Canonical Criterion for Third-Order Transitions

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.

Figures (4)

• Figure 1: Roadmap for the energy-cumulant diagnostic and its relation to MIPA. (a) MIPA classifies higher-order transitions via derivatives of $S(E)=k_B\ln g(E)$, where $g(E)$ is the DOS, identifying the dependent transition as a precursor and the independent one as an ordered-side reorganization. (b) $\Xi(T)$ establishes $N^2\Xi(T)\to s^{(3)}(e^*)$ in the single-saddle regime, linking signed extrema to fluctuations and providing a canonical-theory basis for third-order transitions. (c) Unlike MIPA, $\Xi(T)$ applies to both canonical and microcanonical ensembles. Benchmarks: Ising model, Potts models and nonreciprocal Ising model.
• Figure 2: Canonical third-order transitions and geometric corroboration in the 2D Ising model. (a) Thermodynamic-limit $\Xi(T)$ from Onsager's exact free energy. The divergence at $T_c=2.269$ marks the critical transition; the positive minimum at $T_{\rm ind}=2.229$ (independent third-order, ordered-side reorganization) and negative maximum at $T_{\rm dep}=2.567$ (dependent third-order, disordered-side precursor) are exact and fully resolved. (b,c) Complementary geometry-sensitive observables further corroborate panel (a). (b) Isolated-spin number $n_1$ shows a clear feature near $T_{\rm ind}$, consistent with the independent transition. (c) Cluster-averaged area change rate $\mathrm{d}A/\mathrm{d}T$ shows fastest variation near $T_{\rm dep}$, consistent with the dependent transition. Vertical lines mark $T_{\rm ind}$ (green) and $T_{\rm dep}$ (orange); dark lines indicate exact thermodynamic-limit values, lighter lines finite size estimates.
• Figure 3: Finite size robustness of the canonical diagnostic $\Xi(T)$ in the square-lattice $q=8$ Potts model. From reweighting of a replica Wang--Landau DOS estimate, $\Xi(T)$ retains robust signed extrema around the first-order transition despite finite size and "phase" coexistence. The negative peak at $T_{\mathrm{dep}}(L)$ on the high-$T$ side identifies a precursor-like dependent third-order transition, whereas the positive dip at $T_{\mathrm{ind}}(L)$ on the low-$T$ side identifies an independent ordered-side reorganization. Vertical dashed lines mark the $L=56$ values listed in the figure. Solid lines indicate the exact transition temperature $T_{\mathrm{c}}/J = 1/\ln(1+\sqrt{q})$.
• Figure 4: Two-dimensional driven nonreciprocal Ising model at fixed $K_m=0.3$. $\Xi_R(J)$ is extracted from nonequilibrium steady-state time series of the synchronization observable $R(t)$ after discarding initial transients and performing block averaging. For $L=10,15,20,30$, $\Xi_R(J)$ develops a reproducible signed extremum and changes sign within a narrow $J$ window associated with the finite size onset of apparent synchronization and oscillatory transients. With increasing $L$, the feature gradually shifts and weakens, consistent with a crossover in two dimensions rather than a thermodynamic phase transition. This shows that the cumulant-ratio diagnostic remains fully operational without microcanonical input.