Papers

We formalize the gauge structure of the Navier–Stokes equation for incompressible fluids, interpreting $\nabla\cdot\mathbf{v}=0$ as a gauge fixing analogous to the Coulomb gauge in electromagnetism. We construct the Martin–Siggia–Rose action and its full BRST symmetry, introducing BRST doublets for the Gribov parameter $\gamma$ and the monodromy phase $\theta$. Through a Higgs mechanism, $\gamma$ acquires a vacuum expectation value $\gamma_{0}$, generating a mass scale for the vorticity and the ghosts; we compute the one-loop effective potential and analyze vacuum stability. Intermittency — measured by the exponents $\zeta_{n}$ — is described by fluctuations of the Higgs field $\sigma$ around the condensate, via a Gribov-inspired log-Poisson hierarchy that satisfies exactly $\zeta_{3}=1$ (Kolmogorov's four-fifths law). Fits to DNS data favor filamentary vorticity structures, with $D_{f}\approx1$ (one parameter) or $D_{f}\approx2.1$, $\Delta\approx0.44$ (two parameters). The one-loop anomalous dimension in $d=3$, $\Delta_{\sigma}\approx0.50$, supports the identification $D_{f}=2\Delta_{\sigma}$. The formalism unifies classical hydrodynamics with gauge theory and spontaneous symmetry breaking, opening the study of turbulence and intermittency to quantum field theory methods.

Biological molecular machines are mesoscopic systems that act repeatedly and periodically to perform important cellular tasks while contending with strong fluctuations and operating in an overdamped regime. Optimal control theory is a tool that can be used to understand the design principles behind efficient operation of these machines; however, most studies on optimal control of classical mesoscopic systems have focused on control problems that do not repeat periodically. Here, we automatically differentiate Fokker-Planck simulations to design efficient nonequilibrium control strategies for simple models of periodic molecular machines with and without explicit changes in the machine's chemical state. The designed protocols and theoretical analysis provide insight into the design principles governing efficient driving in these nonequilibrium systems. Designed control protocols should seek to reduce mechanical heat by rotating the entire angular probability distribution at a constant speed without changing its shape, and should reduce chemical heat by reducing the proportion of chemical transitions with large heat.

Nested sampling (NS) resolves the thermodynamics of an atomistic system from a single simulation, but its practical reach is limited by the Markov-chain updates needed to decorrelate walkers within each likelihood-constrained ensemble. Flow-based NS has reduced this bottleneck for gravitational-wave (GW) inference, yet its transfer to atomistic systems is not merely a change of application. Comparing a GW150914-like binary-black-hole likelihood with an eight-particle two-dimensional Lennard-Jones (LJ) system of comparable dimensionality, we show that the two landscapes differ fundamentally: atomistic multimodality is discrete and combinatorial, generated by particle permutations separated by hard collision walls, and its coordinate coupling is dense and collective, whereas the GW posterior exhibits smooth degeneracies and localized parameter coupling. Guided by this diagnosis, we introduce NS-Flows: a single conditional normalizing flow, conditioned on the NS energy bound and trained on a sliding window of recent live sets, that replaces MCMC by direct parallel draws corrected by importance-weighted rejection resampling. Live sets supply the data self-consistently, allowing flow training without structured priors or a pre-existing dataset. For LJ disks under periodic boundary conditions, the algorithm can reduce energy evaluations by two orders of magnitude and wall-clock time by about 30%, an advantage that becomes more favorable as the cost of the potential grows. The flow's generative efficiency further acts as a physical diagnostic: it varies non-monotonically along the annealing trajectory, is lowest in the dense disordered regime, and is quantitatively captured by the constrained ensemble's internal mode complexity together with target drift across the training window, identifying liquid-like ensembles, not prior-target separation, as the hard case for current flow architectures.

We derive a new set of bounds controlling the speed-up of the survival rate of trajectory-dependent observables induced by the application of a perturbation. Via the study of survival probabilities in two different contexts, continuous-time Markov jump processes and overdamped diffusions, we obtain different fashions of the bound on the achievable boost. These bounds, obtained by a nonlinear response theory of constrained path probabilities, are explicitly linked to thermodynamics. For arbitrary time-dependent survival rates, we find a linear bound that depends on both the perturbation strength and the entropy production rate prior to the perturbation. For rare processes characterized by a constant survival rate, the bound is exponential and is determined by the excess heat generated by the perturbation. We exemplify the inequalities on two prototypical model systems, namely, a unicyclic network and forced diffusion in a bistable potential.

