Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.21003

Criticality Beyond Nonanalyticity: Intrinsic Microcanonical Signatures of Phase Transitions

Loris Di Cairano

Published Feb 24, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2602.21003

Related papers

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

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.

arXiv:2605.14198•May 13, 2026•0 citations

Microcanonical analysis of small systems

Michel Pleimling, H. Behringer

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.

Phase Transitions•Sep 1, 2005•18 citations

Microcanonical analysis of small systems

Michel Pleimling, Behringer, Hans

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.

arXiv:cond-mat/0504018•Apr 1, 2005•1 citation

Phase Transitions in the Two-Dimensional Ising Model from the Microcanonical Perspective

Kedkanok Sitarachu, Michael Bachmann

Abstract The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.

arXiv:2005.02240•Journal of Physics Conference Series•Feb 1, 2020•10 citations

Inflection points of microcanonical entropy: Monte Carlo simulation of q state Potts model on a finite square lattice

E. Praveen, S. V. M. Satyanarayana

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.

AIP conference proceedings•Jan 1, 2014•1 citation

Phase transitions in finite systems = topological peculiarities of the microcanonical entropy surface

D. H. E. Gross, Evgeny V. Votyakov

It is discussed how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be defined and classified for finite systems from the topology of the energy surface $e^{S(E,N)}$ of the mechanical N-body phase space or more precisely of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ without taking the thermodynamic limit. The first calculation of the entire entropy surface $S(E,N)$ for a q=3-states Potts lattice gas on a 50*50 square lattice is shown. There are two lines, where $S(E,N)$ has a maximum curvature $\sim 0$. One is the border between the regions in \{$E,N$\} with $D(E,N)>0$ and with $D(E,N)<0$, the other line is critical starting as a valley in $D(E,N)$ running from the continuous transition in the ordinary q=3-Potts model, converting at $P_m$ into a flat ridge/plateau (maximum) deep inside the convex intruder of $S(E,N)$ which characterizes the first order liquid-gas transition. The multi-critical point $P_m$ is their crossing.

arXiv:cond-mat/9904073•Apr 6, 1999•3 citations
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.
A microcanonical approach to criticality in the mean-field ϕ4ϕ^4ϕ4 model: evidence of intrinsic microcanonical structure before the thermodynamic limit
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.
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.
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.
Abstract The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.
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.
It is discussed how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be defined and classified for finite systems from the topology of the energy surface eS(E,N)e^{S(E,N)}eS(E,N) of the mechanical N-body phase space or more precisely of the curvature determinant D(E,N)=∂2S/∂E2∗∂2S/∂N2−(∂2S/∂E∂N)2D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2D(E,N)=∂2S/∂E2∗∂2S/∂N2−(∂2S/∂E∂N)2 without taking the thermodynamic limit. The first calculation of the entire entropy surface S(E,N)S(E,N)S(E,N) for a q=3-states Potts lattice gas on a 50*50 square lattice is shown. There are two lines, where S(E,N)S(E,N)S(E,N) has a maximum curvature ∼0\sim 0∼0. One is the border between the regions in \{E,NE,NE,N\} with D(E,N)>0D(E,N)>0D(E,N)>0 and with D(E,N)<0D(E,N)<0D(E,N)<0, the other line is critical starting as a valley in D(E,N)D(E,N)D(E,N) running from the continuous transition in the ordinary q=3-Potts model, converting at PmP_mPm​ into a flat ridge/plateau (maximum) deep inside the convex intruder of S(E,N)S(E,N)S(E,N) which characterizes the first order liquid-gas transition. The multi-critical point PmP_mPm​ is their crossing.