# Validity of the factorization approximation and correlation induced by nonextensivity in $N$-unit independent systems - Condensed Matter > Statistical Mechanics

Validity of the factorization approximation and correlation induced by nonextensivity in $N$-unit independent systems - Condensed Matter > Statistical Mechanics - Download this document for free, or read online. Document in PDF available to download.

Abstract: We have discussed the validity of the factorization approximation FA andnonextensivity-induced correlation, by using the multivariate $q$-Gaussianprobability distribution function PDF for $N$-unit independent nonextensivesystems. The Tsallis entropy is shown to be expressed by $S q^{N} =S {q,FA}^{N}+ \Delta S q^{N}$ where $q$ denotes the entropic index,$S {q,FA}^{N}$ a contribution in the FA, and $\Delta S q^{N}$ a correctionterm. It is pointed out that the correction term of $\Delta S q^{N}$ isconsiderable for large $| q-1 |$ and-or large $N$ because the multivariate PDFcannot be expressed by the factorized form which is assumed in the FA. Thisimplies that the pseudoadditivity of the Tsallis entropy, which is obtainedwith PDFs in the FA, does not hold although it is commonly postulated in theliteratures. We have calculated correlations defined by $C m= < \delta x i\:\delta x j^m > q -< \delta x i^m > q\: < \delta x j^m > q$ for $i eqj$, where $\delta x i=x i -< x i > q$ and $<\cdot > q$ stands for $q$-averageover the escort PDF. It has been shown that $C 1$ expresses the intrinsiccorrelation and that $C m$ with $m \geq 2$ signifies correlation induced bynonextensivity whose physical origin is elucidated within the superstatistics.PDFs calculated for the classical ideal gas and harmonic oscillator arecompared with the $q$-Gaussian PDF. A discussion on the $q$-product PDF ispresented also.

Author: ** Hideo Hasegawa Tokyo Gakugei Univ.**

Source: https://arxiv.org/