We study the relativistic transformation of temperature for effective cosmological fluids with negative pressure within a covariant thermodynamic framework. The Lorentz transformation of temperature is derived for barotropic fluids with $p=w\rho$ and for the (generalized) Chaplygin gas with $p=-A/\rho^\alpha$. For barotropic fluids with constant negative equation-of-state parameter, we show that negative pressure suppresses the growth of temperature under relativistic boosts. In contrast, the Chaplygin gas exhibits a qualitatively different behavior due to its density-dependent pressure, leading to a monotonic increase of the transformed temperature with velocity. The dependence of the relativistic temperature on the relative velocity and background energy density is illustrated through representative examples.

We explore the phases and active-to-absorbing state phase transitions (AAPT) in a two-species model, where the species A and its mutant B are asymmetrically or nonreciprocally coupled. We identify a multicritical point that connects the global absorbing state of both the species and its mutant, with a uniform active state. The critical dynamics at this multicritical point is studied within the lowest order perturbation theory. This asymmetric coupling between species A and mutant B leads to unequal and distinct upper critical dimensions $d_c^A$ and $d_c^B$ respectively for A and B dynamics. We show that the multicritical point in this model is characterised by an unusual breakdown of scale-invariance by fluctuation-induced logarithmic modulations with power law behaviour of the order parameter and correlation lengths of the species A at all dimensions $d<d_c^A$. Above $d_c^A$, conventional scale-invariance is restored. The dynamics of the mutant species B belongs to the DP universality class with an upper critical dimension of $d_c^B=4$

We show that the $E_8$ and Leech lattices generate the stealthiest isometry-invariant, locally square-integrable point processes of intensity $1$ in $\mathbb{R}^8$ and $\mathbb{R}^{24}$, respectively, and that they are the unique stealthiest processes. We also apply the proof technique to sphere packing and energy minimization, and we construct a $20$-dimensional periodic configuration that is stealthier than any known $20$-dimensional lattice with the same particle density. This periodic configuration is a formal dual of Vardy's $20$-dimensional sphere packing.

Linear regression is one of the simplest and most widely used tools to learn patterns from data: it fits a set of coefficients so that a linear combination of predictors best matches observed responses. The quality of the fit is measured by the residual sum of squares, the total squared mismatch between predictions and data, whose minimum defines the training loss. We consider Gaussian design and noise, with teacher coefficients independently drawn from a general distribution $p(\beta)$, and a general class of separable regularizers, including Ridge and Lasso. Using the zero-temperature replica method, we compute analytically the large-deviation statistics of the minimum training loss for large numbers $P$ of predictors and $N$ of observations, with $r=P/N$ fixed. The rate function we compute governs rare sample-to-sample fluctuations of the optimal loss. Extensive numerical simulations are in excellent agreement with our theory and clearly show a pronounced deviation from the Gaussian regime of typical fluctuations in the tails.

Schrödinger bridges and entropic optimal transport are usually formulated as stochastic interpolation problems between initial and final probability distributions. In their computational form, the bridge potentials are obtained by Sinkhorn or iterative proportional fitting, and are often regarded as auxiliary scaling functions. We show that for continuously monitored quantum systems, these potentials acquire a direct measurement-theoretic meaning. The mathematical structure of conditional quantum trajectory theory induces a Fokker–Planck equation on quantum state space. Conditioning this diffusion on a terminal distribution or a terminal measurement effect produces a Doob/Sinkhorn potential whose directional derivative along a unitary control vector field is the imaginary part of a generalized weak value. The same construction connects the Schrödinger-bridge viewpoint to the optimal-path framework for continuously monitored trajectories: the backward bridge potential plays the role of an effect-like costate, and its weak-value directional derivative gives the local control signal. By specifying the desired endpoint distribution, the induced drift produced by the Schrödinger bridge solution is the control solution that minimizes the quadratic cost feedback law, guiding the distribution to its desired endpoint. We quantify the control score of the available control Hamiltonian as a logarithmic directional derivative of the bridge potential, or an imaginary weak value. Three explicit examples identify weak measurement as a natural entropic regularization mechanism for quantum state transport and gives a route from Sinkhorn scaling to quantum feedback and Hamiltonian control synthesis for practical optimal control.

