Mechanical Proof of the Second Law of Thermodynamics Based on Volume Entropy

Campisi, Michele (2007) Mechanical Proof of the Second Law of Thermodynamics Based on Volume Entropy.

Abstract

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.