According to Born (\emph{Atomic Physics, page 103}), spectra are \emph{``quantum phenomena, which from a classical standpoint are perfectly unintelligible''}. However we illustrate results on classical calculations of infrared spectra of ionic crystals (actually LiF) which show that the situation is much more complex. Indeed it turns out that: 1) At room temperature and at higher ones (up to 1060 K) the classical computations reproduce the experimental data, even better than the \emph{presently available} quantum ones do; 2) At lower temperatures (even at 7.5 K), the classical computations reproduce pretty well the data, if one accepts the idea advanced in 1916 by Nernst (the inventor of the third principle) that zero-point energy has room in classical physics too. It is eventually pointed out that the mentioned results might be regarded as a first step towards an implementation of the Einstein Classical Program, which aims at deducing quantum physics (admittedly the correct theory) from a realistic theory. In fact, we are considering the Einstein classical program in the extreme version in which the realistic theory is just (\emph{essentially, see below}) classical electrodynamics of matter in bulk, involving phase space orbits, solutions of Newton equations. An Appendix is devoted to illustrate the Nernst approach, which concerns also the relation between equipartition and Planck's law.

Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.

Characterising the universal aspects of non-equilibrium quantum many-body dynamics is one of the key goals of this century's physics research. Progress, however, is hindered by the lack of general theoretical frameworks for studying interacting quantum matter far from equilibrium. A recent breakthrough has been the realization that several key non-equilibrium quantities, such as the rate of growth of entanglement or the fluctuations of conserved charges within finite subsystems, can be related to equilibrium properties through a space-time duality that effectively exchanges the roles of space and time. This observation effectively enables the study of non-equilibrium phenomena using tools and concepts borrowed from equilibrium statistical mechanics and thermodynamics. A first proof of principle of this framework, dubbed space-time duality approach (SDA), was provided by interacting integrable systems, where thermodynamic properties can often be characterized exactly, while dynamical quantities typically remain beyond analytical reach. Subsequent developments, however, revealed that the SDA suffered from an intrinsic ambiguity, restricting its applicability to homogeneous quenches and to charge fluctuations arising from symmetric initial states. Here we resolve this ambiguity from first principles and derive closed-form predictions for entanglement growth and charge fluctuations after general quantum quenches. We benchmark our results against the exact analytical solution of the Rule 54 quantum cellular automaton and extensive TEBD simulations of the XXZ chain. Moreover we show that, when specialised to the entanglement entropy, our framework naturally reproduces the predictions of the quasiparticle picture.

The Haldane-Shastry spin chain is a prototypical integrable model with long-range interactions, notable for hosting quasiparticles with fractional statistics and serving as a discrete analogue of a conformal field theory. Its remarkable simplicity is closely tied to a full Yangian spin symmetry. While the highest-weight states for this symmetry are known explicitly, a systematic treatment of the descendant states, needed for the computation of various physical quantities, has remained incomplete. In this work, we provide a detailed construction of these descendants in terms of the algebraic Bethe ansatz following recent work of Ferrando et al. In the limit of extreme twist, it includes the Gelfand-Tsetlin basis. As an application, we derive product and determinant formulae for norms and overlaps of these states.

Understanding the time evolution of physical observables in open quantum many-body systems coupled to external environments is a natural and difficult problem, and exact results are still rare. In this work, we study this problem for Wess-Zumino-Witten (WZW) conformal field theories with Lindblad jump operators linear in Kac-Moody current modes. We investigate the exact operator dynamics generated by these Lindbladians, identifying classes of current operators whose Heisenberg equations close and can therefore be solved analytically using the underlying current algebra. In Abelian $U(1)_k$ WZW theories, this closure of operator dynamics holds for arbitrary settings of jump rates and includes exactly tractable cooling dynamics. In contrast, for non-Abelian WZW theories, exact closure occurs only for symmetric current-mode dissipation, where upward and downward current-mode transitions occur with equal rates, and even then it leads to a simple closed evolution only for a single current operator. Generic imbalances, including those needed for cooling, produce additional non-Abelian terms and prevent closure of the opeartor dynamics. Consequently, the current algebra gives rise to a broad family of exactly solvable dissipative dynamics in the Abelian setting, whereas in the non-Abelian case it singles out only a special exactly solvable dynamics corresponding to an infinite-temperature bath.

