Multichannel relaxation in a three-state glass model: AC-calorimetry loss spectra and the hidden fictive temperature of the Kovacs effect

<jats:p>The Kovacs memory effect shows that a single fictive temperature cannot fully characterize an aged glass, yet turning that qualitative statement into a computable, mechanistic account of the missing variable–and connecting it to an independent, frequency-domain observable such as ACcalorimetry loss spectroscopy–has remained an open combination despite closely related recent efforts. We study a minimal three-state master-equation model–the two-saddle-point extension of the two-level glass model introduced by Peyrard and Garden [Phys. Rev. E 102, 052122 (2020)]–driven by the oscillatory temperature protocol of AC calorimetry, and equipped with the exact (Schnakenberg) non-equilibrium entropy-production rate rather than a phenomenological two-body construction or a linear-response approximation. Four results follow. First, the complex heat capacity develops two well-separated loss peaks, one per relaxation channel, where a two-level system gives only one Debye peak; the entropy-production rate does not mirror this doublet, rising monotonically and saturating instead, yet we show the two are exactly related by a Kubo-type identity, ⟨σ⟩(ω) ∝ ωC′′(ω), verified numerically to better than 0.5%–the extra factor of ω, not any independence, is what turns a resonance into a ramp. Second, the Kovacs protocol–a slowly-cooled equilibrium state versus a quenched-and-aged state matched in energy and temperature–shows that the standard, energy-matching fictive temperature reports no difference between the two, while channel-resolved fictive temperatures split immediately and relax back to equality on the time scale of the Kovacs hump itself: a quantitative, mechanistic reading of the hidden variable, in the spirit of–but microscopically derived rather than fit to data, as in–the Kovacs-Aklonis-Hutchinson-Ramos multiparameter description. Third, a conventional cooling ramp shows both channels freezing out as two independent, rate-dependent glass transitions within a single scan, each obeying its own Kissinger-type relation–exactly the double-transition signature reported in a combined modulated-DSC, dielectric, and depolarization-current study of polyethylene naphthalate; a fast modulation superimposed on that ramp, as in modulated DSC, shows a slow channel’s entire equilibrium heat capacity appearing as a “non-reversing” signal whenever the probe frequency suits a faster channel instead–reproducing, from this minimal model, the frequency-dependent reversing heat capacity established experimentally by Boller, Schick, and Wunderlich. Fourth, aging isothermally for a variable duration before reheating produces an enthalpy-recovery overshoot that grows monotonically with aging time, and, right at the instant reheating resumes, a genuinely negative apparent heat capacity–an exact, provable feature of the model, and the same non-equilibrium negativespecific-heat phenomenon reported by Bisquert for a two-level system; we show both effects are mechanistically the same process as the channel-specific fictive temperatures relaxing back toward the aging temperature–a protocol left untouched by the most closely related prior work, which never resumes heating after aging. The model is validated against exact detailed balance, positivity of entropy production, and the analytic two-level Debye susceptibility; the Supplemental Material extends the loss-spectrum mechanism, the Kubo-type identity, and the combinatorics of the hidden fictive temperature to relaxation chains of up to ten states, showing the identity holds unchanged in accuracy at every N tested, and that the same construction produces either N − 1 resolved loss peaks or a single broadened, quasi-continuous band, depending on whether the underlying barriers spread out or stay within a fixed range as N grows.</jats:p>