Non-KAM (Kolmogorov-Arnold-Moser) systems, when subjected to weak time-dependent perturbations, exhibit an abrupt transition to classical chaos through the breakdown of invariant phase-space tori. We showcase the utilization of non-KAM systems in the quantum regime as quantum sensors, leveraging their sensitivity at resonances. Quantum Fisher information (QFI) is a central quantity in quantum parameter estimation theory that measures how much information a quantum state contains about an unknown parameter that is encoded into it. In other words, it quantifies the sensitivity of a quantum state to small changes in that parameter. In this work, through numerical analysis in conjunction with analytical results, we study the performance of the non-KAM systems for quantum sensing applications by computing the QFI. We find that the growth of the QFI is remarkably enhanced when the resonance condition is satisfied. For frequency estimation under Floquet unitary encodings, we derive a transport bound: if the mean excitation number grows as $\langle\hat n(t)\rangle\sim t^{\alpha}$, the QFI obeys $I(t)\lesssim t^{2\alpha+2}$. The quantum kicked harmonic oscillator, a paradigmatic non-KAM system, realizes the full hierarchy: localized dynamics ($\alpha=0$) yield quadratic growth, delocalized diffusion along stochastic webs ($\alpha=1$) yields quartic growth, and translationally invariant resonances ($\alpha=2$) saturate the bound with anomalous hexic growth, $I(t)\sim t^{6}$, established analytically at resonance $R=2$ and numerically at $R=4$. The enhancement stems from resonance-induced translational symmetry rather than exponential instability, identifying non-KAM resonances as a metrological resource distinct from chaos-assisted and criticality-based sensing.

We show that $p$-wave magnets efficiently generate magnetization via a linear thermal Edelstein effect arising from the orbital magnetic moments of their magnons. Furthermore, they can generate perfectly nonreciprocal orbital currents through a purely even-order nonlinear response. This makes them promising candidates for orbital-current rectification. Because these transport phenomena originate from the orbital magnetic moment of magnons, they connect magnonics and orbitronics. More generally, we find that odd-parity-wave magnets with coplanar ground states host magnons with zero spin magnetic moment. Instead the magnons carry a collinear, out-of-plane orbital magnetic moment, realizing an orbital version of antialtermagnetism. We establish these general results using symmetry arguments and demonstrate them explicitly for a coplanar ground state with the minimal number of sublattices, inspired by the ground state of CeNiAsO. Our conclusions hold both in absence and presence of spin-orbit coupling.

It was recently shown by Yeliussizov \cite{Yeliussizov} that integer partitions in dimensions $d \geq 7$ asymptotically grow strictly faster than MacMahon numbers. As MacMahon numbers match with integer partitions in dimensions $d = 1,2$, the comparison of asymptotics of integer partitions with MacMahon numbers in intermediate dimensions ($d = 3,4,5,6$) is an open question. In this work, we perform Markov chain Monte Carlo (MCMC) simulations till $N=15000$ by using adaptive weight learning followed by conventional MCMC steps to numerically estimate the asymptotics of integer partitions in these intermediate dimensions. We numerically establish that in these intermediate dimensions, partitions asymptotically grow faster than MacMahon numbers. More specifically, assuming that the limits exist, we show: $\lim_{n\to\infty}n^{-3/4}\log p_3(n) = 1.8196 \pm 0.0019$, $\lim_{n\to\infty}n^{-4/5}\log p_4(n) = 1.7215 \pm 0.0045$, $\lim_{n\to\infty}n^{-5/6}\log p_5(n) = 1.6521 \pm 0.0059$, and $\lim_{n\to\infty}\log n^{-6/7}p_6(n) = 1.652 \pm 0.021$ for partitions in dimensions $d=3,4,5,$ and $6$ respectively. These numbers are all larger than MacMahon leading order asymptotic coefficients of $1.7898, 1.6614, 1.5737,$ and $1.509$ respectively. Additionally, we also find estimates for some of the sub-leading asymptotic terms in $\log p_d(n)$ in each of the dimensions.

We derive an exact transport equation for the quantum Calogero model, i.e. inverse-square interacting bosons on a line, from microscopic first principles.

