We introduce the word thermalization hypothesis according to which in a literary work there are effective interactions between words that drive them to thermalization where their word frequency distribution is the same as for the phenomenon of Rayleigh-Jeans (RJ) thermalization and condensation. In this concept, the different words play a role of classical oscillators (particles) and their word frequencies, or related tokens, are analogous to certain filling factors over energy state levels. The thermalization process is characterized by the conservation of total system energy, or total number of words/tokens, and total probability norm, or total number of different words. We apply this approach to analyze 16 literary works, including Bible, complete works of Shakespeare, and data from American National Corpus. We show that for these cases the effective system energy is very low and thus there is the RJ condensation phenomenon with a big fraction of total probability concentrated at the lowest energy states corresponding to a big fraction of rare words appearing only once or twice in the whole text. Such a RJ condensation is observed for light propagation in multimode optical fibers and other physical systems. We relate the thermalization of words to the recently proposed wealth thermalization for households in world countries. The economy concepts of Lorenz and Pareto curves, and also Zipf curves, are applied to the word frequency distributions of the selected 16 cases and it is shown that the thermalization theory gives a strikingly good description of real data.
A score-based diffusion model is trained on the two-dimensional classical Heisenberg model, the lattice representation of the $O(3)$ nonlinear sigma model. Neural networks are trained on three different coarse-grainings of the same training set to implement a multiscale sampling scheme that modestly improves both the accuracy and computational cost of direct sampling on the finest lattice. Stein (or Schwinger-Dyson) diagnostics for the two-point correlation function are introduced to assess the quality of the learned score without relying on reverse diffusion sampling. These diagnostics are also incorporated into an additional training loss. We discuss several technical considerations relevant to implementing diffusion models on $S^2$. These include a consideration of lifting the network input to $S^3$, as in the $CP^1$ representation, and developing a boundary-layer approximation to the short-time spherical heat kernel that extends its usefulness to larger diffusion times.
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.
When an AI diffusion-type generator samples a frustrated spin glass, which macroscopic alternatives does it decide first, and when? We study this for bridge-diffusion models of the Path Integral Diffusion (PID) type started from a point source. We show that every isotropic PID is a time-changed Gaussian stochastic localization, and conversely every signal-to-noise schedule is realized by a PID protocol. What a continuous-time generator decides therefore does not depend on the protocol, which only sets when and at what cost. We describe the generator's state by its partial commitments: the probabilities it assigns, at each moment, to nested classes of configurations fixed by the target. The partial commitments are martingales that move only along class contrasts. Their ensemble average gives the information about each level at an exact rate set by squared magnetization contrasts. Each split is decided, approximately, like a binary Gaussian channel at signal-to-noise ratio 1.04/\lambda_P, with \lambda_P the leading eigenvalue of the parent's correlation matrix. The theory is applied to Bethe-lattice and Sherrington--Kirkpatrick (SK) glasses with up to 64 spins. It is observed generically that a trajectory first shifts its weight to one of the two mirror halves of the zero-field measure and then to the glassy classes inside it. These classes appear near the glass transition and keep their identity on cooling. Nested glassy levels have similar \lambda_P and are decided together, so a trajectory commits in one or two distinct events---the mirror choice and a compound glassy decision---which separate further with coupling and size, as the spectra predict. For a generator that takes finitely many steps, the protocol matters through the late, local stage rather than the macroscopic decisions.
Diffusion-based generative modeling has emerged as a general computational framework for simulating high-dimensional, structured data across various domains. Understanding its generative mechanisms is paramount for advancing model training and inference in a principled way. Here, we demonstrate that deterministic sampling trajectories share a universal geometric structure invariant to data modalities and diffusion model implementations. This structure can be largely explained by an exactly solvable model based on the optimal linear denoiser.
