Generalized algebra within a nonextensive statistics

Abstract

By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g., associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element is no more universal within the generalized algebra.

11 pages, no figure, TeX. Reports on Mathematical Physics (2003), in press

References in corpus (23)

• Statistical mechanics in the context of special relativity
• Non Linear Kinetics underlying Generalized Statistics
• Unified first law of black-hole dynamics and relativistic thermodynamics
• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Non-Gaussian equilibrium in a long-range Hamiltonian system
• Inequivalence of ensembles in a system with long range interactions
• Dynamic black-hole entropy
• Axioms and uniqueness theorem for Tsallis entropy
• General pseudoadditivity of composable entropy prescribed by existence of equilibrium
• Long-range interactions and non-extensivity in ferromagnetic spin models
• Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system
• Incomplete statistics and nonextensive generalization of statistical mechanics
• Dynamics and Thermodynamics of Systems with Long Range Interactions: an Introduction
• On the generalized entropy pseudoadditivity for complex systems
• Nonextensive Statistics and Multiplicity Distribution in Hadronic Collisions
• Extensive form of equilibrium nonextensive statistics
• Nonextensive statistics and incomplete information
• Black Hole Entropy: Thermodynamics, Statistical-Mechanics and Subtraction Procedure
• Quantum gas distributions prescribed by factorization hypothesis of probability
• Correlated electrons and generalized statistics
• Many-body q-exponential distribution prescribed by factorization hypothesis
• Chaos in the thermodynamic limit
• Measuring information growth in fractal phase space

Cited by in corpus (83)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive
• Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics
• Generalization of symmetric $\alpha$-stable Lévy distributions for $q>1$
• Numerical indications of a q-generalised central limit theorem
• Probability distribution and entropy as a measure of uncertainty
• A position-dependent mass harmonic oscillator and deformed space
• Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \infty$ limiting distributions
• Continuous growth models in terms of generalized logarithm and exponential functions
• Basic-deformed thermostatistics
• Generalized molecular chaos hypothesis and H-theorem: Problem of constraints and amendment of nonextensive statistical mechanics
• A generalization of the central limit theorem consistent with nonextensive statistical mechanics
• Thermodynamic equilibrium and its stability for Microcanonical systems described by the Sharma-Taneja-Mittal entropy
• Generalized space and linear momentum operators in quantum mechanics
• $\kappa$-generalization of Gauss' law of error
• Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy
• Entropic Forms and Related Algebras
• Generalized exponential function and discrete growth models
• Generalized Shannon-Khinchin Axioms and Uniqueness Theorem for Pseudo-additive Entropies
• Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization
• Physical correlations lead to kappa distributions
• Dual Conformable Derivative: Definition, Simple Properties and Perspectives for Applications
• Arithmetical and geometrical means of generalized logarithmic and exponential functions: generalized sum and product operators
• Influence of global correlations on central limit theorems and entropic extensivity
• On a representation of the inverse Fq transform
• Generalized Metropolis dynamics with a generalized master equation: An approach for time-independent and time-dependent Monte Carlo simulations of generalized spin systems
• Consistent nonadditive approach and nuclear equation of state
• k-Deformed Fourier Transform
• Deformed Fokker-Planck equation: inhomogeneous medium with a position-dependent mass
• Information-theoretic measures for a position-dependent mass system in an infinite potential well
• Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics
• The maximization of Tsallis entropy with complete deformed functions and the problem of constraints
• Generalization of the possible algebraic basis of $q$-triplets
• Algebra and calculus for Tsallis thermostatistics
• Unifying Aspects of Generalized Calculus
• Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem
• Distributivity and deformation of the reals from Tsallis entropy
• Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
• Multiplicative duality, q-triplet and (mu,nu,q)-relation derived from the one-to-one correspondence between the (mu,nu)-multinomial coefficient and Tsallis entropy Sq
• Statistical field theories deformed within different calculi
• A Scale-invariant Generalization of the Rényi Entropy, Associated Divergences and their Optimizations under Tsallis' Nonextensive Framework
• Tsallis entropy induced metrics and CAT(k) spaces
• Long-range interactions, doubling measures and Tsallis entropy
• Vanishing largest Lyapunov exponent and Tsallis entropy
• Non-distributive algebraic structures derived from nonextensive statistical mechanics
• Open Problems within Nonextensive Statistical Mechanics
• Multivariate Generalizations of the q--Central Limit Theorem
• A unified characterization of generalized information and certainty measures
• Some Open Points in Nonextensive Statistical Mechanics
• Rescaling the nonadditivity parameter in Tsallis thermostatistics
• Shannon Entropy Reinterpreted
• A definition of the coupled-product for multivariate coupled-exponentials
• Functional-differential equations for $F_q$%-transforms of $q$-Gaussians
• Entropies from coarse-graining: convex polytopes vs. ellipsoids
• Tsallis entropy approach to radiotherapy treatments
• Generalized entropic structures and non-generality of Jaynes' Formalism
• On the equivalence between four versions of thermostatistics based on strongly pseudo-additive entropies
• Comments on "On q-non-extensive statistics with non-Tsallisian entropy"
• Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions
• Advantages of $q$-logarithm representation over $q$-exponential representation from the sense of scale and shift on nonlinear systems
• Transport Equation for Small Systems and Nonadditive Entropy
• A unified time scale for quantum chaotic regimes
• Tsallis entropy composition and the Heisenberg group
• Nilpotence in Physics: the case of Tsallis entropy
• Generalising the logistic map through the $q$-product
• On the Relativity of Quantumness as Implied by Relativity of Arithmetic and Probability
• One and two side generalisations of the log-Normal distribution by means of a new product definition
• Scaling property and the generalized entropy uniquely determined by a fundamental nonlinear differential equation
• $\alpha$-divergence derived as the generalized rate function in a power-law system
• A generalized Vitali set from nonextensive statistics
• Fisher metric from relative entropy group
• The ${\tau }_q$-Fourier transform: covariance and uniqueness
• Emergent family of Tsallis entropies from the $q$-deformed combinatorics
• Nonextensive Pythagoras' Theorem
• Extensivity and nonextensivity of two-parameter entropies
• Generalized Exponential Function and some of its Applications to Complex Systems
• Two types of $q$-Gaussian distributions used to study the diffusion in a finite region
• Power-law entropies for continuous systems and generalized operations
• The deformed exponential functions of two variables
• Generalized Algebra Grounded on Nonadditive Entropies
• On the $q$-generalised multinomial/divergence correspondence
• Algebraic structures and deformed Schrödinger equations from groups entropies
• An axiomatic characterization of generalized entropies under analyticity condition