Non-monotonic von Neumann entropy in open systems. A conjecture.

In open, thermalizing systems, being in a non equilibrium situation, the von Neumann, or information theoretic Shannon entropy (Sinfo= -Tr ho ln ho) is not identifiable with the Clausius thermodynamic entropy S . Unlike the latter, it is capable of exhibiting a temporal maximum before reaching the final Gibbsian state. The necessary and sufficient conditions for this to happen are conjectured, and verified in numerous multi-state instances but not actually proven. They are written in terms of three energies: (1) Ein, the average energy of the states weighted by the initial occupation probabilities; (2) Edem, the mean energy of the states weighted equally, democratically; (3) Etherm, the energy average weighted by the final, fully thermalized, Gibbsian distribution. They read: Ein < Edem < Etherm.