Generalized Algebra Grounded on Nonadditive Entropies

Abstract

The class of $N$-body complex systems with total number of microscopic states given by $W(N)\sim {\nu }^{N^{\gamma }}(\nu >1,\gamma >0)$ can be thermostatistically handled with the nonadditive entropic functional $S_{\delta }(\{p_i\})=k{\sum }_{i=1}^W{p}_i(\ln \frac{1}{p_i}{)}^{\delta }(\delta >0,S_1=S_{BG})$, $S_{BG}=k{\sum }_{i=1}^W{p}_i\ln \frac{1}{p_i}$ 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 ${\ln }_{q}z=\frac{z^{1-q}-1}{1-q}(z\ge 0,q\in \mathbb{R},{\ln }_{1}z=\ln z)$, satisfactorily handles such systems with $q=1-1/\rho$. Furthermore, for this class, the size of the corresponding admissible phase space is characterized by ${\ln }_q(x{\otimes }_{q}y)={\ln }_{q}x+{\ln }_{q}y,x,y\ge 1,q\le 1$, 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 $S_{q,\delta }(\{p_i\})=k{\sum }_{i=1}^W{p}_i({\ln }_q\frac{1}{p_i}{)}^{\delta }(q\in \mathbb{R},\delta >0)$ 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.

References in corpus (32)

• Black hole thermodynamical entropy
• Option Pricing Formulas based on a non-Gaussian Stock Price Model
• Generalized algebra within a nonextensive statistics
• Thermostatistics of overdamped motion of interacting particles
• Tsallis Fits to pT Spectra and Multiple Hard Scattering in pp Collisions at the LHC
• From QCD-based hard-scattering to nonextensive statistical mechanical descriptions of transverse momentum spectra in high-energy $pp$ and $p\bar{p}$ collisions
• The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics
• Experimental validation of nonextensive scaling law in confined granular media
• Defect turbulence and generalized statistical mechanics
• Generalization of symmetric $\alpha$-stable Lévy distributions for $q>1$
• Influence of the interaction range on the thermostatistics of a classical many-body system
• Fractals, non-extensive statistics and QCD
• Tunneling in a very slow ion-molecule reaction
• Role of dimensionality in complex networks: Connection with nonextensive statistics
• Evidence of $q$-exponential statistics in Greek seismicity
• Uniqueness and characterization theorems for generalized entropies
• Validity and Failure of the Boltzmann Weight
• Scaling behavior in land markets
• Evidence for criticality in financial data
• Tsallis cosmology and its applications in dark matter physics with focus on IceCube high-energy neutrino data
• Constraints on Tsallis Cosmology from Big Bang Nucleosynthesis and the Relic Abundance of Cold Dark Matter Particles
• Statistical characterization of the standard map
• Cauchy distributions for the integrable standard map
• A generalised model for asymptotically-scale-free geographical networks
• Role of dimensionality in preferential attachment growth in the Bianconi-Barabási model
• Connecting complex networks to nonadditive entropies
• Multivariate Group Entropies, Super-exponentially Growing Complex Systems and Functional Equations
• Comparative study of the heavy-quark dynamics with the Fokker-Planck Equation and the Plastino-Plastino Equation
• Scaling properties of d-dimensional complex networks
• Neural complexity -- Statistical-mechanical approach of human electroencephalograms
• Generalization of the Gauss Map: A jump into chaos with universal features
• Statistical mechanical approach of complex networks with weighted links