Nonextensive thermodynamic relations based on the assumption of temperature duality

The nonextensive thermodynamic relations are expressed under the assumption of temperature duality, endowing the "physical temperature" and the "Lagrange temperature" in different physical sense. Based on this assumption, two sets of parallel Legendre transform structures are given. One is called "physical" set, and the other called "Lagrange" set. In these two formalisms, the thermodynamic quantities and the thermodynamic relations are both liked through the Tsallis factor. Application of the two sets of the thermodynamic relations to the self-gravitating system shows that the heat capacity defined in the classical thermodynamics has no relevance to the stability of the system. Instead, the new defined heat capacity can determine the stability of the system.