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cond-mat.stat-mech: explore 27 source-linked works published from 2023 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

Effects of relational graph modularity and depth on the learning performance of neural networks

In recent years, graph-based machine learning techniques, such as reinforcement learning and graph neural networks, have garnered significant attention. While some recent studies have started to explore the relationship between the graph structure of neural networks and their predictive performance, they often limit themselves to a narrow range of model networks, particularly lacking mesoscale structures such as communities. Our work advances this area by conducting a more comprehensive investigation, incorporating realistic network structures characterized by heterogeneous degree distributions and community structures, which are typical characteristics of many real networks. These community structures offer a nuanced perspective on network architecture. Our analysis employs model networks such as random and scale-free networks, alongside a comparison with a biological neural network and its subsets for more detailed analysis. We examine the impact of these structural attributes on the performance of image classification tasks. Our findings reveal that structural properties do affect performance to some extent. Specifically, within moderate-depth architectures, networks featuring coherent, densely interconnected communities demonstrate enhanced learning capabilities. Crucially, we find that this advantage is strictly depth-dependent: extending the architecture to eight layers reverses the effect entirely. This comparison with the biological neural network emphasizes the relevance of our findings to real-world structures, suggesting an intriguing connection worth further exploration. This study contributes meaningfully to network science and machine learning, providing insights that could inspire the design of more biologically informed neural networks.

cs.LG↗
arXiv2026

Dynamical stability for dense patterns in attractor neural networks

Recurrent neural networks are canonical models of biological memory. In these models, memories are represented by distributed patterns of neural activity that are stored in the recurrent connections between neurons, such that they become attractors of the network's dynamics. During memory recall, network dynamics thus converge toward one of these memory patterns when started from a noisy or partial cue. Therefore, memory performance critically hinges on the dynamical stability of the stored patterns. However, previous theoretical approaches only studied dynamical stability under highly restrictive conditions that do not readily apply to biological neural circuits. Here, we develop a theory of the local stability of discrete fixed points in a broad class of networks with graded neural activities and in the presence of noise. Using methods from random matrix theory, we analyze the bulk and outliers of the eigenvalue spectra of the Jacobians that characterize network dynamics around fixed points. We show that either all fixed points are stable or all of them are unstable, depending on whether their number is below a ``critical load for stability'', which is distinct from the classical critical capacity that measures the maximal number of achievable fixed points regardless of their stability. We further analyze the dependence of this critical load for stability on experimentally measurable quantities characterizing the statistics of memory patterns and the activation functions of neurons. Our analysis highlights the computational benefits of sparse-like patterns and threshold-linear activation functions and offers testable predictions for neural circuits supporting memory.

cond-mat.dis-nn↗
arXiv2026

Speed Limit for Information Acquisition in Stochastic Learning Dynamics

Neural networks acquire internal representations through learning. In this work, we formulate stochastic gradient descent (SGD) as a Markovian stochastic process and derive a Fisher-information flow speed limit that bounds the rate at which trainable parameters can acquire information about latent variables in the data-generating process. The resulting inequality decomposes the information flow into drift and noise contributions, thereby quantifying the roles of deterministic learning forces and SGD-induced fluctuations from an information-theoretic perspective. We verify the bound in analytically tractable basis-function linear regression, where the information budget predicted by the bound reproduces the ordering and characteristic time scales with which different latent variables are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for diagnosing when and how different aspects of the data-generating mechanism are acquired during stochastic learning.

cond-mat.stat-mech↗
arXiv2026

Physics of Information Geometry - Part I: Principle of Least Action on the Probability Simplex

We develop a least-action framework for describing how a probability distribution can evolve from an equilibrium state to a prescribed nonequilibrium state under constrained incremental changes. Taking a Gibbs distribution as the equilibrium reference, the framework gives a direct physical meaning to the geometry of the probability simplex: distance from equilibrium corresponds to nonequilibrium free energy, while changes between successive distributions carry an informational kinetic cost. The Pythagorean structure of relative entropy then provides the central insight of the work. It shows that intermediate distributions chosen via sequential information projections can reduce the kinetic cost of large transitions and establishes an energy-conservation-like relation between the kinetic expenditure along a path and the free energy accumulated in reaching the target distribution. Motivated by this geometry, we construct a greedy least-action path through successive information projections, obtain a closed-form characterization of each projection through the Lambert W function, and establish a finite-step performance guarantee. We further show that state-dependent costs can be incorporated naturally by reshaping the underlying Gibbs reference, providing a thermodynamic interpretation of path penalties as modifications of the effective energy landscape. Together, these results provide a unified view of distributional evolution through least action, information geometry, and nonequilibrium thermodynamics.