Cellular automata are discrete dynamical systems defined on a lattice, in which each site carries a finite set of states that evolve in time according to local deterministic rules. An important application of cellular automata is in lattice gas models of fluids, where the cellular automaton framework provides a particle-based microscopic description of hydrodynamic behavior. The macroscopic fluid equations emerge after coarse-graining over many lattice sites and time steps, offering a bottom-up route to hydrodynamics. A celebrated example is the Frisch-Hasslacher-Pomeau (FHP) model, an automaton defined on a two-dimensional triangular lattice that yields the two-dimensional Navier-Stokes equations upon coarse-graining. In this work, we construct a parity-breaking generalization of the FHP model through two modifications: introducing chiral two-body collision rules and systematically rotating particle velocities to mimic the effect of a background magnetic field. We show that this automaton yields a hydrodynamic model with odd viscosity, a transverse transport coefficient that is a hallmark of odd fluids. We verify the analytical transport coefficients using Poiseuille-flow simulations of the chiral FHP automaton. Our results demonstrate that the chiral automaton introduced here provides a bridge between microscopic parity-breaking scattering processes and macroscopic odd-fluid hydrodynamics.

We investigate how interaction geometry shapes the low-energy phases of sparse tunable long-range quantum models. We focus on a class of graphs whose degree grows logarithmically with system size, and show how symmetry and frustration in graph connectivity can drive, suppress, and reshape ground-state phase transitions. The central examples are power-of-$p$ graphs, where even and odd values of $p$ exhibit qualitatively distinct behaviour: even-$p$ graphs inherit the rich phase structure of the power-of-two model, while odd-$p$ graphs are governed by geometric frustration. Fibonacci graphs provide a contrasting case, lacking the discrete self-similarity of the power-of-$p$ family but exhibiting a direct geometric mapping between the short- and long-range limits. Across our models, we find that phase structure and criticality are governed by the same effective-geometry principle, unifying our framework for experimentally motivated long-range quantum systems.

Unlike in closed quantum systems where individual energy eigenstates are understood as physical excitations, open quantum systems have distinct right and left eigenstates of the Liouvillian that decay with time and are difficult to interpret. Here we introduce a physically motivated quasiprobability measure combining the two types of eigenstates that interprets a Liouville eigenmode as a set of coherences. This coherence measure is intimately connected to the return probability and allows one to visualize the modes as quasiprobability distributions in a "doubled" phase space. Using this measure we show that, remarkably, an oscillator retains its quantized "orbits" in phase space for a large class of linear and nonlinear damping, thus providing a formulation of semiclassical quantization for open systems. The orbits have measurable dynamical signatures and are broadened in the presence of a thermal bath, similar to energy levels. For quadratic systems, our results yield an extension of the concept of invariant tori, which play a central role in Hamiltonian systems.

Can intelligence be measured? We propose that intelligence can be defined as the lawful amplification of rare but valid futures: a system increases the probability of outcomes that would be unlikely under passive dynamics but remain admissible under the constraints of the domain. We start with the premise that an intelligent system must model the world and its own place within it. Because the system is part of the world it models, this leads naturally to recursive self-simulation: the system represents futures in which its own actions are part of the trajectory. Our central results give a necessity statement and a conditional near-sufficiency statement connecting this architecture to a precise thermodynamic measure of lawful amplification of rare-valid futures: high rare-valid lift is impossible unless the internal simulation identifies rare-valid futures with high fidelity; conversely, when rare-valid fidelity is high and the simulation contains an effective policy, the achievable lift approaches the actuation-limited optimum. Thus recursive self-simulation is not merely a plausible feature of intelligence but, under the stated assumptions, is necessary and nearly sufficient for high thermodynamic intelligence. The resulting framework makes intelligence measurable on a universal scale, from passive matter and feedback controllers, large language models, and humans as text generators to Maxwell-demon-like information engines.

