Papers

For the hard-sphere Boltzmann equation, we derive an exact nonlinear generating vertex for collisional stress and heat-flux production. The complete two-input Hermite hierarchy for stress resums to a universal traceless tensor times one scalar radial law, while the heat-flux vertex follows from the same law by an exact contraction. Taylor differentiation yields universal bilinear collision tensors with exact stress and heat-flux gradings, without a finite moment closure. DSMC tests, including wall-driven flows and a bimodal Mach-5 shock, recover independently sampled Boltzmann productions and the predicted convergence, while the converged production is independent of the thermal Gaussian foot used for its Hermite representation.

(Suzuki-)Trotter decompositions, splitting methods, (Lie) product formulae... The most common numerical methods for the time evolution of quantum systems come with many names. And they are used practically everywhere with applications ranging from the solution of classical equations of motion and various Monte Carlo simulations to the real and imaginary time evolution on classical as well as quantum computers. Here we review the state of the art of said methods, focussing especially on the progress made over the last few years. We highlight recently discovered efficient time evolution algorithms and explain how best to use them in practice. A central part of this work is the estimation of error bounds that has improved greatly within the past decade. The relevance of time evolution methods for quantum computing is discussed with a focus on noisy hardware. Finally, a comprehensive overview of generalisations, related methods and alternatives to Trotterization is provided. This includes time-dependent Hamiltonian dynamics, processed methods, multi-product formulae, symplectic integrators, TDVP for tensor networks, quantum signal processing, Crouch-Grossman methods and more. The overall perspective in this work is that of a theoretical physicist. All mathematical proofs as well as some technical details are omitted for easier readability. Instead, this review serves as a hands-on guide and, of course, as a starting point for references that provide further details.

Determining which microscopic states remain thermally active when tunneling and interactions compete is a central question in Hubbard physics. Here we develop an entropy measurement protocol for a double quantum dot, extending the charge-based approach established for a single dot, and measure the total entropy of a tunable two-site Hubbard molecule realized in a GaAs double dot. By shifting both dot levels together at fixed detuning, the entropy is correctly probed with only a single charge sensor. As interdot tunneling increases, the system evolves from two atomic-like dots through hybridized molecular states to a merged single dot. Across this evolution, the entropy decreases as tunnel-induced energy splitting exceeds the thermal energy and suppresses the occupation of higher-energy states. The measurements resolve the diminishing contribution of antibonding states and the changing thermal contributions of hybridized singlet and triplet states, in quantitative agreement with a two-site Hubbard model. By distinguishing states with the same charge configuration but different orbital and spin content, entropy reveals how tunneling and interactions determine the thermally active states of the minimal Hubbard system.

Most molecular transitions occur on timescales far beyond direct molecular dynamics simulations. The committor, the probability that a configuration reaches a product state before a reactant state, is a central kinetic statistic, providing a mechanism-independent reaction coordinate and a foundation for transition path theory and the calculation of rates. This review surveys modern approaches for estimating the committor and related kinetic statistics from molecular simulations, with an emphasis on self-supervised methods that learn solutions of their defining dynamical equations rather than relying on labeled shooting data. We develop a common operator viewpoint connecting generator-based partial differential equations, variational principles, Markov state models, dynamical Galerkin approximation, and neural networks. Empirical and theoretical evidence points to the efficiency of these methods. We provide theoretical and practical guidance for realizing their full potential in applications, including strategies for treating non-Markovian effects and for sampling. We conclude by identifying opportunities for further research, including connections to reinforcement learning and generative modeling.