cs.IT↗
arXiv2026

Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering

Energy-Based Models (EBMs) provide an interpretable framework for generative modeling of scientific data, but poor Markov Chain Monte Carlo mixing often limits their reliability. We introduce a training algorithm based on Parallel Trajectory Tempering (PTT), which exploits the continuity of the optimization path to maintain equilibrium sampling throughout learning. This enables stable and fast training on highly multimodal and data-scarce scientific datasets. Combined with reservoir sampling and adaptive optimization, PTT has a computational cost comparable to Persistent Contrastive Divergence, making it a practical replacement for standard training methods. It also provides direct estimates of thermalization times, equilibrium samples from trained models, and accurate log-likelihoods at essentially no additional cost. Experiments on Restricted Boltzmann Machines show that PTT consistently outperforms existing EBM training approaches. On discrete tabular data, it also surpasses state-of-the-art deep generative models, yielding higher-quality samples and greater robustness to overfitting and limited data. Our results make equilibrium maximum-likelihood training of EBMs practical and computationally efficient.

cs.LG↗
arXiv2026

Human mutation field reveals an equilibrium-like structure with irreversible circulation

The evolution of DNA sequences can be viewed as stochastic dynamics on a high-dimensional discrete space, but it is unclear when empirical transition biases reduce to an effective energy landscape versus retain irreducible non-equilibrium circulation. Human context-dependent mutation probabilities offer a direct test: every single-nucleotide substitution in a local context has a reverse substitution, so the logarithm of the forward-to-reverse probability ratio defines an antisymmetric field-the human mutation field. We show this field has a dominant gradient component and a smaller but reproducible curl component. Using seven-base human germline substitution probabilities, we infer an effective mutational landscape with a Siamese neural network constrained to predict only energy differences. This model predicts forward-to-reverse log-ratios for held-out mutations with a correlation of about 0.93, close to both an unconstrained predictive reference (0.948) and the empirical reversible ceiling from Hodge projection (about 0.96). Although trained only on mutation probabilities, the inferred landscape largely recovers short-word genomic composition and Chargaff reverse-complement symmetry for sequences up to length four. Deviations from equilibrium structure reveal a small but detectable nonequilibrium component: a residual irreversible circulation violating the Kolmogorov cycle condition for detailed balance, reproducible across African, Asian, and European populations, and strongest in CpG-linked cycles and CpG-transition edges, consistent with methylcytosine deamination. These results give a thermodynamic decomposition of the human mutation field: most mutation bias is organized by a local equilibrium-like energy landscape aligned with genome composition, while the residual circulation points to specific directional mutational mechanisms.

q-bio.GN↗
arXiv2026

Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem

The recently proposed Microcanonical Hamiltonian Monte Carlo algorithm has not yet been studied in detail from a thermodynamic point of view; this work aims to fill that gap. We demonstrate how thermodynamical state variables and potentials can be derived and thereby demonstrate that the construction of the algorithm formally represents a microcanonical thermodynamic ensemble. In particular, we demonstrate (analytically and numerically) that the algorithm fulfils the Helmholtz theorem, an alternative formulation of the first law of thermodynamics. Furthermore, we construct a new sampling algorithm that extends the original to lower-dimensional inference problems. Finally, we argue that canonical Markov Chain Monte Carlo algorithms are more natural than Microcanonical Hamiltonian Monte Carlo from the thermodynamic and information-theoretic point of view.