We forecast future volatilities and correlations of financial markets based on the current trends in these markets. This complements previous work that models future expected returns by a cubic polynomial of the current trend strength. Empirically, we observe that volatilities and correlations tend to increase day after day in times of strong up- or down-trends. This effect is particularly pronounced in down-trends. It can be accurately quantified by quadratic polynomials of today's trend strengths, which refine common mean-reversion models of volatilities and correlations. Our results improve the prediction of market risk by accounting for market trends. They also support a recent proposal to model financial markets by a lattice gas near its critical point.

This work follows the evolution of financial networks in Norway over a period of nine years at a monthly rate. The data consist of board directors and their affiliations to companies, which we model as simplicial complexes. In this framework, directors are represented as nodes and companies as faces of the complex. To characterize the latter, we focus on three topological measures: the Euler characteristic, computed through the Betti numbers, torsion computed through the reduced determinant of the higher-order Laplacians, and higher-order clustering coefficients. The first two fail to capture the effect of imposed law on representation, unlike our notion of curvature which is a geometrical measure computed from the coefficients of the series expansion of the heat kernel in powers of time, which is our major contribution in this work. In particular, the Euler characteristic integrates curvature, and thus local information is lost. Subsequently, not every topological measure can reliably capture shocks in networks. Further, the number of spanning trees may undergo significant changes at the lowest order, yet these changes need not be reflected in the torsion. Conversely, the change in the curvature revealed variation in the board interlock due to legislation, and serves as a sensitive measure for detecting shocks in networks. Inflection points in curvature are associated with external forcing, and minima with shock arrival times. Sharp transitions are also observed in the components of torsion, while smooth changes are observed in higher-order clustering.

In this article, we present four different $1+1$D quantum models that realize the Yang-Lee (YL) phase transition under a deformation that preserves $PT$ symmetry. These are the antiferromagnetic Ising spin chain in transverse and longitudinal magnetic fields, the massive Schwinger model, the Blume-Capel model, and the three-state quantum clock model. Using the state-operator correspondence, we identify the YL critical point, compute the scaling dimensions of the lowest operators in each model, and find perfect agreement with the exact results for the YL criticality in two dimensions. Using bosonization for the Schwinger model and the Polyakov-Hubbard transformation for the other models, we show that in all of these quantum models the YL critical point is described, as expected, by a massless bosonic field with an $i ϕ^3$ interaction. In the quantum clock model, this critical field interacts with a massive bosonic field, and we identify the massless and massive states in the Hamiltonian spectrum. In addition, we numerically compute the two-point function of $ϕ$ at the Yang-Lee critical point and show that it grows with distance, in agreement with theoretical expectations.

Many industrial applications and biological scenarios involve the interdiffusion of two polymeric species. Motivated by biological subcellular source-driven processes, we study polymer-polymer interdiffusion problems in the absence or the presence of a polymeric source, for both unentangled and entangled scenarios. Utilizing a two-fluid formalism, we arrive at scaling relations, self-similar reductions, and analytical solutions, which are confirmed with one- and two-dimensional numerical simulations. The introduction of a source term breaks the self-similar structure, modifying the boundary conditions and the domain of integration. Nevertheless, we show that the front characteristics of the diffusing droplet exhibit similar spatial structures as in the absence of a source. Our results allow deeper understanding of polymer-polymer interdiffusion and nonlinear transport, especially in the presence of a source.

