Validity of the factorization approximation and correlation induced by nonextensivity in $N$-unit independent systems

Abstract

We have discussed the validity of the factorization approximation (FA) and nonextensivity-induced correlation, by using the multivariate $q$-Gaussian probability distribution function (PDF) for $N$-unit independent nonextensive systems. 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 correction term. It is pointed out that the correction term of $\Delta S_q^{(N)}$ is considerable for large $| q-1 |$ and/or large $N$ because the multivariate PDF cannot be expressed by the factorized form which is assumed in the FA. This implies that the pseudoadditivity of the Tsallis entropy, which is obtained with PDFs in the FA, does not hold although it is commonly postulated in the literatures. 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 $i \neq j$, where $\delta x_i=x_i -< x_i >_q$ and $<\cdot >_q$ stands for $q$-average over the escort PDF. It has been shown that $C_1$ expresses the intrinsic correlation and that $C_m$ with $m \geq 2$ signifies correlation induced by nonextensivity whose physical origin is elucidated within the superstatistics. PDFs calculated for the classical ideal gas and harmonic oscillator are compared with the $q$-Gaussian PDF. A discussion on the $q$-product PDF is presented also.