Thermodynamics of an ideal generalized gas: II. Means of order alpha

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. This generalizes earlier work involving only pure heat conduction and underlines the incomparability of the internal energy and adiabatic potentials when expressed as powers of the adiabatic variable. In an L-potential equilibration, the final state will be one of maximum entropy, whereas in an entropy equilibration, the final state will be one of minimum L. Unlike classical equilibrium thermodynamic phase space, which lacks an intrinsic metric structure insofar as distances and other geometrical concepts do not have an intrinsic thermodynamic significance in such spaces, a metric space can be constructed for the power means: 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 Rényi 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.