Parallel Maxwellian relations and their correlations in nonextensive thermodynamics

We presented a prescript in previous works to solve the issue of the definition of temperature in thermodynamics on the basis of the dual interpretation of temperature concept. A nonextensive thermodynamic formalism consisting of two sets of parallel Legendre transformation structures, namely, physical and Lagrange sets, were proposed by providing further dual interpretations of thermodynamic quantities. In this study, we reconstruct Maxwellian and other thermodynamic relations on the basis of these two parallel formalisms. The correlations between the thermodynamic relations are determined, where an equivalent rule is suggested, by recoursing to the link of the Tsallis factor. This rule shows that the Tsallis factor is invariable with a constant volume or entropy in partial derivative calculations. These correlations could be easily tested in nonextensive statistics, whose applications into adiabatic expansions of nonextensive gas produce interesting results. Lagrange internal energy and generalized heat capacity are negative in some small nonextensive systems due to the enhanced nonextensive effect. The negative definite of generalized heat capacity would not lead to instability, as physical heat capacity is the quantity used to govern the thermodynamic process of nonextensive systems.