Nonextensive Thermodynamics

Abstract

Carnot [1824] fully appreciated that

when the volume of a gas changes at constant temperature the amount of heat absorbed or released by a gas will follow an arithmetical progression when increases or decreases in volume follow a geometrical progression.

Replacing the word “heat” by “entropy” results in the extensivity of the latter. Although this applies to the common case of an ideal gas, we are not bound to it, and in this chapter we consider more generalized mean relations. This has the effect of destroying extensivity of the thermodynamic potentials, except in some special cases, and in so doing, we will obtain nonextensive ones.

Specifically, we supplement the early approach to thermodynamics discussed in Chap. 2 with more general considerations involving general means of temperature and volume, other than arithmetic and geometric mean values of classical thermodynamics (Lavenda 2005). These are the most common since they lead to logarithmic forms for the fundamental relation of entropy in which the entropy is a function of all the extensive independent variables. The Grüneisen equation of state (2.56), which makes the elimination of the chemical potential between the energy and total number of particles superfluous, will identify the free energies as the potentials from which all others can be derived (Einbinder 1948).