Nonextensive

Nonextensive refers to a range of concepts in statistical mechanics and thermodynamics that describe systems where standard Boltzmann-Gibbs (extensive) statistics do not apply. The central idea is that entropy and related thermodynamic quantities may be nonadditive when subsystems interact strongly, exhibit long-range correlations, or have complex, fractal-like phase-space structures.

A common formalism is based on Tsallis entropy, S_q = k (1 - sum_i p_i^q)/(q - 1), where q is a real parameter that quantifies departure from extensivity. When q = 1, Tsallis entropy reduces to the Boltzmann-Gibbs entropy and conventional, extensive thermodynamics is recovered. For independent subsystems, the entropy is nonadditive, following a generalized composition rule S_q(A+B) = S_q(A) + S_q(B) + (1 - q) S_q(A) S_q(B) (conventions vary). This nonadditivity is a hallmark of nonextensive systems.

Nonextensive statistics provides a framework for systems with long-range interactions (such as gravity or unscreened Coulomb forces), persistent memory, multifractal structures, or nonergodic dynamics. It often yields generalized distributions, notably q-exponential and q-Gaussian forms, which can describe power-law tails and anomalous scaling observed in empirical data.

Applications span fields including astrophysics, plasma physics, turbulence, econophysics, and complex networks. The approach remains one of several attempts to model nonadditive phenomena, and its interpretation and scope continue to be active topics of research and discussion.