A Rigorous Derivation of the Entropy Bound and the Nature of Entropy   Variation for Non-equilibrium Systems during Cooling
  We use rigorous non-equilibrium thermodynamic arguments to prove (i) the residual entropy of any system is bounded below by the experimentally (calorimetrically) determined absolute temperature entropy, which itself is bounded below by the entropy of the corresponding equilibrium (metastable supercooled liquid) state, and (ii) the instantaneous entropy cannot drop below that of the equilibrium state. The theorems follow from the second law and the existence of internal equilibrium and refer to the thermodynamic entropy. They go beyond the calorimetric observations by Johari and Khouri [J. Chem. Phys. 134, 034515 (2011)] and others by extending them to all non-equilibrium systems regardless of how far they are from their equilibrium states. We also discuss the statistical interpretation of the thermodynamic entropy and show that the conventional Gibbs or Boltzmann interpretation gives the correct thermodynamic entropy even for a single sample regardless of the duration of measurements. 
P.D. Gujrati
cond-mat.stat-mech cond-mat.mtrl-sci physics.class-ph