The running-time distribution of a randomized algorithm is computed under independent randomness, but independence is rarely what a source guarantees. We study sources $\nu$ with $\nu[w]\le DP[w]^s$ for every finite prefix $w$ of the tape, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$; individual choices may be predictable, even determined by the past. The worst-case probability of any closed tape event $E$ is exactly $\min\{1,DC_s(E)\}$, where $C_s(E)$ is the infimum cost of covering $E$ by prefixes of varying depth. This tree max-flow/min-cut identity makes worst-case probabilities exact and optimizable. Three applications follow. In scheduling, $n$ tasks on one machine are each retried until the next random symbol lies in its success set; every schedule has the same makespan distribution under independent randomness, yet under our constraint two fixed orders miss a linear deadline with worst-case probabilities $1$ and $\exp[-\Theta(n)]$, and a static priority rule is optimal at every deadline. For the standard local walk on a $k$-CNF with an assignment satisfying a majority of each clause, we find the sharp source threshold $s_*$: below it, on some instances one admissible source forces nontermination for every clause selection; at it, with fixed $D$, the expected number of flips is at most cubic in the initial distance, against linear under independence, with matching instances at distances up to a constant fraction of the number of variables; a fixed margin above it restores linear dependence. For noisy predecessor search with prefix-bounded comparison errors, verified backtracking attains any prescribed error probability at or above a source threshold whose sharpness we leave open, with the same cubic-to-linear query bounds. Restarting on the shared tape restores exponential tails exactly when the nontermination event has a prefix cover of cost below one.
Pretrained diffusion models provide powerful learned priors, but in scientific sampling the target distribution often depends on physical context that a single generative model can neither fully represent nor learn from scarce joint-system data. We introduce Generative Gibbs for Physics-Aware Sampling (GG-PA), a training-free framework that composes learned partial priors with explicit physical context through Gibbs sampling. We prove that this sampler is asymptotically exact as the diffusion time vanishes and remains exact at finite diffusion time for quadratic interactions. As a special case, the finite-time result corrects a finite-noise bias in plug-and-play split Gibbs samplers for linear inverse problems at no additional cost. We further introduce replica exchange over diffusion time to accelerate mixing. In experiments, a one-dimensional double-well prior recovers a context-induced distribution shift when coupled to an environment, and copies of such a prior composed on a lattice reproduce the emergent collective behavior of the $\phi^4$ model. In atomistic peptide systems, GG-PA captures a context-induced distribution shift and the spontaneous formation of hydrogen-bonded dimers from two copies of a prior trained only on the monomer.
Learning Gibbs distributions using only sufficient statistics has long been recognized as a computationally hard problem. On the other hand, computationally efficient algorithms for learning Gibbs distributions rely on access to full sample configurations generated from the model. For many systems of interest that arise in physical contexts, expecting a full sample to be observed is not practical, and hence it is important to look for computationally efficient methods that solve the learning problem with access to only a limited set of statistics. We examine the trade-offs between the power of computation and observation within this scenario, employing the Ising model as a paradigmatic example. We show that the couplings of the model can be learned to a desired accuracy from statistics whose order grows linearly with the $\ell_1$ width of the model and only logarithmically with the inverse of the accuracy. In particular, for a fixed accuracy, the required order is proportional to the width. This allows us to infer the model's structure, using statistics whose order grows logarithmically with the inverse of the weakest coupling, and also to learn its magnetic fields. When the width and the accuracy are fixed, the running time is polynomial in the number of variables. We also discuss a setting where prior information about structure of the model is available and show that the learning problem can be solved efficiently with even more limited observational power.
