The eigen microstate entropy SEM=−∑IpIln⁡pIS_{\mathrm{EM}}=-\sum_I p_I\ln p_I, with weights pI=σI2p_I=\sigma_I^2 obtained from the singular value decomposition of an ensemble matrix built from time-series data and normalized so that ∑IpI=1\sum_I p_I=1, was recently introduced as a general spectral diagnostic of phase transitions and shown to act as a precursor signal for liquid-liquid phase separation in living cells and for El Niño events. Its authors propose that the divergence of relaxation times near a critical point (critical slowing down) should manifest as a decreasing SEMS_{\mathrm{EM}} as the eigen-microstate weights concentrate, and they explicitly name glassy dynamics as an application that remains to be pursued: no study has yet applied SEMS_{\mathrm{EM}} to glass-forming liquids, to the detection of the glass transition from time-series data, or to relating SEMS_{\mathrm{EM}} to conventional measures such as the structural relaxation time or the configurational entropy. The problem is to carry out this program: determine whether, and in what sense, SEMS_{\mathrm{EM}} identifies and characterizes dynamical arrest in glass formers across molecular and colloidal glasses.

Background

When a liquid is supercooled below its melting point without crystallizing, its structural relaxation time τα\tau_\alpha (the time for density correlations to decay) grows by many orders of magnitude until, near the laboratory glass temperature TgT_g (where τα\tau_\alpha exceeds the observation time and the liquid falls out of equilibrium), the system appears frozen: this loss of mobility is called dynamical arrest, and the empirical Vogel–Fulcher law suggests τα\tau_\alpha diverges at a finite temperature, though whether the glass transition is a genuine thermodynamic singularity or a purely kinetic phenomenon is itself unresolved. Formally analogous behavior occurs near equilibrium critical points, where the divergence of relaxation times is called critical slowing down (CSD). The configurational entropy scs_c, the entropy associated with the number of distinct amorphous packings, is the central thermodynamic quantity in the Adam–Gibbs and random-first-order-transition theories, which link the growth of τα\tau_\alpha to the shrinking number of accessible amorphous states; Berthier, Ozawa, and Scalliet (2019) review how scs_c is defined and measured in glass-forming liquids. The eigen microstate theory (EMT), developed by Hu, Liu, and collaborators since 2019, provides a data-driven spectral description of complex systems: from MM snapshots of NN observables one builds the ensemble matrix Aiτ=si(τ)/C0A_{i\tau}=s_i(\tau)/\sqrt{C_0} (with C0C_0 the total squared amplitude, so that ∑IσI2=1\sum_I\sigma_I^2=1) and decomposes it by singular value decomposition (SVD), A=∑IσI UI⊗VIA=\sum_I\sigma_I\,U_I\otimes V_I. The orthogonal left singular vectors UIU_I are the eigen microstates, and the squared singular values pI=σI2p_I=\sigma_I^2 form a probability distribution over them; at an equilibrium continuous phase transition one eigen microstate condenses (p1p_1 takes a finite macroscopic value) and plays the role of an order parameter, with renormalization-group and finite-size-scaling analyses (the standard tools of critical phenomena) analogous to those for ordinary phase transitions. The eigen microstate entropy SEM=−∑IpIln⁡pIS_{\mathrm{EM}}=-\sum_I p_I\ln p_I quantifies how many eigen microstates share the statistical weight: it is large when the weight is spread over many independent modes and decreases when the weight concentrates onto few. Liu, Niu, Zhang, Hu, Chen, Zhang, Shi, Li, Tan, Liu, Li, and Chen (2025) introduced SEMS_{\mathrm{EM}} as a metric of complexity, established its scaling behavior at equilibrium critical points (the mean spherical, Ising, and Potts lattice models), and applied it to non-equilibrium data: a significant increase of SEMS_{\mathrm{EM}} precedes biomolecular condensate formation in liquid-liquid phase separation (demixing into droplets) in living cells and appears months before El Niño events. They further state that CSD near a critical point is characterized by a decreasing SEMS_{\mathrm{EM}}, as the system lingers in a smaller subset of states, but this dynamical connection was not demonstrated by direct relaxation-time measurements, and they note that applying the method to CSD in glassy dynamics remains an intriguing direction for future research. Glassy dynamics shares the CSD phenomenology — relaxation times grow dramatically as the system becomes trapped in a subset of configuration-space states — but unlike equilibrium critical points it has no established order parameter or diverging static correlation length, so whether a spectral quantity like SEMS_{\mathrm{EM}} carries a robust, interpretable signature of arrest is genuinely unknown. Subsequent work in 2026 has extended EMT to the Lennard-Jones liquid–vapor critical point (an equilibrium transition), to polymer diffusion in porous media, and to sudden stratospheric warmings (abrupt polar-vortex breakdowns), where a Marchenko–Pastur baseline — the limiting singular-value spectrum of a matrix of uncorrelated noise — was introduced to distinguish genuinely collective modes from noise, but none of it addresses glassy systems.

Problem Statement

Determine whether the eigen microstate entropy can identify and characterize dynamical arrest in glass-forming systems, as proposed by the source paper. Concretely, with one overarching objective — establishing (or ruling out) SEMS_{\mathrm{EM}} as a spectral diagnostic of glassy slowing down — an answer should: (i) compute SEMS_{\mathrm{EM}} from ensemble/time-series data of glass formers, including at least one well-characterized molecular glass-former (from simulation and/or experiment) and one colloidal glass system, and characterize its behavior on cooling toward the glass transition and, if studied, during aging (the slow non-stationary drift of properties below TgT_g after a quench); (ii) establish what, if any, robust signature accompanies the growth of the structural relaxation time τα\tau_\alpha, and clarify at which characteristic temperature scale (TgT_g; the mode-coupling crossover, where mode-coupling theory — a first-principles kinetic theory of slow dynamics — predicts ideal arrest; or the Vogel–Fulcher temperature, where the empirical relaxation-time fit diverges) any change in SEMS_{\mathrm{EM}} occurs; (iii) relate the spectral concentration underlying SEMS_{\mathrm{EM}} to conventional measures, specifically τα(T)\tau_\alpha(T) and the configurational entropy sc(T)s_c(T), addressing whether concentration of eigen-microstate weights reflects the shrinking number of accessible amorphous states posited by configurational-entropy theories of the glass transition; and (iv) demonstrate, with appropriate controls such as random-matrix baselines and surrogate data (randomized replicas that preserve trivial statistics while destroying dynamical structure), that any observed trend is a genuine dynamical-arrest signal rather than a trivial consequence of reduced thermal fluctuations on cooling, and state whether the glass transition can be detected from time-series data via SEMS_{\mathrm{EM}} alone. These components jointly support the single objective, and all four are required; a finding that SEMS_{\mathrm{EM}} shows no robust arrest signal, properly controlled, is as valid an answer as a positive characterization and discharges (ii)–(iv) by reporting the absence of a signature and the corresponding negative conclusions about the characteristic temperature scale, the relations to τα(T)\tau_\alpha(T) and sc(T)s_c(T), and whether the glass transition is detectable from SEMS_{\mathrm{EM}} alone.

References

  1. Teng Liu, Xuezhi Niu, Mingli Zhang, Gaoke Hu, Yuhan Chen, Yongwen Zhang, Rui Shi, Jingyuan Li, Peng Tan, Maoxin Liu, Hui Li, Xiaosong Chen (2025). Phase transition revealed by eigen microstate entropy. arXiv:2512.23086, DOI 10.48550/arXiv.2512.23086. https://arxiv.org/abs/2512.23086
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