cond-mat.stat-mech↗
arXiv2026

Thermodynamic Cyclic Processes with Markov Samplers in Bayesian Inference

The concept of Markov chain Monte Carlo (MCMC) cycles, an analogy to cyclic processes in heat engines, is presented in order to examine Bayesian inference problems. In this effort, we develop adaptive ensemble schedulers that allow the tuning of external parameters of a Bayesian canonical ensemble during an MCMC run, realising the MCMC cycles in practice. We run these cycles on different statistical models. As a fundamental insight, we find (both theoretically and in practice) that such systems can produce a non-zero net work output if and only if the considered model is non-Gaussian. As such, they may serve as a measure of non-Gaussianity in Bayesian inference, which we test on an example from supernova cosmology.

stat.CO↗
arXiv2026

Unsupervised Thermodynamics of Molecular Diffusion Models: Action-Operator Semantics and Auditable Free-Energy Readout

Diffusion models are increasingly utilized for modeling molecular structures and conformational ensembles, yet the thermodynamic meaning of their learned representations and scores remains elusive. To resolve this ambiguity, we introduce a mathematically consistent action-operator framework natively compatible with diffusion models. By defining a fixed molecular environment as a base action $S_0(x)$ and an alchemical perturbation as an operator $O(x)$, standard diffusion noising induces effective noised actions and operators whose gradients and alchemical derivatives are directly represented by the model's learned fields. This rigorous self-consistency enables a ``noisy operator bridge'' capable of reading out free-energy differences ($ΔF$) from endpoint ensembles and per-frame evaluations. In controlled experiments on alanine dipeptide systems, we show that incorporating physical inductive biases enables partial recovery of the base action and perturbation operator. When applied to a challenging C6-H to C6-F ligand-pocket nonbonded perturbation (185L/IND) with negligible phase-space overlap, our supervised bridge estimates the alchemical $ΔF$ within approximately $1\ k_\mathrm{B}T$ of a stable 19-state MBAR reference. Finally, we demonstrate that endpoint coordinates and binary labels alone are sufficient to partially recover the operator shape and a centered free-energy scale without any force or action supervision. This work provides a rigorous path toward transforming generative molecular diffusion models from black-box coordinate samplers into auditable thermodynamic estimators.

physics.chem-ph↗
arXiv2026

Coarse-Graining Hidden Representations: Unsupervised Neuron Selection via Mapping Entropy

Overparameterized neural networks carry far more hidden units than a task nominally requires, raising the question of which neurons are essential and whether that distinction is legible in the representation itself, without labels or gradients. We cast neuron selection as the problem of coarse-graining the hidden layer by retaining a subset of its neurons, and score each putative selection by the mapping entropy (ME). This quantity measures the loss of discriminatory power inherent in discarding part of the network neurons, and the selection that minimises the ME is taken as particularly informative. This criterion is fully unsupervised, in that it depends only on hidden-activation statistics. In teacher-student networks, ME optimisation recovers the minimal teacher-consistent representation and retains extra units in proportion to the hidden layer's residual variability; in a non-linear Gaussian process task, it selects coherent functional-class mappings whose preferred class shifts across training. On this task and on translation-augmented MNIST, ME-selected subnetworks outperform random subsets of equal size, most clearly under strong compression - linking configurational distinguishability to predictive performance.

cs.LG↗
arXiv2026

Entropy-Generated Attention Beyond Softmax and Entmax: Kaniadakis and Reciprocal-Symmetric Abe Operators

We derive two attention operators from generalized statistical entropies. Kaniadakis entropy yields an exact full-support normalization whose weights and low-score sensitivities decay algebraically, rather than exponentially as in Softmax or by exact truncation as in entmax. Classical Abe entropy yields an implicit reciprocal-symmetric operator. With $q=e^ε$, the involution $q\leftrightarrow q^{-1}$ removes every odd correction about Softmax; we obtain the normalized second- and fourth-order terms, including the deformation of the normalization multiplier. These stationary laws follow from a Fisher-metric Lagrangian on the probability simplex, whose Shannon sector recovers scaled dot-product Softmax. We also give a tangent-gradient test for deciding whether changing the entropy changes the attention profile or only its scale. Rényi and two-parameter Sharma--Mittal entropies retain the Tsallis--entmax inverse-gradient shape, but their global moments make the effective temperature input dependent when the external temperature is fixed. Distinguishing profile-shape equivalence from fixed-parameter operator equivalence separates new normalization shapes from adaptive rescalings and organizes the operators by support, tail behavior, and realization complexity.