Clusters that merge and split have no unique genealogy. Tracking the host size of a conserved constituent defines $q_*$ by $\langle(S_{t+\tau}/S_t)^{q_*}\rangle=1$, with $q_*\ne0$. Across coupled map networks, coagulation-fragmentation dynamics, and Vicsek systems, $q_*\simeq\alpha-2$ without fitting, where $c(s)\propto s^{-\alpha}$ on the same size window. Radius scans locate the known Vicsek interaction radius and recover an independently specified clustering scale in experimental active rods. Local scale covariance yields the relation, while a tent map counterexample shows when reducing the stationary balance at each parent size to one moment fails.

Physical decoherence can preserve the microscopic strategic neutrality (i.e. zero effective field and equal potential of the two homogeneous strategic orientations) of a quantum game while changing the thermodynamic regime of the corresponding interacting population. We demonstrate this for an Eisert–Wilkens–Lewenstein (EWL) Stag Hunt embedded as independent nearest neighbor encounters on a square lattice. For the restricted strategies $\mathsf{Q}=i\mathbb{Z}$ and $\mathsf{D}=i\mathbb{Y}$, noisy two-player payoff matrices are determined for phase damping, depolarization, and amplitude damping and mapped exactly to channel dependent Ising parameters $\mathfrak{J}(\Gamma,p)$ and $\mathfrak{H}(\Gamma,p)$. Phase damping and depolarization show the clearest contrast: they share the same microscopic neutrality branch $\mathfrak{H}=0$, while only depolarization suppresses the interaction as $(1-p)^2$. At $\beta=1$, this produces an exact depolarization-driven square-lattice critical point at $p_{*}\approx 0.233460\ldots$, whereas phase damping remains in the ordered coexistence regime along the same neutrality branch. Monte Carlo finite size scaling is consistent with two-dimensional Ising criticality and distinguishes field driven coexistence below $p_{*}$ from a smooth crossover above it. Amplitude damping additionally reveals a strong dependence on channel placement: the post-strategy neutrality branch reaches $\Gamma=0$ at $p=1/3$ and then disappears. Resource negativity further shows that microscopic two-qubit entanglement and collective interaction strength are distinct quantities. The resulting extended lattice remains an ordinary classical Ising system.

We study the minimum-work feedback control of particles subject to active fluctuations. Considering an active Ornstein-Uhlenbeck particle confined by a moving harmonic trap, we derive exact optimal protocols following an initial position measurement. Our results show that nonequilibrium correlations between position and active fluctuations allow work extraction from the activity based on positional information only, i.e., without directly measuring the active degree of freedom, which was the focus of earlier literature. We find that depending on the persistence time, activity can either facilitate or impede transport relative to passive systems. Surprisingly, unlike feedback schemes based on direct measurements of the active fluctuations, positional feedback remains energetically advantageous even in the limit of infinitely persistent activity. Our results provide design principles for information engines and optimal control strategies operating in active environments.

We numerically investigate the mechanical response of amorphous solids to localized force dipoles, referred to as pinching, using athermal quasistatic simulations of model glasses with varying degrees of stability. We employ a control parameter, the imposed extension, corresponding to the increase in rest length between the two pinched particles relative to their initial separation. Increasing this extension allows us to continuously tune the system from the elastic to the plastic regime. For small extensions, the response remains elastic. In this regime, the displacement field induced by pinching is well described by linear elasticity and exhibits a long-range power-law decay consistent with dipolar forcing. Averaged responses display anisotropic, quadrupolar-like patterns, with quantitative agreement between simulations and analytical predictions. This continuum description remains valid down to particle-scale distances. As the imposed extension increases, the response becomes plastic. Pinching can then trigger either localized or system-spanning rearrangements, depending on glass stability. Well-annealed glasses exhibit localized plastic events, whereas poorly annealed systems display delocalized cascades. We introduce a method to extract the principal axis of plastic deformation and analyze the associated displacement fields and plastic activity. Overall, our results demonstrate that pinching provides a minimal local probe of amorphous solids. The resulting response, governed by both glass stability and imposed extension, offers insight into the interplay between elasticity, elementary rearrangements, and the emergence of collective plasticity.

