ON THE EXTENSIVITY OF THE ENTROPY SqFOR N ≤ 3 SPECIALLY CORRELATED BINARY SUBSYSTEMS

Abstract

Many natural and artificial systems whose range of interaction is long enough are known to exhibit (quasi)stationary states that defy the standard, Boltzmann–Gibbs statistical mechanical prescriptions. For handling such anomalous systems (or at least some classes of them), nonextensive statistical mechanics has been proposed based on the entropy [Formula: see text], with [Formula: see text] (Boltzmann–Gibbs entropy). Special collective correlations can be mathematically constructed such that the strictly additive entropy is now Sqfor an adequate value of q ≠ 1, whereas Boltzmann–Gibbs entropy is nonadditive. Since important classes of systems exist for which the strict additivity of Boltzmann–Gibbs entropy is replaced by asymptotic additivity (i.e. extensivity), a variety of classes are expected to exist for which the strict additivity of Sq(q ≠ 1) is similarly replaced by asymptotic additivity (i.e. extensivity). All probabilistically well defined systems whose adequate entropy is S1are called extensive (or normal). They correspond to a number Weffof effectively occupied states which grows exponentially with the number N of elements (or subsystems). Those whose adequate entropy is Sq(q ≠ 1) are currently called nonextensive (or anomalous). They correspond to Weffgrowing like a power of N. To illustrate this scenario, recently addressed [Tsallis, 2004] we provide in this paper details about systems composed by N = 2, 3 two-state subsystems.