Generalized Algebra Grounded on Nonadditive Entropies

Abstract

The class of $N$-body complex systems with total number of microscopic states given by $W(N)\sim \hat{I}{½}^{N^3}(\hat{I}½>1,{\hat{I}}^3>0)$ can be thermostatistically handled with the nonadditive entropic functional $S_{\hat{I}}´(\{p_i\})=k{\sum }_{i=1}^W{p}_i(\ln \frac{1}{p_i}{)}^{\hat{I}}´(\hat{I}´>0,S_1=S_{BG})$, $S_{BG}=k{\sum }_{i=1}^W{p}_i\ln \frac{1}{p_i}$ being the Boltzmann-Gibbs functional. Indeed, $S_{\hat{I}´=1/{\hat{I}}^3}(\{1/W(N)\})=k[\ln W(N){]}^{{\frac{1}{\hat{I}}}^3}\propto N$, as mandated by thermodynamics. Another wide class is that with $W(N)\sim N^{\ddot{I}}(\ddot{I}>0)$ and a generalized statistical mechanics grounded on the nonadditive entropic functional $S_q(\{p_i\})=k{\sum }_{i=1}^W{p}_i{\ln }_q\frac{1}{p_i}(q\in \mathbb{R},S_1=S_{BG})$, with ${\ln }_{q}z=\frac{z^{1-q}-1}{1-q}(z\ge 0,q\in \mathbb{R},{\ln }_{1}z=\ln z)$, satisfactorily handles such systems with $q=1-1/\ddot{I}$. Furthermore, for this class, the size of the corresponding admissible phase space is characterized by ${\ln }_q(x{\otimes }_{q}y)={\ln }_{q}x+{\ln }_{q}y,x,y\ge 1,q\le 1$, and the $q$-product $x{\otimes }_{q}y=[x^{1-q}+y^{1-q}-1{]}_{+}^{\frac{1}{1-q}}(x{\otimes }_{1}y=xy)$ also leads to the definition of a $q$-algebra. The entropic functional $S_{q,\hat{I}´}(\{p_i\})=k{\sum }_{i=1}^W{p}_i({\ln }_q\frac{1}{p_i}{)}^{\hat{I}}´(q\in \mathbb{R},\hat{I}´>0)$ unifies both cases above: $S_{q,1}=S_q$, $S_{1,\hat{I}´}=S_{\hat{I}}´$ and $S_{1,1}=S_{BG}$. In this paper, we generalize the $q$-algebra associated with $S_q$ to a new one associated with $S_{q,\hat{I}´}$, namely the $(q,\hat{I}´)$-algebra.