We investigate the dynamics of entanglement asymmetry in periodically driven two-dimensional conformal field theories with a global U$(1)$ symmetry, using a dynamical-system description based on iterated conformal maps, in particular Möbius maps. Starting from a symmetry-breaking excited state prepared by a local operator insertion, we show that a broad range of nonequilibrium phenomena, including local symmetry restoration, the quantum Mpemba effect, and its inverse, admit a simple geometric description in terms of the invariant data of the conformal map. We explicitly demonstrate this using Möbius maps, for which the invariant structure is described by the conjugacy classes. In particular, the dynamics of entanglement asymmetry is determined by the relative positions of the state-preparing operator insertion, the subsystem, and the fixed points, yielding distinct patterns of growth, decay, oscillation, and saturation. We further argue that this geometric picture extends to general holomorphic maps, establishing the conformal-map dynamics as a natural framework for organizing the phenomenology of entanglement asymmetry in periodically driven two-dimensional conformal field theories.

We show that microscopically inversion-asymmetric, permeable active fluid membranes are statistically flat and effectively inversion-symmetric at large scales. Their fluctuations are governed by an asymptotically exact linear hydrodynamic equation giving orientational long-range order and positional quasi-long-range order. At intermediate scales, their dynamics is described by a Kardar-Parisi-Zhang equation with spatially long-range noise, producing rough membranes with short-range translational and long-range orientational order described by exactly known scaling exponents. Active stresses can destabilize the membrane at long wavelengths, while sufficiently strong active permeation flow can drive finite-wavevector instabilities and pattern formation. These results reveal a novel activity-driven route to the destruction of flat fluid membranes.

Micron-sized particles moving through a fluid are subject to viscous resistance and thermal fluctuations. Beyond steady Stokes friction, colloids also exhibit hydrodynamic memory effects arising from conservation laws of the surrounding fluid. Although fluid-flow problems in complex geometries and confinements are often highly involved, numerical or approximate solutions can be obtained; translating these solutions into closed-form equations of motion for individual colloids, however, is rarely possible. Here, we develop a theoretical framework that provides both underdamped and overdamped colloidal dynamics from the solution of an arbitrary flow problem: (i) Using the Lorentz reciprocal theorem, we express the colloidal equation of motion in terms of the geometry's Green's function and the fluid-flow profile. This formulation yields both the corresponding Stokes drag as well as a Basset-like hydrodynamic memory contribution. (ii) As colloid inertia is often negligible, we furthermore derive the corresponding overdamped (colloid-inertia-less) limit, which inherits the Stokes and Basset-like resistances and additionally gives rise to a spurious drift. We propose applications of this framework to passive and active particles subject to involved confinements and problem geometries.

Micron-sized colloid particles diffusing through a fluid experience hydrodynamic inertial and memory effects, the latter causing velocity autocorrelation to decay as a power law. At the same time, many theoretical descriptions treat colloidal diffusion in a fluid using overdamped Langevin dynamics for its convenience, eliminating velocity, omitting inertia, but also ignoring the power-law memory. In this Letter, we show that hydrodynamic memory survives in the overdamped (colloid-inertialess) limit. By identifying the dimensionless parameter controlling the crossover between early- and late-time dynamics, we derive in closed form the overdamped Langevin equation in the presence of hydrodynamic memory. This provides a theoretical framework for realistically describing colloidal dynamics in a fluid, and establishes a rigorous basis for the positional memory observed in high-resolution experiments. Our theory predicts that hydrodynamic memory becomes increasingly pronounced for smaller particles and under stronger external forcing, and offers experimental probes for the crossover from conventional exponential relaxation to memory-dominated power-law dynamics.

The impact of projective measurements on a many-body quantum state is tightly linked to its underlying entanglement structure. Critical states are particularly sensitive, as long-range entanglement allows local measurements to have global consequences. This has been extensively studied in the context of critical states described by a conformal field theory (CFT), where measurements can alter critical properties in post-measurement quantities ('measurement-altered criticality') and induce long-range entanglement with universal features ('measurement-induced entanglement'). By contrast, little is known about the role of measurements on critical states with quenched disorder, which admit no CFT description. Here, we develop a theory of measurements on one-dimensional critical states governed by infinite-randomness fixed points (IRFPs), focusing on two closely related examples: the random XXZ chain and the random transverse-field Ising model. We show that the measurement-induced entanglement decays as a power-law with universal exponent $(3-\sqrt{5})/2$ in both models, and that the minimum number of measurements required to establish this entanglement obeys a universal scaling form. When only a finite density of measurements is made, we show that the critical properties of the state are remarkably robust: `measurement-altered criticality' is absent, with measurements either preserving the critical exponents, or destroying the criticality entirely. These results establish IRFPs as an analytically tractable arena for measurements on critical states, complementary to the CFT case. The outcome randomness that necessitates replica methods in clean systems becomes trivial at infinite randomness, with the universal response governed by the statistics induced by the quenched disorder.

