A microcanonical approach to criticality in the mean-field $\phi^4$ model: evidence of intrinsic microcanonical structure before the thermodynamic limit

TL;DR

This paper presents a microcanonical approach to understanding criticality in the mean-field $4$ model, revealing intrinsic finite-size structures that predict thermodynamic critical points.

Contribution

It introduces microcanonical inflection-point analysis (MIPA) as a finite-size critical marker and demonstrates its effectiveness on the mean-field $4$ model.

Findings

MIPA accurately reconstructs critical behavior from finite-size data.

The microcanonical inflection points converge to the thermodynamic critical point.

Finite-size critical structures are intrinsic and measurable, not just finite-size effects.

Abstract

Collective critical behavior is often identified with thermodynamic nonanalyticities and divergences emerging only in the infinite-size limit. Here we adopt a complementary viewpoint: criticality is a structural property due to the rearrangement of the interactions among system's constituents that already exists at finite size and becomes singular only asymptotically. We show that the microcanonical entropy derivatives provide a natural finite-$N$ arena where such structure is encoded in intrinsic extremal/inflection morphologies, and that microcanonical inflection-point analysis (MIPA) turns these morphologies into a unique finite-size critical marker and a well-defined critical trajectory. Using the mean-field $\phi^4$ model as a stringent benchmark, we reconstruct $\beta_N(\varepsilon)$ and $\gamma_N(\varepsilon)$ from microcanonical simulations, validate them against analytic results, and demonstrate that the MIPA trajectory converges to the exact thermodynamic critical point while simultaneously organizing the approach of other observables to their asymptotic behavior. Our results elevate finite-size criticality from a rounded remnant of the thermodynamic limit to a measurable and predictive object in its own right, with direct relevance to modern finite-system platforms and numerical studies.