conformations: G-quadruplex, Hairpin, and Duplex. Our comprehensive molecular dynamics (MD) simulations demonstrate that the time-series of interaction energies for DNA self, DNA-water and DNA-ion show 1/f noise with a universal decay exponent $\beta \simeq 0.90$. We show that DNA topology and structural rigidity modulate its dynamic coupling with the surrounding hydration shell and mobile counter-ions. This coupling critically shapes the 1/f noise profile: DNA self energies in flexible structures (G-quadruplex, Hairpin) display pronounced low-frequency modes absent in the rigid Duplex. These findings reveal how the interplay between DNA topology and environmental coupling governs different stochastic behavior in biologically relevant forms of DNA.

Landau diamagnetism is usually derived from the grand canonical potential, a calculation that yields the correct susceptibility but little physical insight, while the rule that extremal cross sections govern the de Haas-van Alphen (dHvA) oscillations is commonly asserted rather than exhibited. We present an elementary zero-temperature construction in which the occupied states of a free-electron gas are grouped, interval by interval along the field direction, into the Landau levels onto which they condense. Within each interval the field-induced energy cost reduces to a transfer of states between two congruent triangles, and center-of-mass arithmetic yields the Landau susceptibility. The construction fails only in a narrow neighborhood of an extremal cross section of the Fermi surface; there the contribution oscillates with the dHvA period, one oscillation for each Landau level that passes through the extremal cross section. Direct zero-temperature state counting locates the oscillation: the residual is concentrated in the few intervals nearest the extremal cross section, the summed contribution of the distant ones being negligible on the scale of the peak oscillation amplitude.

Higher-order interactions couple three or more elements in ways that cannot be decomposed into pairwise contributions. In diverse complex systems they produce striking collective behaviours such as explosive synchronization, discontinuous transitions and altered stability. Neither such behaviour nor a description written in many-body terms establishes a many-body mechanism, since both depend on how the system is represented. This review distinguishes three questions: what mechanism governs the system, what behaviour does it produce, and how is either represented? Representation covers both which variables are included and how relations among them are encoded. We examine direct many-body mechanisms, nonlinear but pairwise mechanisms that mimic them, and indirect routes through shared environments or eliminated variables, together with temporal aggregation, an artefact of observation rather than a mechanism. The same behaviour can arise from different mechanisms, and a given higher-order mechanism has no universal behavioural consequence. We then formulate reducibility, the question of whether a higher-order model can be replaced by a lower-order one, as a comparison between a source model and an admissible class of lower-order models, relative to a target and a tolerance. The target may be a microscopic rule, an observable or a qualitative feature. Information-theoretic and inferential methods are reviewed as complementary tools for detecting higher-order dependence, testing lower-order alternatives and reconstructing interaction structure. However, observational data alone generally cannot establish the mechanism. We close by identifying open problems in linking mechanisms to measurable signatures, coarse-graining, identifiability and causality, and in extending reductions beyond mean field. (Abridged)

Kramers-Wannier duality of the quantum Ising chain is naturally realized as a noninvertible transformation on a finite periodic chain. Upon incorporating appropriate symmetry-twisted sectors, however, the duality can be promoted to an invertible unitary transformation. In this Letter, we classify when the Kramers-Wannier duality admits an invertible realization on a finite Hilbert space. We show that this is determined by the representation of the Ising bond algebra and establish a necessary and sufficient condition in terms of its two central elements. This representation-theoretic criterion unifies the conventional noninvertible realization with invertible constructions involving symmetry-twisted sectors and applies equally to other models with the same underlying bond algebra. As an explicit example, we find an order-disorder duality in a non-trivial realization of the Ising bond algebra that admits an invertible Kramers-Wannier duality.

We study the off-diagonal matrix elements of real-space observables in time-reversal-invariant single-particle chaotic systems. By analyzing the semiclassical expression for the off-diagonal variance derived from Berry's conjecture, we show that the banded structure of the observable matrix emerges naturally. For local observables, we identify a characteristic bandwidth associated with a late-time timescale inversely proportional to the particle velocity. We further show that, for systems with steep-wall confinement, the predicted magnitude follows the entropy scaling of the eigenstate thermalization hypothesis (ETH), multiplied by an additional kinetic-energy-dependent factor that is independent of spatial dimension and is not captured by conventional many-body ETH. We illustrate these results through a case study of quantum billiards and verify the semiclassical scaling numerically in a generalized quarter-Sinai billiard. Our results elucidate the dynamical implications of Berry's conjecture and provide a comparison between single-particle eigenstate thermalization and many-body ETH.