We revisit the charge-1/3 anyon fluid obtained by doping the $ν= 2/3$ Jain fractional Chern insulator (FCI). In the standard composite fermion description, the doped anyons fractionalize into three translation-related flavors of secondary composite fermions, whose gauge-mediated interactions drive a robust inter-flavor pairing instability. In our previous work, we analyzed a flavor-asymmetric paired state and obtained a charge-ordered Fermi liquid. Inspired by a recent paper, we consider an alternative flavor-symmetric paired state and show that it is an SC* state: a charge-$2e$ condensate that coexists with residual $\mathbb{Z}_2$ topological order. The weak and strong pairing regimes share the same intrinsic topological order but differ in chiral central charge, giving $c_- = 7/2$ and $c_- = 2$. We further show how other proposed effective field theories fit within the same composite fermion description, and argue that across the doping driven FCI-to-superconductor transition, localized anyons evolve into Bogoliubov quasiparticles rather than vortices. At low doping, we identify an approximate SU(3)-symmetric regime in which the system instead realizes a non-Fermi liquid $\mathbb{Z}_3$ Orthogonal Metal with three charge-1/3 fermion pockets and no sharp electron quasiparticle. Finally, we comment on the energetics of various possible ground states and discuss implications for experiments in moire materials.

We study single-file transport of driven overdamped colloidal particles on a periodic path with deep potential wells. In the small trap limit (i.e., trap size smaller than particle size), the particle current transitions from zero to finite as the number of particles on the path exceeds a critical number $n_c$. Beyond this threshold, $n_c$ particles cluster behind the trap, demonstrating collective correlated motion. The remaining `extra' particles circulate, giving a finite current. We study this phenomenon numerically using overdamped Brownian dynamics simulations, and present an experimental realization of this behaviour for micron-scale colloidal particles driven in an optical vortex. Using our experimental observations, we present results characterizing potential wells as deep as several hundred $k_BT$.

We formulate a thermodynamic model of a nano-reactor containing charge-regulated macroions within an electrolyte-permeable enclosure. The model is then formalized within the Poisson-Boltzmann electrostatics augmented by the consistent inclusion of the charge dissociation of molecular groups residing on the surface of the entrapped macroions via charge regulation formalism. By solving the basic equilibrium equations in the linearized Debye-Hückel type approximation, we analyze the salient features of the inhomogeneous electrolyte distribution and macroion charge. We found that the surface charge asymmetry/symmetry of the macroions strongly affects the spatial profile of electrostatic potential. The effective screening length shows the non-monotonic behavior, arising from the complex interplay between the bathing external solution and macroion effective charges, which govern charge regulation equilibria. The total pressure at the nano-reactor enclosure boundary decreases monotonically as the enclosure radius and the ionic bulk salt concentration increase. Also, the resulting pressure is strongly influenced by the surface charge densities of the nano-reactor and the number of confined macroions.

We study the collective dynamics of self-propelled spherocylinders carrying magnetic dipole moments in two dimensions. Magnetic interactions are modeled as two opposite monopoles $\pm Q$ separated by a distance $\ell$ along the particle director, a dumbbell model that remains well-defined at short range and introduces an explicit geometric lever arm for the magnetic torque. This approach, combined with the elongated particle geometry, produces a torque that competes with steric alignment in a manner inaccessible to point-dipole or disk models. By independently varying monopole separation and dipole strength (parameters that map directly onto the geometry and magnetization of cylindrical magnets) we show that the system navigates a rich landscape of collective states: gas, polar flock, chain, vortex-alignment, and locked-dimer phases. Our results establish that particle elongation and distributed magnetic charge together provide a minimal, experimentally accessible set of tuning knobs for controlling coherent states in magnetic active matter, with direct implications for the design of self-organized magnetic microswimmers and active colloidal assemblies.

Rydberg atom simulators, in both analog and digital modes, have attracted significant recent interest due to their versatile geometric reconfigurability. In this work, leveraging this feature, we propose two complementary approaches, one for each mode, to characterize emergent fermions in critical quantum many-body systems. In the analog mode, we assemble the Rydberg atoms in a "developable" (namely, preserving local couplings) Möbius band geometry to realize antiperiodic boundary conditions, where fermionic states reside. Spectroscopic measurement in this sector then reveals universal energy ratios of the bosonic and fermionic states. In the digital mode, we carry out a fermionic version of Kibble-Zurek ramping with a quantum circuit, directly addressing the fermionic scaling form. Reconfigurability allows an exponential speed-up of this task, with an $O(\log L\log\log L)$ circuit-depth overhead. Our work establishes the Rydberg atom simulator as a uniquely powerful platform to attack the notoriously difficult issue of experimentally probing emergent fermions that are nonlocally defined in a bosonic system.

