statistical mechanics

This work proposes the first framework that formalizes the discovery of structural mappings from statistical mechanics solvable models as an AI agent task. Addressing the challenge of automatically determining whether a given partition function can be mapped to a known solvable Ising-type model, the authors introduce StatMechBench-v0—a benchmark comprising six problem classes—and design a multi-layer verification mechanism integrating large language models, numerical simulations, symbolic computation, and Pfaffian analysis. Experiments reveal that while current LLM-based agents can leverage feedback to correct code and recover partition functions, they frequently misclassify solvability categories or underestimate problem complexity. These findings highlight fundamental limitations in structural reasoning by existing models and underscore the necessity of incorporating symbolic reasoning and structural invariant verification into AI-driven scientific discovery pipelines.

This paper addresses the lack of physical interpretability in distribution shift and generalization error within machine learning. Methodologically, it systematically reconstructs thermodynamic laws within the exponential family framework, modeling log-loss minimization as a maximum-entropy-driven statistical mechanical process and establishing rigorous correspondences between thermodynamic quantities—such as work, heat, and thermodynamic cycles—and learning dynamics. Key contributions include: (i) the first formulation of universal thermodynamic laws—zeroth through fourth—for exponential families; (ii) a thermodynamic characterization of distribution shift, yielding a principled generalization error bound under shift-induced dynamics; and (iii) a computable “statistical heat engine” evaluation framework grounded in information geometry and log-loss optimization. Collectively, these results provide a novel information-physical perspective on AI foundations and introduce quantitative analytical tools for characterizing learning behavior under distributional change.

Emergent phenomena in complex systems lack predictable, universal regularities. Method: We propose and rigorously validate the “Mesoscale Peak Principle of Causal Efficacy”: causal efficacy—quantified via effective information (EI)—peaks not at microscopic or macroscopic scales, but at a characteristic mesoscale, revealing an intrinsic scale-selection mechanism by which local interactions generate global behavior. Our framework integrates maximum-entropy interventions with mutual information to construct a multiscale causal measure, and combines statistical model selection to identify causal structure. Results: Robust, non-monotonic peaks in causal efficacy are consistently observed across both the Ising model and multi-agent collective-behavior models, confirming the principle’s universality. This work establishes a foundational, computationally tractable basis for modeling emergence, constructing effective theories, and performing cross-scale causal inference.

Bayesian inference under non-Gaussian likelihoods remains challenging due to the absence of a rigorous thermodynamic interpretation. Method: This work establishes a formal statistical-mechanical analogy for Bayesian inference, rigorously mapping Bayesian updating onto thermodynamic ensemble transformations. It systematically identifies partition functions, relative entropy, and thermodynamic quantities—work, heat, and free energy—with probabilistic update operations; introduces the concept of “likelihood work” and derives a Jarzynski-type equality; and defines an effective dimension based on the partition function as a novel model complexity measure. Contribution/Results: We derive an analytical expression for the path-wise relative entropy of sampling trajectories, enabling continuous-time Bayesian updating. The framework is validated on strongly non-Gaussian cosmological inverse problems, demonstrating superior modeling accuracy and computational robustness. This provides a unified thermodynamic interpretation of information-theoretic inference and delivers practical tools for statistical learning.

This study investigates the trade-offs among collective output, stability, and adaptability in multi-agent systems to achieve optimal order. To this end, it proposes a unified analytical framework grounded in agent influence (power) and response functions, incorporating task dependency and system relativity into formal measures of order, entropy, and information. The framework reveals an intrinsic trade-off between synchrony and system fragility. By integrating multi-agent modeling, macroscopic variable derivation, and risk-preference parameterization, the work optimizes system utility, thereby enhancing the predictability and controllability of collective behavior. It further delineates the precise conditions under which collective intelligence emerges and optimal order is attained.

A unified theoretical framework characterizing the macroscopic generative dynamics of LLM-driven agents remains lacking; it is unknown whether universal, architecture- and prompt-agnostic physical laws govern such dynamics. Method: We model LLM text generation via the principle of least action, integrating large-scale statistical estimation of state-transition probabilities with trajectory-level analysis, and conduct the first empirical test of detailed balance in LLM generative dynamics. Contribution/Results: We empirically demonstrate that LLMs implicitly learn an underlying potential function governing token transitions, and that their state transitions strictly satisfy detailed balance—a property robust across diverse models and prompts. This establishes the first quantifiable physical foundation for the macroscopic generative process of LLMs, advancing AI agent research from empirical engineering toward a measurable, predictive scientific paradigm.

This work aims to construct a conceptual bridge between statistical physics and deep learning for researchers without a physics background. By recasting statistical physics as a natural extension of probability theory, it systematically elucidates the Boltzmann–Gibbs distribution, Ising models, spin glasses, and phase transition theory, while uncovering their intrinsic connections to Hopfield networks and restricted Boltzmann machines (RBMs). The key insight lies in demonstrating the equivalence between integrating out hidden units in RBMs and the renormalization group transformation, thereby revealing a physically grounded mechanism underlying multilayer deep networks. This framework not only deepens the theoretical understanding of deep learning but also provides a clear physical interpretation of the developmental trajectory of large language models.

This work addresses the challenge of reconciling power-law distributions in strongly correlated complex systems with thermodynamic consistency. By constructing a thermodynamic framework for power-law statistics based on the renormalized entropy $s_{2-q}$, it integrates macroscopic variational principles with microscopic superstatistics. Introducing the concept of varentropy—the variance of entropy—it unifies macroscopic and microscopic perspectives and reveals the physical origin of the nonextensivity parameter $q$, establishing the thermodynamic relation $|q - 1| \simeq 1/C$, where $C$ denotes the heat capacity. Through asymptotic analysis of the $q$-order generalized factorial, the study achieves a finite and stable thermodynamic limit of order $O(N^0)$, thereby providing a self-consistent and thermodynamically sound foundation for power-law behavior in systems with finite heat capacity.

This work addresses the challenge of constructing effective mesoscopic dynamical equations for complex multiscale systems by proposing a hypothesis-driven modeling paradigm grounded in the generalized Onsager principle. Within a class of hypotheses that satisfy prior theoretical constraints—such as global well-posedness and asymptotic stability—the framework unifies the description of dissipative and conservative processes, integrating energy dissipation structure analysis with data-driven techniques to identify concrete models. Validated on both continuous PDE benchmarks and microscopic chain model data, the approach not only accurately reconstructs unknown mesoscopic dynamics but also yields physically interpretable diagnostic insights, thereby achieving a balanced trade-off among accuracy, robustness, and interpretability.

This study addresses the limitations of traditional macroeconomic analysis, which relies on microfoundations and struggles to capture complex systems. Bypassing explicit microeconomic modeling, the authors employ a Thermodynamic Macroeconomics framework to construct simulated exchange economies via computational methods and directly measure their entropy functions through an approach analogous to calorimetry in physics. For the first time, they successfully quantify the entropy of a macroeconomic system without recourse to microfoundations, empirically verifying its path independence and concavity—key properties of a state function. The results align with analytical solutions across multiple simulated economies, demonstrating that thermodynamic methods remain valid and effective even in complex economic systems where microfoundational models are intractable, thereby substantially expanding the applicability of thermodynamic approaches in economics.