Using thermal analyticity, we derive directional large-frequency bounds on the smooth functions entering full eigenstate thermalization in continuum quantum field theory. The strongest directions reproduce the scales previously identified for lattice systems, while the complete analyticity domain reveals a broader asymmetric multifrequency structure. In thermal two-dimensional conformal field theory, exact two- and three-point functions saturate the corresponding bounds. At fourth order, where genuine dynamical information first appears, we show that the holographic CFTs studied here saturate the strongest full-ETH bounds. These results establish a direct connection between full ETH, free cumulants, and gravitational scrambling, and are consistent with the emergence of asymptotic freeness realized by the gravitational scrambling algebra.

Using thermal analyticity, we derive directional large-frequency bounds on the smooth functions entering full eigenstate thermalization in continuum quantum field theory. The strongest directions reproduce the scales previously identified for lattice systems, while the complete analyticity domain reveals an asymmetric multifrequency structure. Thermal conformal correlators saturate the second- and third-order bounds, and the leading gravitational Regge contribution saturates the strongest fourth-order direction.

Starting from a general Hamiltonian describing the quantum dynamics of probe particles interacting with a set of environment degrees of freedom, we derive the quantum master equation with which the reduced density matrix of a probe particle evolves, the general equilibration condition that they satisfy when the environment is prepared in a thermal state – encoded in a KMS property for line operators extended in real time – and the Langevin description that emerges. The novelty of our results resides in their generality: We do not assume a specific form of the environment self-coupling, of the environment operator that couples the particle to it, of the statistics of its correlation functions, whether the statistics of the momentum transfer is Gaussian or not, or whether the probe particles move relativistically. Our results only require that spacetime translations, parity and time reversal be symmetries of the Hamiltonian, and that there exists a separation of scales between those characterizing the dispersion relation of the probe particle and those of the environment. We also discuss the limiting case in which Gaussian Brownian motion emerges. Overall, our results constitute a first-principles derivation of what properties of the underlying quantum field theory govern the energy loss and momentum fluctuations of probe particles.

Spherical p-spin models provide a mean-field framework for glassy dynamics. Coupling such systems opens a controlled way to probe how they evolve jointly, develop cross correlations, and possibly share common aging scales. We study spherical p-spin systems with site-independent inter-system couplings and generally correlated quenched disorders. We derive closed multi-subsystem two-time equations for correlations, responses, and spherical constraints. A marginal one-step replica-symmetry-breaking construction provides static benchmarks for overlap plateaux, aging crossover scales, and effective temperatures. We then specialize to two ferromagnetically coupled p=3 systems and compare dynamical data with marginal 1RSB predictions. Independent quenched disorders reduce the equal-time correlation between the systems below its disorder-free value, while correlated disorders enhance it. For sufficiently strong coupling, finite-time fluctuation-dissipation estimates within and between the systems are compatible with a common effective temperature even when one or both subsystems would equilibrate as a paramagnet in isolation. This common value lies above the effective temperatures that the constituent systems would have separately and increases with disorder correlation and coupling strength. The marginal 1RSB value T/m captures these trends but is slightly below the finite-time dynamical estimate. For weak coupling, the dynamical results instead suggest that two aging regimes develop for each system, with partial equilibration within each. We perform a marginal replica analysis with two-step breaking, whose predictions compare favorably with the numerical measurements. Our numerical accuracy is insufficient to determine whether the observed finite-time crossover will eventually become a sharp transition.