The recent success of foundation models and deep learning has far outpaced our theoretical understanding of how large-scale neural networks learn and generalize. Statistical physics provides a natural language for describing such high-dimensional, nonlinear, and strongly coupled systems, offering concepts such as mean-field theory, phase transitions, energy landscapes, and stochastic dynamics. In this topical review, we adopt a statistical-physics way of thinking -- from concepts including phases and criticality, and free-energy viewpoints, to methods such as kernel and Gaussian-process limits, mean-field and dynamical mean-field theories, random-matrix tools, and Fokker-Planck formalisms, and to a mindset emphasizing universality, scaling, and tractable limits with explicit validity regimes. We survey recent progress on understanding modern deep learning through this lens, including mean-field analyses of wide networks, spectral and geometric perspectives on loss landscapes, stochastic-gradient-based training viewed as a dynamical process, and phase-transition analogies that organize phenomena such as overparameterization, generalization, and representation change. Our goal is to synthesize these threads into a coherent framework that clarifies what each approximation explains, when it is expected to hold, and how apparently competing narratives relate or diverge. Together, these perspectives form a unified physical framework that views representation learning as a collective phenomenon in high-dimensional systems, bridging deterministic and stochastic regimes of gradient-based optimization.
Modern distributed networks, notably transformers, acquire a remarkable ability (termed `in-context learning') to adapt their computation to input statistics, such that a fixed network can be applied to data from a broad range of systems. Here, we provide a complete mechanistic characterization of this behavior in transformers trained on a finite set $S$ of discrete Markov chains. The transformer displays four algorithmic phases, characterized by whether the network memorizes and generalizes, and whether it uses 1-point or 2-point statistics. We show that the four phases are implemented by multi-layer subcircuits that exemplify two qualitatively distinct mechanisms for implementing context-adaptive computations. Minimal models isolate the key features of both motifs. Memorization and generalization phases are delineated by two boundaries that depend on data diversity, $K = |S|$. The first ($K_1^\ast$) is set by a kinetic competition between subcircuits and the second ($K_2^\ast$) is set by a representational bottleneck. A symmetry-constrained theory of a transformer's training dynamics explains the sharp transition from 1-point to 2-point generalization and identifies key features of the loss landscape that allow the network to generalize. Put together, we show that transformers develop distinct subcircuits to implement in-context learning and identify conditions that favor certain mechanisms over others.
Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits through their ability to support rich time-dependent behaviour. Whilst high-dimensional by construction, they can be used to encode low-dimensional dynamical systems, for example by restricting to low-rank connectivity, hence mirroring the emergence of latent neural manifolds found in empirical neural data. Here, we exploit a relationship between RNNs and a class of universal approximators to train networks that can represent arbitrary, nonlinear stochastic differential equations (SDEs), with given drift and diffusion coefficients, within a low-dimensional latent subspace. We show that RNNs can faithfully embed the drift and diffusion of a given SDE, including nonlinear and nonequilibrium dynamics such as chaotic attractors. We apply this approach by constructing RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results show that neural populations can implement a broad class of stochastic dynamics within low-dimensional manifolds, unifying ideas from the dynamics of neural computation, the theory of neural manifolds, associative memory, and nonequilibrium statistical mechanics.
Masked diffusion models generate text by reversing a stochastic masking process. Unlike autoregressive LLMs, which generate text one token at a time, they can reveal many tokens in parallel, at the price of an error, since tokens revealed together are drawn independently. We show that this trade-off maps onto the statistical physics of driven many-particle systems. (i) For any data, the error of parallel generation is the entropy production of the generation process relative to the reversed masking process. (ii) For Markov data, generation is a random sequential adsorption process with labelled gaps, which we solve exactly for any number of steps. (iii) For many steps, the error becomes the dissipation of a slowly driven system: the unmasking schedule plays the role of the driving protocol, the number of steps that of its duration, and the error is set by a thermodynamic metric that we obtain in closed form from the mutual information between tokens. Optimal schedules are geodesics of this metric. The exact solution shows how correlations in the text shape both the error and the quality of the output. The longer the correlations, the more the error concentrates at the start of generation. Schedules that start with small steps then gain more, both in path error and in perplexity, and need fewer steps to reach their asymptotic behaviour.
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 data-generating parameters. The resulting inequality decomposes the upper bound on 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 data-generating parameters are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for understanding which properties of the data-generating process are acquired, in what order, and how rapidly they can be encoded in the trainable parameters.
