Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2004-11-03
Physics
Condensed Matter
Statistical Mechanics
10 pages including 7 figures. Invited paper to appear in a special issue of the International Journal of Bifurcation and Chaos
Scientific paper
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), {\it nonextensive} statistical mechanics has been proposed based on the entropy $S_{q}\equiv k (1-\sum_{i=1}^Wp_i^{q})/(q-1)$, with $S_1=-k\Sigma_{i=1}^{W} p_i \ln p_i$ (Boltzmann-Gibbs entropy). Special collective correlations can be mathematically constructed such that the strictly {\it additive} entropy is now $S_q$ for an adequate value of $q \ne 1$, whereas Boltzmann-Gibbs entropy is {\it 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 $S_q (q\ne 1)$ is similarly replaced by asymptotic additivity (i.e., extensivity). All probabilistically well defined systems whose adequate entropy is $S_{1}$ are called {\it extensive} (or {\it normal}). They correspond to a number $W^{\it eff}$ of {\it effectively} occupied states which grows {\it exponentially} with the number $N$ of elements (or subsystems). Those whose adequate entropy is $S_q (q \ne 1)$ are currently called {\it nonextensive} (or {\it anomalous}). They correspond to $W^{\it eff}$ growing like a {\it power} of $N$. To illustrate this scenario, recently addressed, we provide in this paper details about systems composed by $N=2,3$ two-state subsystems.
Sato Yuzuru
Tsallis Constantino
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