Consider a branching random walk (BRW) of depth $n$ with standard Gaussian increments, conditioned on all endpoints lying in an interval of radius $r$ (which may depend on $n$), i.e. a small ball event. We prove that this conditioning results in exponential decay of correlations between the endpoints, with correlation length $O(r)$. Our proof is based on the renormalization group, which in this context captures the effect of the endpoint conditioning on earlier steps in the walk. This effect can be separated into so-called relevant and irrelevant parts, and in our setting the relevant part makes the conditioned BRW behave like a branching Ornstein-Uhlenbeck walk for the first $n-O(r)$ steps, leading to the correlation decay. The irrelevant part shrinks in a nontrivial manner, and the simplest argument to bound it uniformly only yields correlation length $O(r^2)$. To achieve the optimal $O(r)$ bound, we establish that the irrelevant part shrinks horizontally in space before decaying uniformly. The BRW is a prototypical example of a log-correlated Gaussian field, and it forms the basis of many hierarchical field theories in the mathematical formulation of quantum field theory. As such, we present our results in a general form which is applicable to other hierarchical field theories with confining potentials, of which the small ball conditioned BRW is an example. We apply our general result to two other models: first, the sinh-Gordon model which has a strictly convex potential, so at the level of covariances the model behaves like one with a quadratic potential, i.e. the model has a positive mass. Second, we consider generalizations of the $\phi^4$ model with any power $\alpha \geq 2$; as $\alpha$ increases from $2$ to $\infty$, these power potentials naturally interpolate between the massive (quadratic) potential and the infinite square well potential of the small ball event.

We investigate the relaxation of an approximately conserved charge in interacting quantum systems close to local equilibrium. To this end, we provide a pedagogical review of Zubarev's non-equilibrium statistical operator approach in the minimal setting of a single non-conserved charge and apply it to the problem at hand. We explicitly highlight the physical assumptions that lead to a local relaxation law: weak charge violation, a separation between the short timescale of microscopic correlations and the much longer timescale of charge relaxation, and the resulting loss of microscopic memory. Under these conditions, the leading relaxation rate is determined by an equilibrium correlation function of the charge-violating operator. We show that the same result follows from a simpler local-equilibrium construction based on the system's evolution over an intermediate timescale, providing a direct alternative for practical calculations and making the common physical ingredients of the two approaches explicit. Beyond the decay law itself, we relate the relaxation rate to the equilibrium diffusion of the same charge. We then allow the charge density to vary in space, which leads to a diffusion-relaxation equation. Finally, we illustrate the formalism through electroweak $\mathrm{B+L}$ washout and a perturbative scalar model, where agreement with the linearized Boltzmann equation establishes a direct connection between equilibrium-correlator and kinetic descriptions.

Living systems compute with chemistry by mapping environmental signals onto specific internal chemical states. Despite recent advances in molecular programming, it remains unclear which physicochemical features control the computational expressivity of chemical systems. Here we inverse-design thermodynamically consistent chemical reaction networks whose steady-state response to an environmental input computes a target nonlinear function. Using implicit differentiation we train the free-energy landscape directly: standard chemical potentials, transition-state energies and thermodynamic drives. Increasingly large networks generated by elementary ligation and cleavage steps fit increasingly complex nonmonotonic polynomial functions, with expressivity scaling logarithmically with network size, predicted primarily by the number of reactions. Training individual energetic parameter classes reveals that internal thermodynamic drives, capable of breaking detailed balance, dominate trainability, with comparable performances achieved only by pairs of parameter classes. These results identify nonequilibrium drive as the most effective single resource for steady-state computational expressivity in chemical reaction networks.

