Sensitivity of the population of states to the value of q and legitimate range of q in Tsallis statistics

In the framework of the Tsallis statistical mechanics, we study the change of the population of states when the parameter q is varied, for some model systems; the results show that the difference between predictions of the Boltzmann-Gibbs and Tsallis statistics can be much smaller than the precision of any existing experiment. Also, the relation between privilege of rare and frequent events and the value of q is restudied. It is shown that positive q privilege frequent and negative q privilege rare events. Finally, the convergence criteria of the partition function of some simple model systems, in the framework of Tsallis statistical mechanics, is studied. Based on this study, we conjecture that q = 1, in the thermodynamic limit.