Creating images from noise is image generation; reconstructing fine details from coarse inputs is super-resolution. Despite their practical differences, both can be understood as reversing information loss across scales. We introduce $\textbf{SKILD}$, a $\textbf{S}$cale-invariant $\textbf{K}$-Space $\textbf{I}$mage $\textbf{L}$earning $\textbf{D}$iffusion model that unifies generation and continuous super-resolution within a single unconditional framework. Both natural images and critical physical systems exhibit scale invariance, and we leverage it to design a forward process that attenuates image content from fine to coarse scales while injecting spectrum-matched Gaussian noise, making scale an explicit coordinate of the diffusion dynamics. The same trained reverse process performs generation and continuous super-resolution by varying only the starting timestep: $\textit{no task-specific architecture, no conditioning branch, no classifier-free guidance, no retraining per scale factor}$. Empirically, SKILD reaches FID $2.61$ and Inception Score $9.65$ on unconditional CIFAR-10, performs $2\times$--$8\times$ super-resolution on ImageNet from a single unconditional checkpoint while outperforming conditional models across perceptual metrics, and reconstructs critical Ising models whose connected four-point correlations closely track the ground truth.
Large language models excel at factual recall, yet the fundamental limits of storing input--output associations with neural networks remain unclear. We study these limits in a minimal setting: a linear associative memory that maps $p$ input embeddings in $\mathbb{R}^d$ to their $d$-dimensional targets via a linear layer, requiring each mapped input to score its own target above all other outputs. Since all inputs compete for the same outputs, these constraints are strongly correlated, which makes the storage capacity $p_c$ --- the largest number of associations that can be stored --- hard to characterise. We address this in three steps. First, we introduce a decoupled model, in which each input competes with its own independent set of outputs, and provide numerical and analytical evidence that it is equivalent to the original model, both in capacity and in the structure of the learnt solution. Second, using tools from statistical physics, we show that the decoupled model has capacity $p_c \log p_c / d^2 = 1/2$, extend this computation to linear two-layer networks, and go beyond the capacity by predicting the spectrum of the learnt weights and the storage mechanism. The latter explains how the optimal solution outperforms a na\"ive Hebbian rule: rather than boosting input--output alignments with broad fluctuations, it raises the correct scores just above the extreme-value threshold set by the competing outputs. Third, we prove the decoupled model's capacity thresholds rigorously, including in the non-convex two-layer case; combined with a recent proof for the original model, this shows that both models have the same capacity for full-rank weights. These findings give a sharp characterisation of factual storage in linear networks, and suggest decoupling as a general tool for memorisation and learning problems in which each input must be matched to one of an extensive number of outputs.
Computing molecular free-energy landscapes over collective variables (CVs) is central to understanding molecular mechanisms, but poses a difficult learning problem without training data available a priori. Samples of the marginal CV distribution must be generated by molecular dynamics (MD), whose trajectories can remain trapped in metastable states and rarely cross the free-energy barriers governing conformational changes or chemical reactions. We introduce VESPER, which couples the generation of these samples to neural-network-driven free-energy learning: MD supplies configurations for learning, while the evolving model biases the MD to accelerate exploration and generate further training data. VESPER minimizes a single variational objective that is convex in the free energy and can combine several tempering levels, each flattening the free-energy landscape to promote barrier crossing. Every level has the same physical free energy as its minimizer, allowing data from all levels to train one shared model without reweighting. Learning uses the difference in mean parameter gradients between biased MD configurations and auxiliary walkers that explore the flattened model in CV space. We prove conditional convergence of the coupled overdamped population dynamics without replica exchange under log-Sobolev and Polyak-Lojasiewicz assumptions. VESPER matches or improves on established enhanced-sampling methods on alanine dipeptide and yields self-consistent free-energy reconstructions from three to six CVs on alanine tetrapeptide. For methane dehydrogenation on Ni(511) with a machine-learned interatomic potential, it recovers two-CV Boltzmann marginals and a reactant-to-saddle free-energy difference along a three-CV path, both qualitatively consistent with the lower-dimensional reference. It also samples RMSD-defined folded and unfolded regions of chignolin while biasing seventeen CVs.
