Statistical mechanical proof of the second law of thermodynamics based on volume entropy

In a previous work [Campisi, M. (2005). On the mechanical foundations of thermodynamics: The generalized Helmholtz theorem. Studies in History and Philosophy of Modern Physics, 36  , 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 (Sf⩾Si)(Sf⩾Si) 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 are considered. Based on the principles of microscopic (quantum or classical) mechanics we prove here that, provided the initial equilibrium satisfies the natural condition of decreasing ordering of probabilities, the expectation value of the volume entropy cannot decrease for arbitrary transformations performed by some external source of work on an 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 how it improves on 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.