We investigate the power-law decay of the survival probability of a Brownian particle diffusing in an infinite planar wedge whose two sides are partially reactive and described by Robin boundary conditions. We employ matched asymptotic analysis to relate the long-time asymptotics to a stationary harmonic Robin problem near the apex. This approach determines the persistence exponent for arbitrary opening angles and yields the prefactor explicitly for the family of wedges with $\alpha=\pi/n$ ($n=1,2,\ldots$). A simple extension of the apex prefactor to arbitrary angles is conjectured and supported numerically. This asymptotic behavior describes the crossover to the well-known result for perfectly absorbing wedges. As immediate applications, we also deduce the long-time behavior of the probability density function of the boundary local time on wedge sides, as well as the probability density function of the associated first-crossing time.

Spatial temperature fields in Brownian heat engines are commonly prescribed a priori, and the resulting transport and thermodynamic properties are then calculated. Here we formulate the complementary inverse-design problem: determining the temperature profile and barrier height that optimize a chosen thermodynamic objective. We consider an overdamped Brownian particle in a symmetric triangular periodic potential under a constant opposing load and derive the exact stationary current and probability density for an arbitrary bounded temperature field, $\Tc\le T(x)\le\Th$. In the quasistatic limit, the efficiency becomes an exact functional of two inverse-temperature integrals over the uphill and downhill branches. Its rigorous global maximum under the pointwise temperature bounds is $\eta_{\max}=1-\Tc/\Th$, attained uniquely, up to sets of measure zero, by the hot-uphill/cold-downhill piecewise-constant profile. At finite current, however, the optimization changes qualitatively because the current is determined jointly by the cycle affinity and a nonlocal transport resistance. We derive the exact functional gradient and the corresponding box-constrained optimality conditions, showing that the current- or power-maximizing profile generally differs from the quasistatic efficiency optimum. For any prescribed temperature field, the current-maximizing barrier satisfies an exact balance between the marginal gain in thermal rectification and the marginal increase in transport resistance, with the characteristic estimate $U_0^*\simeq T_{\rm act}$, where $T_{\rm act}^{-1}=(2/L)\int_0^{L/2}\dd x/T(x)$.

The yielding transition marks the onset of irreversible plastic deformation in amorphous solids and plays a central role in determining the mechanical stability and failure of metallic glasses, colloidal suspensions, and biological assemblies. Despite extensive research, the microscopic factors governing the nature of yielding, particularly the transition between brittle and ductile mechanical responses, remain poorly understood. Recent studies have identified kinetic fragility as a key parameter governing yielding in passive glasses; whether this connection persists in active glasses remains open. Here, using molecular dynamics simulations of a Kob-Andersen glass former doped with Run-and-Tumble (RTP) active particles under oscillatory shear, we show that activity systematically reduces kinetic fragility and consequently alters the mechanical response. The common yield point $\gamma_c$ decreases monotonically with activity and exhibits a power-law dependence on the Arrhenius activation barrier. Increasing activity suppresses the dependence of the yield strain on thermal history and transforms the response from brittle-like to increasingly ductile, with smoother stress relaxation and reduced stress discontinuities. The timescale to reach steady state near yielding retains a critical power-law divergence, indicating that activity does not alter the underlying critical character of the transition. Active glasses also develop broader, more diffuse shear bands. Our results establish kinetic fragility as a unifying parameter governing yielding in both passive and active glasses and demonstrate that activity offers a powerful route to tune the mechanical response of amorphous materials.

Hyperdensity functional theory facilitates the investigation of the equilibrium behavior of a general order parameter or statistical mechanical observable in spatially inhomogeneous classical many-body systems. The approach is based on applying the exact Mermin-Evans classical density functional mapping to an extended ensemble. Here we present the multivariate generalization for investigating simultaneously the properties and interrelations of several different hyperobservables of choice. The resulting framework gives rise to a systematic characterization and prediction scheme for general many-body phenomena. All pertinent equilibrium averages, variances, and covariances constitute universal density functionals, as we demonstrate explicitly. Associated one-body hyperfluctuation profiles quantify the degree of correlation of the local density with first- and second-order combinations of hyperobservables. These multivariate hyperfluctuation profiles are accessible in many-body simulations and they satisfy exact hyper-Ornstein-Zernike equations, which we derive from the minimization principle in the extended multivariate ensemble. The formal structure of the theory integrates naturally with supervised machine learning, which renders all hyperdensity functionals accessible in practice via training of neural networks on simulation data. We demonstrate all salient techniques using the illustrative case of clustering in confined hard rod fluids, thereby choosing the total number of particles and the largest cluster size as the representative hyperobservables of interest. Our numerical methodology enables the efficient and successful prediction of all statistical quantities induced by the chosen hyperobservables, which we verify via comparison to test data and which we attribute to the tight interplay of first-principles and machine-learning concepts that our general approach combines.