Quantum quenches drive edge states far from equilibrium, yet whether the memory of a topological initial state survives in a non-integrable, interacting system has remained largely unexplored. We study this question in the bond-alternating XXZ chain -- an interacting Su--Schrieffer--Heeger model hosting symmetry-protected topological edge modes with markedly enhanced boundary magnetization -- and analyze quenches across all combinations of single-particle and many-body initial and final Hamiltonians. The results organize by a single distinction as we rigorously establish in this work: whether the post-quench Hamiltonian is free or genuinely interacting. For a free post-quench Hamiltonian, the dynamics is solved exactly by a correlation-matrix approach; the boundary-mode return amplitude decays as $t^{-3/2}$, and initial interactions enter only through a dressed one-body density matrix. For a genuinely interacting post-quench Hamiltonian, finite-time stability bounds prove that away from local resonances the first-dimer magnetization remains stable on time windows growing as arbitrarily large powers of the inverse inter-dimer coupling. Matrix product state simulations across all four protocols show that interactions in the final Hamiltonian markedly extend finite-time boundary memory -- with local suppression near the isotropic $SU(2)$ point -- revealing a many-body protection mechanism in a non-integrable system where scrambling would otherwise wash out initial-state memory fast.

When stochastic field dynamics are cast into a path-integral formulation, perturbation theory becomes systematic but the resulting expansion quickly grows combinatorially large. The setting targeted here includes multi-component, multi-dimensional fields with matrix propagators, tensor-valued couplings, and non-Gaussian driving noise specified by arbitrary $n$-point cumulants. Wick pairings grow factorially, and component indices must be routed through the tensor-valued vertices. The useful output is not a raw contraction list, but a diagram table: one entry per topology, with multiplicities, coupling sums, signs, and causal constraints resolved. We present sft-wick, an open-source Python package that constructs these diagram tables and computes their integrals numerically. Given an action and an observable, it enumerates topologically distinct Feynman diagrams, derives their algebraic coefficients, and evaluates the resulting diagram integrals from user-supplied response and cumulant functions. The core algorithm enumerates spatial topologies before routing component indices, avoiding contraction-by-contraction Wick expansion. Response-field constraints, including vanishing response-response contractions, the ito prescription, and the absence of causal response loops, are enforced during enumeration. Predictions are validated against direct Langevin simulation, agreeing to within the simulation's statistical noise.

Stochastic resetting has emerged as a versatile protocol to drive quantum many-body systems to non-equilibrium steady states by interspersing unitary dynamics with measurements and resets at random times. In spite of this, a quantum hardware validation of such non-equilibrium steady states is still missing. Here, we achieve this goal by first formulating a noisy discrete-time theory where unitary gates alternate with noisy mid-circuit projective measurements and conditional resets. This noisy conditional resetting theory is then demonstrated on a superconducting quantum processor for up to $N=7$ qubits. We consider, as a paradigmatic case, the unitary dynamics of the interacting Floquet transverse-field Ising model. The stationary state of the noisy conditional resetting agrees quantitatively with the experiments, and it shows crossover behavior related to the equilibrium quantum phase transition of the model. Our results might thus pave the way for the preparation of collective stationary states on noisy quantum devices and for further developments of quantum algorithms involving mid-circuit measurements.

We develop indexed Bellman information complexity, a representation-level theory of interactive decision making centered on information indices and reference histories. The representation strips away problem-specific syntax and retains only the ingredients needed for dynamic programming and information accounting, thereby unifying the earlier framework of indexed algorithmic information ratios (AIR). On the upper-bound side, regret is controlled by Bellman supersolutions or potential identities whose gradient bracket is paid for by indexed information. Upper-confidence-bound (UCB), estimation-to-decision/decision-estimation-coefficient (E2D/DEC), and adaptive-minimax-sampling or exploration-by-optimization (AMS/EBO) methods appear as three relaxations of this same identity. On the lower-bound side, the posterior-reference trajectory supplies both the information telescope and the ghost quantile of small-regret trajectories. The resulting critical radius in the lower bound is an effective-dimension-scale quantity, as in Fano and local-prior-mass lower bounds, rather than the constant radius of a two-point Le Cam argument. The examples show that DEC is best viewed as a one-step relaxation of indexed Bellman information complexity, not as a universally tight conversion mechanism. We illustrate the framework through several applications, with particular emphasis on kernel bandits. In this setting, the active action marginal provides a concrete basis for comparing UCB, E2D, and AMS/EBO.

