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arXiv:2605.14198

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

Loris Di Cairano, Roberto Franzosi

Published May 13, 2026• 0 citations
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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 $ϕ^4$ model as a stringent benchmark, we reconstruct $β_N(\varepsilon)$ and $γ_N(\varepsilon)$ 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.

DOI
10.48550/arxiv.2605.14198

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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-NNN 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 ϕ4ϕ^4ϕ4 model as a stringent benchmark, we reconstruct βN(ε)β_N(\varepsilon)βN​(ε) and γN(ε)γ_N(\varepsilon)γN​(ε) 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.
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 NNN -- we systematically identify these structures in the energy profiles of entropy derivatives that encode the transition. An inflection point in the inverse temperature βN(ε)=∂εSNβ_N(ε)=\partial_εS_NβN​(ε)=∂ε​SN​ and a pronounced peak in its derivative γN(ε)=∂ε2SNγ_N(ε)=\partial^2_εS_NγN​(ε)=∂ε2​SN​ define a well-controlled pseudocritical trajectory whose controlled sharpening and drift culminate in the macroscopic cusp at the critical energy εcε_cεc​ in the thermodynamic limit. This establishes an intrinsic, order-parameter-free notion of criticality that precedes its singular asymptotic representation.
The basic quantity for the description of the statistical properties of physical systems is the density of states or equivalently the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are for example related to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions already in finite systems. In this paper we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities which announce a continuous phase transition in the infinite system are characterised by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analyzing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
We develop a scaling theory for the finite-size critical behavior of the microcanonical entropy (density of states) of a system with a critically divergent heat capacity. The link between the microcanonical entropy and the canonical energy distribution is exploited to establish the former, and corroborate its predicted scaling form, in the case of the 3d Ising universality class. We show that the scaling behavior emerges clearly when one accounts for the effects of the negative background constant contribution to the canonical critical specific heat. We show that this same constant plays a significant role in determining the observed differences between the canonical and microcanonical specific heats of systems of finite size, in the critical region.
Traditional definition of phase transition involves an infinitely large system in thermodynamic limit. Finite systems such as biological proteins exhibit cooperative behavior similar to phase transitions. We employ recently discovered analysis of inflection points of microcanonical entropy to estimate the transition temperature of the phase transition in q state Potts model on a finite two dimensional square lattice for q=3 (second order) and q=8 (first order). The difference of energy density of states (DOS) Δ ln g(E) = ln g(E+ ΔE) −ln g(E) exhibits a point of inflexion at a value corresponding to inverse transition temperature. This feature is common to systems exhibiting both first as well as second order transitions. While the difference of DOS registers a monotonic variation around the point of inflexion for systems exhibiting second order transition, it has an S-shape with a minimum and maximum around the point of inflexion for the case of first order transition.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.