We investigate many-body systems with time-delayed self-interactions mediated by a memory-feedback mechanism. We show that such temporal interactions generate non-equilibrium orders and unique topological defects absent in equilibrium-specifically, helical vortices wherein opposite vorticities propagate in reverse directions along domain walls. In one dimension, memory feedback stabilizes true long-range order against weak noise, thereby circumventing the Mermin-Wagner theorem and resulting in an unconventional finite-temperature phase transition with a dynamical exponent $z = 4$. Physical realizations of these memory-driven non-equilibrium systems using active mechatronic metamaterials has also been proposed. These results demonstrate engineered temporal interactions as a powerful paradigm for non-equilibrium many-body physics.

Topological defects are ubiquitous in physics, arising from condensed matter physics to the early universe. Although there exist many universal scaling laws for correlations between topological defects, such as Porod scaling, the fingerprint of topological defect cores has remained largely unexplored. Here, we discover a universal scaling law in the region $k>1/\xi$, where $\xi$ is the healing length of the topological defects, taking the scaling of the form factor $S_f \propto k^{-(d+p+2)}$, where $d$ is the spatial dimension and $p$ is the defect codimension. We analytically prove that this exponent originates from a universal V-shaped cusp at the defect core and is independent of the underlying system and dynamics. Numerical simulations verify this scaling law in four typical frameworks: the time-dependent Ginzburg-Landau and Gross-Pitaevskii equations in the weak-coupling regime, the gauge/gravity duality model in the strong-coupling regime, and the Klein-Gordon equation in the Friedmann-Robertson-Walker background in cosmology. Our work provides a new probe for studying topological defects in systems ranging from superconductors to cosmological phase transitions.

Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schrödinger–Nagasawa pair. Its centered forward–backward kernel $C_{\rm FB}=\mathbb E_\omega[\psi_F\psi_B]$ resolves an organized connected sector, with weight $\rho_{\rm BSM}=-C_{\rm FB}(x,x)$ on the anticorrelated branch. Here we investigate how this sector forms, localizes, and propagates under Bohm/Fisher feedback. We retain the branching covariance on the prescribed background as the bare noise kernel and truncate the nonlinear interaction to Bohm/Fisher drift vertices. A Martin–Siggia–Rose–Janssen–de Dominicis formulation and a causal two-loop two-particle-irreducible (2PI) closure determine response and correlation functions self-consistently. While the free connected theory exhibits secular growth and ultraviolet accumulation, the dressed theory develops a finite relative screening length. A reduced numerical evolution shows bounded formation of the localized sector, and a self-similar Fisher construction defines a saturated information velocity $c_\star$. A Born–Oppenheimer separation connects this internal organization to collective propagation. The instantaneous adiabatic kernel has two pole families at fixed internal momentum: a gapless difference branch and a gapped sum branch. Projecting both inverse response kernels onto the same localized internal profile defines their collective coefficients. If both propagation speeds match $c_\star$ and the projected gap matches $mc_\star^2/\hbar$, the gapped branch takes the infrared Klein–Gordon form. These matching conditions define a candidate relativistic fixed point whose dynamical realization remains to be tested.

In this article, we study a generalisation of Rényi's car-parking problem in which isosceles trapeziums are sequentially deposited on a strip. We derive an explicit formula for the parking constant in terms of the lengths of the two bases, recovering the classical rectangular and triangular models as special cases. We further obtain quantitative finite-size asymptotics for both the expected number and the variance of deposited trapeziums, with convergence rates that depend explicitly on the geometry of the deposited particle. In particular, although the recursive construction involves two different substrate geometries, their variances have the same leading asymptotic density. Finally, we show that the parking constant depends non-monotonically on the ratio of the two base lengths and possesses a unique minimiser, so that the least efficient shape is a genuine trapezium rather than a triangle.

We study the time-dependent currents arising in periodically driven stochastic systems. In the linear regime of small driving, a closed expression is obtained for the impedance/admittance associated with these currents. This expression leads directly to an interpretation in terms of an equivalent electrical circuit. For a stochastic system with $N$ states, the electrical circuit consists of precisely $N$ parallel branches, with each branch comprising a resistor and a capacitor in series. The low- and high-frequency limits of these currents are calculated, and the results are generalized to more general current expressions. We demonstrate our findings across several archetypal settings, illustrating the potential to detect specific broken symmetries or the graph topology associated with the stochastic process.