By consuming multiple copies of an unknown qubit state, one can modify its purity while preserving the direction of its Bloch vector. We determine the maximum linear rate at which qubit states of different purities can be interconverted, allowing a nonzero error, quantified, for instance, by the trace distance, provided that it vanishes in the limit of infinitely many copies. Interestingly, the optimal conversion rate is determined by the two eigenvalues of the complex right-logarithmic-derivative (RLD) Fisher information matrix associated with $\mathrm{SU}(2)$ rotations of the qubit state. When the output qubits have higher purity, corresponding to concentration, the optimal rate is given by the ratio of the maximum eigenvalues of the input and output RLD matrices. In contrast, when the output qubits have lower purity, corresponding to dilution, the optimal rate is given by the ratio of their minimum eigenvalues. Remarkably, both concentration and dilution can be implemented using SWAP tests as the only nontrivial two-qubit measurement primitive, together with ancillary qubits initially prepared in maximally mixed states, without requiring any additional two-qubit gates. Our work thus provides a novel operational interpretation of the full complex RLD Fisher information matrix. Crucially, its antisymmetric, purely imaginary part encodes geometric information beyond the statistical distance between density operators and plays an essential role in determining the optimal state-conversion rates.

When a classical Brownian particle is subjected to a temperature gradient it can undergo thermophoresis, i.e., the particle gets transported towards the colder regions of the environment, which bears relevant consequences to nonequilibrium self-organization. Here, we investigate how the quantum Brownian motion is affected by a temperature gradient. We employ a recently proposed generalized system-plus-reservoir model, where one assumes a continuum field of thermal baths. As our main result, we obtain the quantum influence functional, within the Feynmann-Vernon path integral approach. To test convergence, we derive an effective Langevin equation for the center of mass of the quantum wavepacket, and show that it coincides with the classical Langevin equation at high temperatures. Both the fully quantum (low temperatures) and the semiclassical (high temperature) limits show signatures of quantum thermophoresis in the quantum Brownian motion, a so far unexplored effect.

Inferring directed effective interactions from neuronal spike trains is a central inverse problem in statistical physics and computational neuroscience. Kinetic Ising models provide a tractable framework for this task, but their application to neural data typically requires binning spike trains into binary activity variables, discarding within-bin timing and conflating collective network dynamics with single-neuron history effects. We introduce SpiKIsing, a latent-variable model that separates these two levels of description. A discrete-time asymmetric kinetic Ising model describes collective network activity, while continuous-time, history-dependent point processes generate the observed spikes conditional on the latent states, accounting explicitly for refractoriness and post-spike recovery. We derive a variational mean-field expectation-maximization scheme in which the point-process likelihood enters as an effective observation field, enabling joint inference of latent activity, network couplings, and emission parameters. The framework extends naturally to maximum-a-posteriori inference with structured priors, including sparsity and a hierarchical extension favouring Dale-consistent outgoing interactions. We validate parameter recovery on matched synthetic data and test the method on spike trains generated by a recurrent conductance-based leaky integrate-and-fire network. In this setting, SpiKIsing recovers sparse connection structure and correctly classifies all excitatory and inhibitory neurons despite the substantial mismatch between the data-generating dynamics and the SpiKIsing model.

Using the two-dimensional Ashkin-Teller (AT) model, we compare criticality on three non-periodic lattices: the Smith-hat aperiodic tiling, Voronoi-Delaunay (VD) random triangulations, and uncorrelated diluted square lattices. The decay of the block-averaged coordination fluctuation $\sigma_Q$ with exponent $\alpha$ is used to describe the connectivity disorder. The first two lattices share the same exponent $\alpha$, which differs from that of the third. We consider the regime where the correlation-length exponent $\nu<1$, where randomness is relevant according to the Harris criterion $d \nu \le 2$, but should be irrelevant in cases of the Smith-hat tiling and VD triangulations, where $\alpha \nu >1$, according to the Harris–Barghathi–Vojta (HBV) criterion. For the diluted lattice, we indeed find that the clean universality behavior breaks down along the entire critical line, indicating a crossover to a fixed line dominated by disorder, in line with both the Harris criterion and the HBV criterion. In contrast, both VD and Smith-hat lattices display critical exponents consistent with the clean AT universality class, as validated by a Coulomb-gas self-consistency check, violating the Harris criterion while conforming to the HBV criterion.