Introduction: The increasing availability of large-scale textual corpora has opened new possibilities for data-driven, quantitative approaches to historical analysis using Natural Language Processing (NLP). However, diachronic corpora with historical relevance from the pre-digital era remain scarce and often incomplete. Methods: We present a quantitative approach to historical analysis based on the reconstruction and exploration of a diachronic corpus of around 600,000 articles from the Italian newspaper "La Repubblica", covering all the articles published from the first of January 1985 to the 31st of December 2000 - a period of major political, social, and geopolitical change in Italy and globally. Using NLP techniques, we analyze the text at both lexical and semantic levels; we then apply tools from complex systems and statistical physics to trace shifts in media discourse over time. Results: The word usage in the corpus is bursty, with a heavy-tailed distribution of Fano factors that is robust across timescales. Changepoints in word salience and the block structure of semantic similarity allow us to detect key transition periods, such as the transition from the First Republic to the Second Republic in Italy, or major international conflicts like the Gulf War or the Kosovo War, without relying on prior labeling. Periods of war correspond to lower entropy in the semantic space, indicating a more focused discourse. Discussion: The results show how combining computational linguistics with ideas from complex systems can offer new quantitative insight into historical changes, opening up new paths for studying the dynamics of media and society through large-scale textual data.
While state of the art for image and video generation, the fundamental reasons for the effectiveness of generative diffusion remain unclear. In particular, the observed advantage of following an entire reverse process with respect to single-shot or autoregressive methods, and for what type of data it may hold, is not well understood. In this work, we aim to better understand the role of the reverse process by considering a complex yet analytically tractable data model, the Gibbs measure of a high-dimensional Gaussian random potential with long-range correlations. Focusing on the exact score, we show that the speciation threshold, in which a trajectory commits to one class of the data, becomes a speciation window, in which the reverse process progressively resolves the Parisi tree of pure states of the full replica-symmetry breaking measure, committing to ever finer branches down to a single pure state. Importantly, these commitments take place at noise levels where the noisy sample distinguishes the pure states only through an energy of order one, beyond the reach of standard replica saddle points. Constructing a Larkin argument suited to such $O(1)$ energy gaps, we describe the score as a foam, with slab-like cells separated by shock-like boundaries. In the variance-exploding process we consider, this window is followed by a shrinkage of the samples towards the data support, and finally by reconstruction within the selected pure state. The denoising error of the exact score follows these stages, dropping during the speciation window with a profile set by the Parisi function alone, and then within the selected state. We believe that the generative foam we describe, and in particular the boundaries between cells, could play an important role in determining where the errors of trained diffusion models redistribute the mass of the generated samples when the data are long-range correlated.
Using large deviations theory, we solve and obtain a general expression for the free energy functional for a broad class of associative memories, including dense associative memories. We illustrate the method by reproducing classical results for the Hopfield model. For a finite number of patterns, we derive the temperature-dependent free energy functional for dense associative memories featuring polynomial interactions and Log-Sum-Exponential (LSE) activation. We also obtain the disorder-averaged ground-state energy landscape of these systems in the extensive limit. Our analytical framework reveals how memory retrieval depends on the initial state in higher-order dense networks, and gives the exact error-free threshold for the LSE model. This method provides a systematic procedure for analyzing diverse, complex architectures in associative memory.
This paper argues that dataset structure is important in image recognition tasks (among other tasks). Specifically, we focus on the nature and genesis of correlational structure in the actual datasets upon which DNNs are trained. We argue that DNNs are implementing a widespread methodology in condensed matter physics and materials science that focuses on mesoscale correlation structures that live between fundamental atomic/molecular scales and continuum scales. Specifically, we argue that DNNs that are successful in image classification must be discovering high order correlation functions. It is well-known that DNNs successfully generalize in apparent contravention of standard statistical learning theory. We consider the implications of our discussion for this puzzle.