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 $\beta_N(\epsilon)=\partial_\epsilon S_N$ and a pronounced peak in its derivative $\gamma_N(\epsilon)=\partial^2_\epsilon S_N$ define a well-controlled pseudocritical trajectory whose controlled sharpening and drift culminate in the macroscopic cusp at the critical energy $\epsilon_c$ in the thermodynamic limit. This establishes an intrinsic, order-parameter-free notion of criticality that precedes its singular asymptotic representation.