Work

Description length of canonical and microcanonical models

Year: 2025

Type: article

Abstract: The (non)equivalence of canonical and microcanonical ensembles is a fundamental question in statistical physics, concerning whether the use of soft and hard constraints in the maximum-entropy construction leads to the same description of a system. Despite the fact that maximum-entropy models are also commonly used in statistical inference, pattern detection, and hypothesis testing, a complete understanding of the effects of ensemble nonequivalence on statistical modeling is still missing. Here, we study this problem from a rigorous model selection perspective by comparing canonical and microcanonical models via the minimum description length principle, which yields a trade-off between likelihood, measuring model accuracy, and complexity, measuring model flexibility and its potential to overfit data. We compute the normalized maximum likelihood (NML) of both formulations and find that (1) microcanonical models always achieve higher likelihood but are always more complex; (2) the optimal model choice depends on the empirical values of the constraints—the canonical model performs best when its fit to the observed data exceeds its uniform average fit across all realizations; (3) in the thermodynamic limit, the difference in description length per node vanishes when ensemble equivalence holds but persists otherwise, showing that nonequivalence implies extensive differences between large canonical and microcanonical models. Finally, we compare the NML approach to Bayesian methods, showing that (4) the choice of priors, practically irrelevant in equivalent models, becomes crucial when an extensive number of constraints are enforced, possibly leading to very different outcomes.

Source: Physical Review Research

DOI: 10.1103/1sd2-rxmy

Authors: Francesca Giuffrida, Tiziano Squartini, Peter Grünwald, Diego Garlaschelli

Institutions: Leiden University, IMT School for Advanced Studies Lucca, Scuola Normale Superiore, Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni, Centrum Wiskunde & Informatica

Language: English

Topic: Statistical Mechanics and Entropy

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,625

Funders: European Commission, NextGenerationEU