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We use rigorous non-equilibrium thermodynamic arguments to prove (i) theresidual entropy of any system is bounded below by the experimentally(calorimetrically) determined absolute temperature entropy, which itself isbounded below by the entropy of the corresponding equilibrium (metastablesupercooled liquid) state, and (ii) the instantaneous entropy cannot drop belowthat of the equilibrium state. The theorems follow from the second law and theexistence of internal equilibrium and refer to the thermodynamic entropy. Theygo beyond the calorimetric observations by Johari and Khouri [J. Chem. Phys.134, 034515 (2011)] and others by extending them to all non-equilibrium systemsregardless of how far they are from their equilibrium states. We also discussthe statistical interpretation of the thermodynamic entropy and show that theconventional Gibbs or Boltzmann interpretation gives the correct thermodynamicentropy even for a single sample regardless of the duration of measurements.

A Rigorous Derivation of the Entropy Bound and the Nature of Entropy Variation for Non-equilibrium Systems during Cooling