Comparison between two kinds of Correlations: Statistic and Microscopic Correlations

In the recent years, the nonextensive Tsallis statistics1 have been attracted more attention indifferent fields because of its ability in prediction of some phenomona which they couldn'tpredict in Boltzman Gibbs statistics. The most important property of this statistics is thenonextensivity in entropy and energy. It has been known that for two independentsubsystems,A and B,with probabilities, PAi, and PBj, respectively, the probabilitiescorresponding to AUB satisfy B AUB ij= P AI PBj. This implies that ??????????????? In view of the probability function defined in Tsallis statistical mechanics [1] which is ??????????????? In fact in spite of independent probabilities, the combined energy of these two systems is notthe sum of their individual energies. There is always a statistic correlation between them dueto the nonextensivity of Tsallis statistics, dynamic correlation. There is also anothercorrelation in statistics which it comes into account via the microscopic interactions. In thisstudy, we will be compared these two different correlations, statistic (dynamic) andmicroscopic (interaction) correlations and the effect of them on the the internal energy will bestudied. To do so we consider a system with N independent and identical harmonicoscillators in Tsallis statistical framework, as an example of a system with statisticcorrelation, in which nonextensivity originates from Tsallis statistic in definition of theentropy, by entropic index, q. A metal nanocluster with many body and long rangemicroscopic interactions have been selected to study the effect of microscopic correlation inthe values of internal energy. The main question is if it is possible to present a microscopiccorrelation via a statistic correlations? Our results show that in some cases it is possible to present the microscopic correlations viadefining an entropic index, q, in terms of statistic correlations. For example in this study wehave shown that it is possible to use the nonextensive vibration partition function of harmonicoscillator in Tsallis statistical mechanics,with q slightly more than unity predict- the behaviorof the metal nanoclusters which it has microscopic nonextensivity.