cs.LG↗
arXiv2026

Data Driven Equation Discovery for Phase-Ordering Dynamics : From Allen Cahn to the Ising Model

Data-driven discovery of governing equations from spatiotemporal data offers a promising route to obtaining coarse-grained descriptions of complex dynamical systems. Here, we investigate the performance of PDE-SINDy for discovering phase-ordering dynamics using the Allen--Cahn equation as a benchmark and the Ising model with Glauber spin-flip dynamics as a microscopic system. We systematically analyze the effects of data availability, size of the candidate library, and noise on the efficiency of the equation discovery. We find that stability-selection PDE-SINDy can robustly identify the relevant terms in the governing dynamics even under limited or noisy data, while the recovered coefficient values are substantially more sensitive to these factors. We further show that enlarging the candidate library can strongly affect both term identification and coefficient recovery. Incorporating library bagging with stability selection reduces this sensitivity and improves the efficiency of equation discovery. For the Glauber spin flip Ising model dynamics, the resulting coarse-grained equation reproduces the characteristic phase-separation and coarsening dynamics of the underlying microscopic system. Overall, our results demonstrate the potential of PDE-SINDy for phase-ordering systems while highlighting the importance of carefully assessing the factors that influence the efficiency of equation discovery.

cond-mat.soft↗
arXiv2026

Percolation Dynamics in Optimization: Variance Cascades and Nested Symmetry

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.

cs.LG↗
arXiv2026

Generative Nested Sampling of Atomistic Thermodynamic Landscapes

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.

cond-mat.stat-mech↗
arXiv2026

Evolutionary design of thermodynamic logic gates and their heat emission

Landauer's principle bounds the heat generated by logical operations, but in practice the thermodynamic cost of computation is dominated by the control systems that implement logic. CMOS gates dissipate energy far above the Landauer bound, while laboratory demonstrations of near-Landauer erasure rely on external measurement or feedback systems whose energy costs exceed that of the logic operation by many orders of magnitude. Here we use simulations to show that a genetic algorithm can program a thermodynamic computer to implement logic operations in which the total heat emitted by the control system is of a similar order of magnitude to that of the information-bearing degrees of freedom. Moreover, the computer can be programmed so that heat is drawn away from the information-bearing degrees of freedom and dissipated within the control unit, suggesting the possibility of computing architectures in which heat management is an integral part of the program design.

cond-mat.stat-mech↗
arXiv2026

FrOGS: Discrete Neural Sampler for Independent Alloy Configurations Across Chemical Conditions

Predicting the thermodynamic properties of an alloy requires sampling its configurations across many chemical conditions and recovering free energies on a common absolute scale. Markov chain Monte Carlo (MCMC) is the standard tool, but it requires separate simulations at different conditions, and auxiliary free-energy methods such as thermodynamic integration are used to place results on a common absolute scale. Modern discrete neural samplers typically use reverse KL divergence as the objective and can be mode-seeking or biased. We present Free energy Offering Generative Sampler (FrOGS), a hybrid discrete neural sampler that couples an autoregressive model to a continuous-time Markov chain (CTMC) to be trained jointly under a single shared loss. FrOGS draws i.i.d. configurations, returns an unbiased estimate of the partition function, and gives consistent estimates of thermodynamic observables. We train a single model across a wide range of chemical conditions to produce estimates on a common absolute free-energy scale. FrOGS matches exact finite-size results on the 2D Ising model and reference phase diagrams for AgPd and CuAu, without mode collapse. We additionally compare to SEGAL, a published autoregressive baseline, and find that only FrOGS recovers the stability range of the CuAu$_3$ phase.

