Microcanonical entropy inflection points: Key to systematic understanding of transitions in finite systems

We introduce a systematic classification method for the analogs of phase transitions in finite systems. This completely general analysis, which is applicable to any physical system and extends towards the thermodynamic limit, is based on the microcanonical entropy and its energetic derivative, the inverse caloric temperature. Inflection points of this quantity signal cooperative activity and thus serve as distinct indicators of transitions. We demonstrate the power of this method through application to the long-standing problem of liquid-solid transitions in elastic, flexible homopolymers.

💡 Research Summary

The paper introduces a universal, microcanonical‑based framework for identifying and classifying phase‑like transitions in finite systems. Traditional canonical analyses, which average over thermal fluctuations, often obscure the subtle signatures of transitions when system size is limited, making it impossible to assign a unique transition temperature. By contrast, the authors start from the microcanonical entropy S(E)=k_B ln g(E), where g(E) is the density of states, and define the inverse caloric temperature β(E)=dS/dE as the fundamental temperature variable. The curvature of β(E), γ(E)=dβ/dE=d²S/dE², is then examined. An inflection point of β(E) corresponds to a maximum of γ(E). The sign of this maximum determines the order of the transition: γ>0 signals a non‑monotonic “back‑bending” of β(E), indicating coexistence of two phases and a finite latent heat Δq – a first‑order transition. Conversely, γ<0 denotes a monotonic β(E) with no latent heat, i.e., a second‑order transition. The transition temperature follows directly from T_tr=1/β(E_tr), and the energetic width of the back‑bending region provides Δq.

To demonstrate the method, the authors study an elastic, flexible polymer model consisting of Lennard‑Jones non‑bonded interactions and finitely extensible nonlinear elastic (FENE) bonds. Using high‑precision density‑of‑states estimates obtained via multicanonical and Wang‑Landau Monte‑Carlo simulations, they compute S(E), β(E), and γ(E) for a polymer with N=102 monomers. Four structural phases are identified: two solid icosahedral states (Mackay and anti‑Mackay), a globular liquid, and a random coil. The analysis reveals a second‑order transition between the two solid phases (γ<0) and a first‑order liquid‑solid transition (γ>0) characterized by a clear back‑bending of β(E) and a non‑zero latent heat. The transition energies and temperatures are extracted directly from the inflection points.

The authors then extend the study to a series of polymers with chain lengths N ranging from 13 to 309. Transition temperatures T_tr(N) for both liquid‑solid and solid‑solid changes are plotted versus N. “Magic” chain lengths (N=13, 55, 147, 309, …) exhibit especially pronounced icosahedral ordering; at these sizes the solid‑solid and liquid‑solid transition lines converge, merging into a single first‑order transition. For non‑magic lengths, solid‑solid transitions appear as separate second‑order events at lower temperatures, while the liquid‑solid transition remains first‑order. The size dependence of the transition temperatures follows a N^{-1/3} scaling, allowing an extrapolation to the thermodynamic limit where only the liquid‑solid transition survives.

Beyond polymers, the authors argue that the β‑γ inflection‑point scheme is applicable to any finite system where a density of states can be estimated, such as atomic clusters, protein folding landscapes, spin models, and even astrophysical aggregates. Because the method relies solely on the microcanonical entropy, it retains all finite‑size information that canonical averaging discards, enabling precise determination of transition order, transition temperature, and latent heat. Consequently, this approach provides a powerful, model‑independent tool for probing cooperative phenomena in small systems and for bridging the gap between finite‑size behavior and the thermodynamic limit.