Generalized Algebra Grounded on Nonadditive Entropies

The class of $N$-body complex systems with total number of microscopic states given by $W(N) \sim \nu^{N^\gamma}\;(\nu >1, \,\gamma > 0)$ can be thermostatistically handled with the nonadditive entropic functional $S_\delta(\{p_{i}\}) = k\sum_{i=1}^W p_i \Bigl(\ln \frac{1}{p_i} \Bigr)^\delta \;(\delta>0,\,S_1=S_{BG})$, $S_{BG}=k\sum_{i=1}^W p_i \ln \frac{1}{p_i}$ being the Boltzmann-Gibbs functional. Indeed, $S_{\delta=1/\gamma}(\{1/W(N)\})=k[\ln W(N)]^{\frac{1}{\gamma}} \propto N$, as mandated by thermodynamics. Another wide class is that with $W(N) \sim N^\rho\;(\rho>0)$ 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 $\ln_q z =\frac{z^{1-q}-1}{1-q}\; (z\geq0,\;q\in\mathbb{R},\;\ln_1 z=\ln z)$, satisfactorily handles such systems with $q=1-1/\rho$. Furthermore, for this class, the size of the corresponding admissible phase space is characterized by $\ln_q (x\otimes_q y) =\ln_q x + \ln_q y,\, x,y\geq1,\,q\leq 1$, 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 $S_{q,\delta}(\{p_{i}\})=k\sum_{i=1}^W p_i \Bigl(\ln_q \frac{1}{p_i} \Bigr)^\delta\;(q\in\mathbb{R},\delta>0)$ unifies both cases above: $S_{q,1}=S_q$, $S_{1,\delta}=S_\delta$ and $S_{1,1}=S_{BG}$. In this paper, we generalize the $q$-algebra associated with $S_{q}$ to a new one associated with $S_{q,\delta}$, namely the $(q,\delta)$-algebra.