We investigate an integrable boundary quench, in which one integrable boundary condition is suddenly switched to another. We develop a general framework for analyzing the resulting real-time dynamics based on form factors of bulk and boundary-changing operators. We first study the problem at the conformal point of the Lee-Yang model and then extend the analysis to its massive perturbation, where we examine the time evolution of the pre-quench vacuum and compute the vacuum-to-vacuum matrix elements of local operators inserted after the quench. The analytical results are validated by numerical calculations using the truncated conformal space approach adapted to boundary-changing situations.

Fluctuational electrodynamics relies on the assumption that thermal fluctuations in distinct bodies are statistically independent. It is shown that this approximation breaks down in the extreme near-field regime, where hybridization of surface phonon-polaritons across nanometric vacuum gaps generates finite fluctuating-current cross correlations between opposite interfaces. Using a microscopic coupled-oscillator model combined with a Green-tensor formulation of the Poynting vector, the resulting correlation-induced correction to radiative heat transfer is derived. For polar materials, these correlations become significant when the hybridization energy approaches the intrinsic damping rate and can substantially modify conventional fluctuational-electrodynamics predictions at subnanometric separations.

We study phase transitions on dynamic triadic hypergraphs, in which a continuous formation field evolves under stochastic Ginzburg--Landau dynamics with a cubic three-body coupling $g_τϕ_iϕ_jϕ_k$, while a discrete opinion variable $s_i\in\{-1,+1\}$ undergoes Kawasaki exchange under a Hamiltonian with pairwise alignment and an irreducible three-body energy $-λ_τ\prod_{a\inτ}s_a$. Near the formation critical point the cubic coupling is subleading and the transition remains continuous, controlled at leading order by a pairwise Ising baseline with renormalized coupling $J_{\rm eff}=J+γw$. The dominant observable is the triadic formation correlator $Ψ_{\rm form}\equiv\langleϕ_iϕ_jϕ_k\rangle$, a $k=3$ composite operator built over the underlying $\mathbb{Z}_2$-symmetric order parameter. Composite-operator scaling yields the effective exponents $β_{\rm TF}=3/2$ and $γ_{\rm TF}=-1$. The susceptibility conjugate to $Ψ_{\rm form}$ vanishes at the critical temperature $T_c$ rather than diverging, in contrast to the divergence characterizing scalar (pairwise) order parameters. The exact partition function of the minimal triad on $\{-1,+1\}^3$ identifies a crossover scale $T^*=4J_{\rm eff}/\ln 3$. A field-theoretic two-point function argument reproduces the same vanishing susceptibility. Restoring the three-body coupling ($λ\neq0$) makes the transition first-order, with a critical endpoint at $λ=0$. The exponent relations $β_{\rm TF}=3β_{\rm Ising}$ and $γ_{\rm TF}=γ_{\rm Ising}-4β_{\rm Ising}$ hold exactly on dense hypergraphs via cluster decomposition, and the vanishing-susceptibility signature persists for $d\geq3$ but fails in $d=2$. A Mori--Zwanzig memory kernel yields a continuously tunable dynamical exponent $z_{\rm TF}$, completing the composite-operator scaling regime.