The mechanical response of a moving probe in an active bath reflects both the dynamics of the bath and how the probe couples to it. For a probe moving at velocity $\mathbf V$ and interacting weakly with an ideal active bath, we show that the trajectory statistics of a free bath particle determine a density response from which the bath force on the probe follows. The probe-particle interaction determines how this response is weighted over wave vector $\mathbf q$, while the probe motion selects frequencies $\omega_{\mathbf q}=\mathbf q\cdot \mathbf V$. This trajectory-response relation separates the bath dynamics from the wave-vector weighting, so neither the stationary state nor force correlations need to be recalculated for each probe interaction and velocity. Applying this relation to active-particle models, we prove that positive power transfer is excluded for active Ornstein–Uhlenbeck particles in all dimensions and for complete-reset run-and-tumble particles in $d\geq2$, while one-dimensional run-and-tumble particles and two-dimensional active Brownian particles possess modes that can transfer positive mechanical power. For a given probe, this response predicts drag reversal, spontaneous probe motion, and motion-induced density distortions, all in quantitative agreement with simulations. Varying the probe geometry changes how the same bath response is weighted, thereby selecting different moving states. Beyond the leading weak-probe, ideal-bath limit, higher orders in probe strength involve multi-interval trajectory statistics, while finite-density corrections involve interacting-particle dynamics. Our results connect the trajectory statistics of active bath particles to the mechanical response of a passive probe, providing a route to predict probe response from single-particle trajectory statistics measured in the absence of the probe.

We study mass-conserving Markov jump processes on a square lattice, where masses hop with a preferred rotational sense, thus breaking both time-reversal and mirror symmetries. We consider closed systems with both periodic and open (reflecting) boundaries, the latter supporting a steady-state edge current. We show that odd diffusion, arising from the chiral transport, generically provides a mechanism for the emergence of scale-invariant two-point density correlations in nonequilibrium steady states even in the presence of lattice rotation symmetry – a mechanism that is qualitatively distinct from the well-known mechanism of anisotropic hopping. We exactly calculate the steady-state equal-time density-density correlations, which exhibit an algebraic decay, $C(\mathbf{r}) \sim |\mathbf{r}|^{-4}$ for $|\mathbf{r}| \gg 1$. This algebraic behavior results from the interplay between the off-diagonal components of the diffusion and mobility tensors, demonstrating that chiral transport alone can generate power-law correlations in isotropic driven systems. Remarkably, the structure factor in the zero-wavenumber limit and the amplitude of the power laws depend nontrivially on chirality. While increasing chirality initially suppresses large-scale density fluctuations, the fluctuations beyond a threshold odd-diffusion strength vary nonmonotonically with chirality and develop a cusp singularity.

The geometric phase (GP) of a periodically driven symmetricspin-boson model (SBM) is investigated using the process-tensor time-evolving matrix product operator (PT-TEMPO) method. A longitudinal monochromatic field drives the qubit along the same coordinate that couples to an Ohmic bosonic environment. The gauge-invariant kinematic GP is evaluated from the reduced-statetrajectory, and a circular non-additivity measure quantifies the combined effects of driving and dissipation relative to driven-unitary and undriven-dissipative references. Amplitude, frequency, coupling-strength, and bath-cutoff sweeps reveal regimes in which periodic driving reduces or enhances environmental phase deformation relative to the unitary counterpart. The one-cycle GP exhibits pronounced nonmonotonic drive frequency dependence, while its bath-induced shift depends sensitively on both coupling strength and cutoff. These trends are interpreted qualitatively in terms of Floquet dressing and bath spectral structure. A two-cycle echo provides a complementary reversibility diagnostic, whose persistence signal does not generally measure the Tong GP. The results characterize the non-additive influence of driving and dissipation on geometric-phase dynamics and identify parameter regimes of reduced bath-induced deformation.

