From internal state variables to fluctuations in glass-forming materials: The linear dynamic heat capacity,...

Several macroscopic-phenomenological theories exist that account for dissipative effects in the dynamic behavior of glass-forming materials. Thermodynamics with internal state variables, for example, provides a rationale for assessing the multiplicity of relaxation mechanisms observed in the linear dynamic heat capacity, expansivity, and compressibility. However, the convoluted formalism of these theories often obstructs the assessment of microscopic characteristics associated with material behavior that can otherwise be studied within linear response theory. In this work, we address this problem by deriving memory functionals of the entropy and volume from a set of ordinary differential equations posed by the normal-coordinate transform of the internal state variables. A generalized susceptibility matrix that represents the linear dynamic material behavior in the frequency domain is the result. In agreement with microscopic reversibility, the generalized susceptibility matrix and its time-domain transform are symmetric, thus substantiating the hypothesis of equal distribution of relaxation times in pressure-jump entropy relaxation and temperature-jump volume relaxation. The Prigogine-Defay ratio in terms of equilibrium fluctuations is obtained from the imaginary part of the generalized susceptibility matrix via the fluctuation-dissipation theorem. By combining fundamentals from classical irreversible thermodynamics and rational thermodynamics to derive the memory functionals, this work contributes to a theoretical framework for assessing macroscopic-phenomenological and microscopic characteristics of glass-forming materials associated with the linear dynamic heat capacity, expansivity, and compressibility. In addition, a compilation of the generalized susceptibility and its time-domain transform, both evaluated for the relaxation-time distributions that lead to the Debye, Kohlrausch, Cole-Cole, Davidson-Cole, and Havriliak-Negami relaxation functions, is provided.