  The property that power means are monotonically increasing functions of their
order is shown to be the basis of the second laws not only for processes
involving heat conduction but also for processes involving deformations. In an
$L$-potentail equilibration the final state will be one of maximum entropy,
while in an entropy equilibrium the final state will be one of minimum $L$. A
metric space is connected with the power means, and the distance between means
of different order is related to the Carnot efficiency. In the ideal classical
gas limit, the average change in the entropy is shown to be proportional to the
difference between the Shannon and R\'enyi entropies for nonextensive systems
that are multifractal in nature. The $L$-potential, like the internal energy,
is a Schur convex function of the empirical temperature, which satisfies
Jensen's inequality, and serves as a measure of the tendency to uniformity in
processes involving pure thermal conduction.