The computation of transport coefficients in interacting quantum many-body systems is rarely analytically accessible. Here, we develop a new Krylov-space mechanism that makes the leading density dependent correction to Green-Kubo diffusivity exactly calculable in the strong-dissipation regime of a one-dimensional noisy spin-$\frac{1}{2}$ XXZ chain. This is done by showing that the density-polarization-dressed bond coherence generates a Krylov subspace where repeated action of the dissipator closes exactly on an explicitly identifiable operator family. Within this subspace, the dissipative dynamics is then mapped onto a self-similar semi-infinite chain with a boundary defect. Surprisingly, all Lanczos coefficients of the Krylov chain and its boundary Green's function can be determined exactly. The latter then determines the exact leading density-dependent correction to the diffusivity. Finally, we calculate analytically the full boundary-to-bulk Green's function and find that it decays exponentially along the emergent Krylov chain.

Black holes in the Universe span nearly ten orders of magnitude in mass over a large cosmic timescale, dating back to at least the Universe's early history, when it was only a few hundred million years old. The formation and evolution of these objects remain a mystery because no direct probe can trace each black hole across the full mass range and cosmic time. This work shows, for the first time, a statistical mechanical description of black hole evolution. I show that though each black hole and its evolution are governed by stochastic processes, one can make a macroscopic description of the distribution function of black holes in the Universe, and its evolution over the entire mass range and cosmic time can be expressed in terms of the Kramers-Moyal expansion. This bridging between statistical mechanics and black hole evolution in terms of the Kramers-Moyal expansion enables the discovery of the underlying physical processes that play a role in the evolution of black holes. In the future, the application of this technique to data accessible from different messengers, such as gravitational waves and electromagnetic waves, which probe black holes in the Universe, can lead to a data-driven understanding of the underlying physical processes by this statistical mechanics technique despite the ignorance of the underlying microscopic stochastic processes.

We investigate how unsupervised machine-learning methods can characterize the active-absorbing phase transition in the contact process in one and two spatial dimensions. Our analysis focuses on principal component analysis (PCA) and variational autoencoders (VAEs), and we show that direct applications of these methods encounter a central difficulty: contact process configurations are intrinsically positive-definite. In the PCA case, standard centering procedures fail to isolate the order parameter in the leading component. To address this, we augment the data with sign-reversed configurations, creating a balanced dataset with zero mean while preserving the density information required to describe the transition. This preprocessing allows the dominant principal component to recover the order parameter and reproduce the expected finite-size scaling near criticality. For VAEs, we find a related limitation: a fixed Gaussian prior does not provide sufficient latent regularization across the full range of control parameters and system sizes for positive-definite data. We overcome this by adopting a heteroscedastic Gaussian prior that adapts to the control parameter, leading to a substantial improvement in the finite-size scaling of reconstruction-based observables near the critical threshold. Taken together, these results highlight a key lesson: the success of machine learning in phase-transition problems does not rely on algorithmic complexity alone, but on tailoring the representation to respect the physical structure of the system, including its intrinsic constraints and symmetries.

The pace of evolution depends on how rapidly new phenotypes arise. We show that neutral Wright-Fisher evolution exhibits anomalous first passage despite diffusive mutations. The mean time for the first individual to reach a prescribed phenotypic distance scales approximately as $(\sigma^2)^{-3/2}$ with mutation variance $\sigma^2$. Two crossovers bound this regime, with inverse-variance scaling on either side. Combining coalescent theory with Kardar-Parisi-Zhang (KPZ) fluctuations at the dilute population edge, we develop an edge-KPZ theory of all three regimes. The anomaly persists under weak selection.

