Criticality Beyond Nonanalyticity: Intrinsic Microcanonical Signatures of Phase Transitions

Abstract

Phase transitions are conventionally defined by nonanalyticities of thermodynamic potentials in the thermodynamic limit. In this Letter, we show that the singularity is not the definition of criticality but its asymptotic outcome: criticality is already written in the microcanonical entropy derivatives at any finite size as intrinsic morphological structures -- inflection points and extrema. The singularity is then the endpoint of a sharpening process that evolves with increasing system size. Combining microcanonical inflection-point analysis (MIPA) with the Berlin-Kac spherical model -- for which the microcanonical density of states is known in closed form at every finite $N$ -- we systematically identify these structures in the energy profiles of entropy derivatives that encode the transition. An inflection point in the inverse temperature $β_N(ε)=\partial_εS_N$ and a pronounced peak in its derivative $γ_N(ε)=\partial^2_εS_N$ define a well-controlled pseudocritical trajectory whose controlled sharpening and drift culminate in the macroscopic cusp at the critical energy $ε_c$ in the thermodynamic limit. This establishes an intrinsic, order-parameter-free notion of criticality that precedes its singular asymptotic representation.

Figures (3)

• Figure 1: Finite-size first- and second-order derivatives of entropy in the mean field Berlin-Kac model. Microcanonical inverse temperature $\beta_N(\varepsilon)=\partial_\varepsilon s_N(\varepsilon)$ (top) and its derivative $\gamma_N(\varepsilon)=\partial_\varepsilon\beta_N(\varepsilon)=\partial_\varepsilon^2 s_N(\varepsilon)$ (bottom), shown for increasing system sizes $N$ (colored curves) together with the thermodynamic-limit prediction $N\to\infty$ (black curve). (a.1) Family $\{\beta_N(\varepsilon)\}_N$ converging to the infinite-size curve, which exhibits a cusp at the critical energy $\varepsilon_c=J/2=1/2$ (inset highlights the sharpening near $\varepsilon_c$). (a.2) A representative finite size (here $N=32000$) revealing an intrinsic inflection point in $\beta_N$ at $\varepsilon_\star$ (vertical guide). (b.1) Corresponding family $\{\gamma_N(\varepsilon)\}_N$ showing the characteristic finite-$N$ "Z-like" structure in the critical region. (b.2) Zoom on the negative peak of $\gamma_N$; its location $\varepsilon_\star(N)$ (colored disks) drifts toward $\varepsilon_c$ while the peak sharpens with $N$, demonstrating that the macroscopic cusp/discontinuity emerges as the singular endpoint of a fully trackable finite-$N$ microcanonical structure.
• Figure 2: Finite-size drift of the microcanonical precursor toward the critical energy.(a) Pseudo-critical energy $\varepsilon_\star(N)$ defined as the location of the negative peak of $\gamma_N(\varepsilon)=\partial_\varepsilon^2 s_N(\varepsilon)$, plotted versus $N^{-1/2}$. The fit $\varepsilon_\star(N)=a+b\,N^{-1/2}+c\,N^{-1}$ yields $a\simeq0.4973$, consistent with $\varepsilon_\star(N)\to \varepsilon_c=1/2$ as $N\to\infty$. (b) Peak height $M(N)=\gamma_N(\varepsilon_\star(N))$ approaching the thermodynamic-limit value $M(\infty)=-1$.
• Figure 3: Validation of the MIPA from numerical simulations. Numerical $\gamma_N(\varepsilon)=\partial_\varepsilon^2 s_N(\varepsilon)$ from microcanonical molecular dynamics compared with exact analytical curves. (a) Simulated $\gamma_N(\varepsilon)$ for $N=16\,000$ (crosses), $N=64\,000$ (squares), and $N=128\,000$ (circles), showing the max--min structure around $\varepsilon_c$ (red dashed line). (b) Detail of the negative peak region; the rounded extremum is the MIPA second-order marker. (c) Overlay with the thermodynamic-limit curve (black), showing systematic sharpening toward the nonanalytic profile. (d) Geometric quantification: the secant connecting local max and min near $\varepsilon_c$ steepens with $N$ (increasing angle $\varphi$), measuring convergence to the singularity.