cond-mat.mtrl-sci↗
arXiv2026

Bellman-sufficient Information Complexity

We introduce Bellman-sufficient information complexity for minimax analysis of sequential decision problems. A Bellman-sufficient state retains enough of the history to close the controlled recursion, while an index $Y=χ(Ω)$ specifies the decision-relevant information being charged. The upper bound is a log-penalized Bellman program; the lower bound is a Bellman--Fano comparison along an algorithm-dependent reference trajectory. If the two values match at a common localization scale and the stated admissibility, calibration, and growth conditions hold, they form an information-risk sandwich. UCB, E2D, and AMS/EBO control or relax the upper Bellman bracket in different ways. For the main application, we give a negative answer to a widely studied form of the GP--UCB minimax-optimality question. For every $0<α<1/4$, we construct one bounded continuous kernel whose minimax regret is $Θ(T^{1-α})$ along an infinite sequence of horizons, while two globally calibrated GP--UCB rules incur linear regret under one fixed truth. An epochwise finite-marginal action-index AIR Bellman policy, implemented through robust AIR/AMS/EBO control, attains the minimax order. The construction separates realized information from the cost of uniform optimism: many low-value directions inflate the exploration multiplier and change the trajectory. Through the canonical RKHS feature map, it also yields a finite-horizon polynomial minimax separation for the specified maximal-information-calibrated LinUCB rule. A reproducible experiment illustrates the mechanism.

cs.LG↗
arXiv2026

Breakdown of Edgeworth Expansion in Finite-Blocklength Regime and Exact Absorption via $q$-Deformation

This paper addresses the structural breakdown of the Edgeworth expansion in the finite-blocklength (FBL) regime, where conventional asymptotic approximations yield unphysical negative probabilities in the deep-tail region. We propose a $q$-deformed framework that resolves this inconsistency by replacing additive polynomial perturbations with a geometric deformation of the information density space. Motivated by the linearization of nonlinear dynamics, we prove that dynamically scaling the $q$-logarithmic parameter exactly absorbs the third-order skewness while preserving global nonnegativity. We establish a universal asymptotic matching, demonstrating that the framework encapsulates higher-order asymptotic scales. Numerical results confirm that the proposed method matches the state-of-the-art precision of the Cornish-Fisher bound without the risk of negative probabilities. The framework offers a robust and computationally stable foundation for evaluating operational limits in ultra-reliable communications such as 6G and URLLC.

cs.IT↗
Compare source metadata on this page
WorkPublishedSource identifierSource
Effects of relational graph modularity and depth on the learning performance of neural networks2026-09-082507.10005arxiv
Dynamical stability for dense patterns in attractor neural networks2026-09-082507.10383arxiv
Speed Limit for Information Acquisition in Stochastic Learning Dynamics2026-09-082609.08219arxiv
Physics of Information Geometry - Part I: Principle of Least Action on the Probability Simplex2026-09-082609.08285arxiv
Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering2026-09-072607.27077arxiv
Human mutation field reveals an equilibrium-like structure with irreversible circulation2026-09-072609.07500arxiv
Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem2026-09-072609.07620arxiv
Thermodynamic Cyclic Processes with Markov Samplers in Bayesian Inference2026-09-072609.07660arxiv
Unsupervised Thermodynamics of Molecular Diffusion Models: Action-Operator Semantics and Auditable Free-Energy Readout2026-09-062606.30687arxiv
Coarse-Graining Hidden Representations: Unsupervised Neuron Selection via Mapping Entropy2026-09-042609.05126arxiv
Entropy-Generated Attention Beyond Softmax and Entmax: Kaniadakis and Reciprocal-Symmetric Abe Operators2026-09-032602.08216arxiv
Data Driven Equation Discovery for Phase-Ordering Dynamics : From Allen Cahn to the Ising Model2026-09-032608.20404arxiv
Percolation Dynamics in Optimization: Variance Cascades and Nested Symmetry2026-09-272609.02373arxiv
Generative Nested Sampling of Atomistic Thermodynamic Landscapes2026-10-072609.03193arxiv
Evolutionary design of thermodynamic logic gates and their heat emission2026-09-012602.13410arxiv
FrOGS: Discrete Neural Sampler for Independent Alloy Configurations Across Chemical Conditions2026-09-212609.02948arxiv
Bellman-sufficient Information Complexity2026-08-312606.11171arxiv
Breakdown of Edgeworth Expansion in Finite-Blocklength Regime and Exact Absorption via $q$-Deformation2026-08-312608.23303arxiv

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