We study the dynamics of Stochastic Gradient Descent (SGD), which is known to steer deep neural networks toward invariant sets that correspond to simpler subnetworks. How this steering unfolds over time remains poorly understood. We answer this by modeling the stochastic gradient flow (SGF) as a percolation process, in which nested architectural symmetries force subnetworks to merge in discrete blocks rather than by single-edge attachment. These structural transitions register as variance spikes in a macroscopic order parameter echoing physical phase transitions. We further state sufficient conditions under which the trapping argument carries over to Adam and AdamW under heavy-tailed gradient noise and measure them on a trained Transformer.

We study search processes in a complex environment using the strategy of stochastic resetting. A common occurrence in these environments is targets with finite reactivity. Stochastic resetting has emerged as a powerful mechanism for optimizing search processes by curtailing long, unproductive excursions inherent to diffusive dynamics. Most theoretical studies, however, assume perfectly absorbing targets – an idealization that overlooks the finite reactivity commonly encountered in realistic chemical and biological systems. In this work, we investigate the interplay between stochastic resetting and finite-target reactivity in reaction - diffusion processes. Considering a one-dimensional system with multiple reactive targets, we demonstrated that the target reactivity modifies the optimization landscape. We uncover distinct regimes in which resetting enhances transport efficiency towards specific targets based on their chemical kinetics. As a consequence, the optimal resetting rate becomes intrinsically reactivity dependent. Our results identify target reactivity as a crucial control parameter governing stochastic transport and provide insights into reaction-diffusion processes in complex media.

We identify the slowest-decaying nonequilibrium perturbations near the three-dimensional conserved Ising critical point and determine their impact on finite-size observables. We study two classes of perturbations: a field-dependent noise-to-mobility ratio $\Theta(\phi)=D(\phi)/M(\phi)$ and the gradient activity of Active Model B+. Starting from the Martin-Siggia-Rose-Janssen-De Dominicis action, we compute the linearized flow using the functional renormalization group. The transport sector is block triangular, with leading odd eigenvalue $y_{\Theta_1}=-\Delta_\phi$, where $\Delta_\phi=(d-2+\eta)/2$. In the gradient sector, removing the detailed-balance direction leaves two genuinely nonequilibrium modes, chemical and current-like. Two smooth regulators give $y_{\Theta_1}\simeq-0.52$, $y_J\simeq-0.56$, and $y_{\rm ch}\simeq-0.89$. A translation Ward identity expresses the current operator as the divergence of the stress tensor. Together with conservation and Itô causality, this forbids chemical operators from generating the current mode, making the nonequilibrium stability matrix triangular; an independent two-loop calculation in $d=4-\varepsilon$ finds no additional current contact counterterm. Within the FRG truncation, $y_J-y_{\Theta_1}=-\eta$; beyond it, this relation requires the absence of an additional contact anomaly. Using the 3D Ising value $\eta=0.0362978(20)$ [Kos et al., 2016] gives $y_{\Theta_1}\simeq-0.5181$ and $y_J\simeq-0.5544$. Because these exponents nearly coincide, single-power fits yield amplitude-dependent apparent exponents and crossover lengths may exceed accessible system sizes. We derive the resulting finite-size scaling rules: odd block observables respond linearly to activity, while even observables receive only quadratic corrections.

A departure of a mixed quantum state from a local Gibbs description is generally invisible to local observables but can be detected by the conditional mutual information (CMI). Here we study the relationship between the renormalization group (RG) and CMI, and in particular, how RG constrains CMI. We first show that the CMI between nonadjacent regions $A$ and $C$, conditioned on the buffer region $B$, is UV-finite whenever the state admits a locally reversible RG with a fixed on-site Hilbert space dimension. We then study two hierarchical models that have long-range CMI and yet admit a simple RG description. The first model has a divergent Markov length at every temperature $0<T<\infty$ but nevertheless flows to an infinite-temperature product state under RG. The second model satisfies the local Markov condition while violating the global one and is stable against weak noise. At the critical noise strength, the two-point CMI decays only polynomially as a function of the system size, while the two-point mutual information vanishes.