Quantum-inspired tensor networks algorithms have shown to be effective and efficient models for machine learning tasks, including anomaly detection. Here, we propose a highly parallelizable quantum-inspired approach which we call SMT-AD from Superposition of Multiresolution Tensors for Anomaly Detection. It is based upon the superposition of bond-dimension-1 matrix product operators to transform the input data with Fourier-assisted feature embedding, where the number of learnable parameters grows linearly with feature size, embedding resolutions, and the number of additional components in the matrix product operators structure. We demonstrate successful anomaly detection when applied to standard datasets, including credit card transactions, and find that, even with minimal configurations, it achieves competitive performance against established anomaly detection baselines. Furthermore, it provides a straightforward way to reduce the weight of the model and even improve the performance by highlighting the most relevant input features.

There has been recent interest in the relaxational modes of small-scale fully connected systems of aligning self-propelled particles (Spera et al., Phys. Rev. Lett. {\bf 132}: 078301 (2024)). We revisit the classical connection between Fokker-Planck and Schrödinger equations to address this by means of exact diagonalization, allowing for rigorous analytical insight into the full spectrum. This allows us to extract exact results which we compare to the existing result from linearized statistical field theory. We derive asymptotically correct analytical results that improve upon the prior approximations. We show that this methodology can fruitfully be extended to the case of non-reciprocal interactions which gives rise to a non-Hermitian Schrödinger problem akin to those in open quantum mechanics. While the non-reciprocity can be chosen such as not to alter the stationary distribution, it fundamentally changes the nature of the steady state which we quantify via the entropy production. We discuss the case of low particle numbers as well as the emergence of mean-field dynamics at large numbers.

We study the free energy and dynamics of a closed elastic filament (a one-dimensional curve in two dimensions) coupled to a scalar concentration field representing, for example, an absorbed species. The density variable has a tendency to phase-separate whereas the local spontaneous curvature is concentration-dependent. We address analytically and by simulation both the free energy landscape and the dynamics (the latter comprising a coupled Willmore flow and Cahn--Hilliard gradient flow on the full differential geometry of a closed filament), addressing issues that previous work typically sidestepped by restricting to the Monge gauge. Specifically we find that the closure constraint for a deformable filament qualitatively changes the free energy landscape compared with either a rigid closed filament or an open elastic one, admitting metastable and stable states with more than one domain of each type. By numerical global free energy minimization we explore equilibrium morphologies across a wide range of model parameters. For selected parameter values we present fully dynamical results, tracking the time evolution of the various contributions to the free energy and confirming the emergence of both metastable and equilibrium multi-domain morphologies.

Cellular signal processing performs reliably despite molecular noise. Mutual information (MI) is widely used to quantify signaling fidelity, capturing how well outputs discriminate input states. However, it fails to capture whether the output preserves the statistical structure of the input, a property crucial in morphogen patterning and dose-dependent signaling. To address this gap, we introduce the 2-Wasserstein (2-WD) distance, which provides a geometric basis for comparing input and output distributions. We define MI as informational fidelity (INF) and the inverse of the 2-WD as geometric fidelity (GMF). Applying this dual-fidelity framework to canonical regulatory motifs under Gaussian channel approximation reveals topology-dependent trade-offs: coherent feed-forward loops can perform well in both dimensions, whereas feedback architectures sacrifice INF to enhance GMF. Experimental analysis of tumor necrosis factor signaling supports the predicted role of feedback regulation. Analysis of RAS-MAPK signaling shows that intracellular signal relay is better described by a balance between INF and GMF than by information transmission alone. Our results demonstrate that reliable signaling need not maximize information alone, but can arise from balancing information transmission with distributional correspondence. Thus, GMF represents a distinct dimension of signaling fidelity and provides a framework for analyzing natural networks and designing task-specific synthetic circuits.

We present efficient and practical protocols to measure the second Rényi entropy, whose exponential is known as the purity. Our approach is based on expressing the purity in terms of transition probabilities generated by an echo-type forward-backward evolution sequence, making it applicable to quantum many-body systems. Notably, our approach does not rely on random-noise averaging, a feature that can be extended to protocols to measure out-of-time-order correlation functions, as we demonstrate. By way of example, we show that our protocols can be practically implemented in superconducting qubit-based platforms, as well as in cavity-QED trapped ultra-cold gases.