Nonextensive Bose and Fermi state is well studied with counting near black-hole horizons in a one-function, nonextremal, static, spherically symmetric class. The calculation is organized around a $q$-deformed grand ensemble whose total partition function is a $q$-product of single-level partitions. This identifies the bosonic occupation-number sum as the primary level partition and allows us to separate two operations that are often conflated: replacing that sum by a geometric-series form and applying the ordinary reciprocal-log identity to a $q$-logarithm. A finite-energy tunneling prescription yields a leading near-horizon state-counting term controlled by the surface gravity and a horizon-response factor. Entropy matching then gives an energy scale proportional to the Hawking temperature; the geometry factor cancels from normalized nonextensive corrections and from the Bose–Fermi ratio. We derive the first-order coefficients analytically and evaluate the unexpanded $q$-integrals numerically. The linear approximation is accurate to better than one percent when $q$ lies within roughly two percent of unity. All prescriptions recover the extensive limit, but the exact bosonic level sum and the geometric approximations give opposite signs for the leading deformation of the conventional $7/8$ Bose–Fermi relation. The result isolates a robust geometry-factorized structure and quantifies a distinct $q$-algebra systematic uncertainty.

Visible dynamics and mean dissipation need not determine the collective phases of rare histories. We construct local Markov models with identical ordinary visible path distributions, stationary states, mean entropy production, mean jump activity and driving affinities, yet different phases when histories are weighted by their full entropy production. Visible configurations modulate hidden driven cycles, whose exact elimination turns kinetic correlations into interactions. A positive spectral reduction separates hidden relaxation from collective modes. Periodic binary models retain opposite phases with entropy variances matched at every observation time. For conserved particles, a pair matches the first two joint cumulants of entropy and hidden activity at every observation time, yet selects segregation or a fluctuating state with suppressed long-wavelength density noise. Bound-state transport and fermionic evolution determine their contrasting dynamics. At a clean Ising critical midpoint, the entropy source couples quadratically to the thermal perturbation, making the fourth cumulant the first to detect the critical susceptibility. Its rate per site grows logarithmically, with an exact cutoff set by system size and observation time. For two quenched disorder ensembles, exact sample-response formulas and numerical evidence support a finite fourth-cumulant density despite very slow relaxation; a weak-disorder continuum response admits a controlled Borel sum. A common positive hidden eigenvector characterizes autonomous selected dynamics. When this condition fails, a finite network develops visible memory while preserving its ordinary visible process and mean diagnostics. These results identify kinetic information missed by aggregate thermodynamic observations.

The sign-structure of a quantum many-body ground state is a central quantity in fixed-node approaches to mitigating the Fermion sign problem. For continuum electronic-structure Hamiltonians, the correct sign-structure is sufficient to compute exact ground state properties; on the other hand, lattice fixed-node approaches retain an additional dependence on trial wave-function amplitudes which are improved upon non-optimally while preserving their sign structure. Here we consider an iterative map obtained by repeatedly replacing the fixed-node trial state with the ground state of its associated lattice fixed-node Hamiltonian (independent of whether and how a practical algorithm could implement this iteration). We show the fixed points of this map are eigenstates of the Hamiltonian and reduced Hamiltonian resulting in at most one fixed point in the interior of each sign chamber. We prove that the ground state fixed point is stable and that the boundary of the ground state sign chamber is repulsive. This shows the amplitude dependence beyond having the correct ground state sign-structure disappears under self-consistent iteration. In other sign chambers, energy descent can be obstructed by the imposed sign structure; this tension leads to the iteration driving select amplitudes to zero producing `support collapse' onto chamber boundaries and corresponding to reduced Hamiltonians. Excited-state fixed points are therefore provably unstable and flow toward lower-energy boundary fixed points. We show sign chamber boundaries have a directional stability; boundary points attractive in one direction are repulsive under a sign flip. By revealing the geometry and stability structure of iterative fixed-node dynamics, our results clarify the foundations of a central approach to the Fermion sign problem and motivate potential new algorithmic strategies.