Canonical Criterion for Third-Order Transitions

TL;DR

This paper introduces a fluctuation-based canonical framework for identifying third-order phase transitions using energy cumulants, providing a practical and broadly applicable method that links microcanonical and canonical descriptions.

Contribution

It establishes a new canonical criterion for third-order transitions based on cumulant ratios, connecting microcanonical and canonical formalisms without requiring density-of-states reconstruction.

Findings

The criterion accurately detects third-order transitions in various models.

It remains operational in nonequilibrium steady states.

The framework is validated on multiple statistical physics models.

Abstract

Microcanonical inflection-point analysis (MIPA) identifies third-order transitions from derivatives of the microcanonical entropy, but whether such transitions admit a direct canonical formulation has remained unclear. Here we establish a fluctuation-based canonical framework for third-order transitions through a cumulant-ratio criterion whose signed extrema define their canonical counterparts and, in the single-saddle regime, are asymptotically linked to microcanonical classification. Because the criterion depends only on energy cumulants, it avoids explicit density-of-states reconstruction and remains operational in nonequilibrium steady states. Physically, it reveals dependent and independent third-order transitions as fluctuation reorganizations around low-order transitions, namely disordered-side precursors and ordered-side restructuring. Benchmarks on Onsager's two-dimensional Ising solution, finite size Potts models, and a driven nonreciprocal Ising model show that the framework is theoretically grounded and broadly applicable.