The equivalence of isothermal and non-isothermal power law distributions with temperature duality

The concept of temperature duality states that the physical temperature and Lagrange temperature both have physical sense in the nonextensive system. By use of this concept, the isothermal power law distribution and the non-isothermal power law distribution are equivalent to each other when the detailed balance is satisfied. Also, the polytropic equation in stellar system and self-gravitating gaseous system can be deduced from both of these two distributions. This indicates that the polytropic system exhibits some 'equilibrium' configuration which, in the stellar system, is probably the result of so called 'violent relaxation'.