  In a previous work (M. Campisi. Stud. Hist. Phil. M. P. 36 (2005) 275-290) we
have addressed the mechanical foundations of equilibrium thermodynamics on the
basis of the Generalized Helmholtz Theorem. It was found that the volume
entropy provides a good mechanical analogue of thermodynamic entropy because it
satisfies the heat theorem and it is an adiabatic invariant. This property
explains the ``equal'' sign in Clausius principle ($S_f \geq S_i$) in a purely
mechanical way and suggests that the volume entropy might explain the ``larger
than'' sign (i.e. the Law of Entropy Increase) if non adiabatic transformations
were considered. Based on the principles of microscopic (quantum or classical)
mechanics here we prove that, provided the initial equilibrium satisfy the
natural condition of decreasing ordering of probabilities, the expectation
value of the volume entropy cannot decrease for arbitrary transformations
performed by some external sources of work on a insulated system. This can be
regarded as a rigorous quantum mechanical proof of the Second Law. We discuss
how this result relates to the Minimal Work Principle and improves over
previous attempts. The natural evolution of entropy is towards larger values
because the natural state of matter is at positive temperature. Actually the
Law of Entropy Decrease holds in artificially prepared negative temperature
systems.
