A possible deformed algebra and calculus inspired in nonextensive thermostatistics

Abstract

We present a deformed algebra related to the q-exponential and the q-logarithm functions that emerge from nonextensive statistical mechanics. We also develop a q-derivative (and consistently a q-integral) for which the q-exponential is an eigenfunction. The q-derivative and the q-integral have a dual nature, that is also presented.

7 pages, no figures

References in corpus (9)

• Statistical mechanics in the context of special relativity
• Non Linear Kinetics underlying Generalized Statistics
• Generalized algebra within a nonextensive statistics
• Nonextensive Pesin identity. Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map
• Deformed exponentials and logarithms in generalized thermostatistics
• A new one parameter deformation of the exponential function
• Is Tsallis thermodynamics nonextensive?
• Mixing and equilibration: Protagonists in the scene of nonextensive statistical mechanics
• Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization

Cited by in corpus (137)

• Generalized algebra within a nonextensive statistics
• 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
• Displacement operator for quantum systems with position-dependent mass
• Generalization of symmetric $\alpha$-stable Lévy distributions for $q>1$
• Group entropies, correlation laws and zeta functions
• Morse potential derived from first principles
• 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
• Fractal derivatives, fractional derivatives and $q$-deformed calculus
• 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
• Thermodynamic equilibrium and its stability for Microcanonical systems described by the Sharma-Taneja-Mittal entropy
• Thermostatistics of deformed bosons and fermions
• A generalization of the central limit theorem consistent with nonextensive statistical mechanics
• On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric
• 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
• On q-Gaussians and Exchangeability
• Variational Approach and Deformed Derivatives
• Generalized exponential function and discrete growth models
• $\kappa$-Deformed quantum and classical mechanics for a system with position-dependent effective mass
• Physical correlations lead to kappa distributions
• Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization
• From the Boltzmann equation with non-local correlations to a standard non-linear Fokker-Planck equation
• Arithmetical and geometrical means of generalized logarithmic and exponential functions: generalized sum and product operators
• Nonextensive Quantum H-Theorem
• Dual Conformable Derivative: Definition, Simple Properties and Perspectives for Applications
• On a representation of the inverse Fq transform
• When Shannon and Khinchin meet Shore and Johnson: equivalence of information theory and statistical inference axiomatics
• 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
• On the average uncertainty for systems with nonlinear coupling
• Information-theoretic measures for a position-dependent mass system in an infinite potential well
• The maximization of Tsallis entropy with complete deformed functions and the problem of constraints
• Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics
• Generalization of the possible algebraic basis of $q$-triplets
• Algebra and calculus for Tsallis thermostatistics
• Unifying Aspects of Generalized Calculus
• A nonextensive approach to the stellar rotational evolution I. F and G type stars
• Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem
• Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
• Distributivity and deformation of the reals from Tsallis entropy
• 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
• Fibonacci Oscillators in the Landau Diamagnetism problem
• Statistical field theories deformed within different calculi
• Inversion of Tsallis' q-Fourier Transform and the complex-plane generalization
• Dynamics in fractal spaces
• Tsallis meets Boltzmann: q-index for a finite ideal gas and its thermodynamic limit
• Tsallis entropy induced metrics and CAT(k) spaces
• A Scale-invariant Generalization of the Rényi Entropy, Associated Divergences and their Optimizations under Tsallis' Nonextensive Framework
• Long-range interactions, doubling measures and Tsallis entropy
• Non-Debye impedance and relaxation models for dissipative electrochemical capacitors
• Vanishing largest Lyapunov exponent and Tsallis entropy
• Mother wavelet functions generalized through q-exponentials
• Bose-Einstein condensation and non-extensive statistics
• q-Stirling's formula in Tsallis statistics
• Non-distributive algebraic structures derived from nonextensive statistical mechanics
• Open Problems within Nonextensive Statistical Mechanics
• $q$-Generalized representation of the $d$-dimensional Dirac delta and $q$-Fourier transform
• Nonextensive Kinetic Theory and H-Theorem in General Relativity
• Multivariate Generalizations of the q--Central Limit Theorem
• Some Open Points in Nonextensive Statistical Mechanics
• A unified characterization of generalized information and certainty measures
• Shannon Entropy Reinterpreted
• Rescaling the nonadditivity parameter in Tsallis thermostatistics
• Generalized Statistics Variational Perturbation Approximation using q-Deformed Calculus
• Reflections on the q-Fourier transform and the q-Gaussian function
• Thermodynamics of exponential Kolmogorov-Nagumo averages
• Position-dependent mass in strong quantum gravitational background fields
• Differential-escort transformations and the monotonicity of the LMC-Rényi complexity measure
• A risk profile for information fusion algorithms
• On the deformed oscillator and the deformed derivative associated with the Tsallis q-exponential
• A definition of the coupled-product for multivariate coupled-exponentials
• Tsallis entropy approach to radiotherapy treatments
• Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions
• Entropies from coarse-graining: convex polytopes vs. ellipsoids
• Functional-differential equations for $F_q$%-transforms of $q$-Gaussians
• Probability distribution of (Schwämmle and Tsallis) two-parameter entropies and the Lambert W-function
• A deformed derivative model for turbulent diffusion of contaminants in the atmosphere
• Generalized entropic structures and non-generality of Jaynes' Formalism
• Effects of correlated variability on information entropies in nonextensive systems
• Nonextensive statistics of relativistic ideal gas
• A Shannon-Tsallis transformation
• The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation
• Non-Gaussian Effects of the Saha's Ionization in the Early Universe
• Transport Equation for Small Systems and Nonadditive Entropy
• Advantages of $q$-logarithm representation over $q$-exponential representation from the sense of scale and shift on nonlinear systems
• The Tsallis-Laplace Transform
• Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions
• A nonextensive view of the stellar braking indices
• A mathematical structure for the generalization of the conventional algebra
• Harmonic oscillators from displacement operators and thermodynamics
• The Relationship between Tsallis Statistics, the Fourier Transform, and Nonlinear Coupling
• Dynamics of Cities
• A unified time scale for quantum chaotic regimes
• Generalising the logistic map through the $q$-product
• Scaled Bregman divergences in a Tsallis scenario
• Morse potential in relativistic contexts from generalized momentum operator, Pekeris approximation revisited and mapping
• Nilpotence in Physics: the case of Tsallis entropy
• One and two side generalisations of the log-Normal distribution by means of a new product definition
• Tsallis entropy composition and the Heisenberg group
• Comment on "Generalized Shannon--Khinchin axioms and uniqueness theorem for pseudo-additive entropies" [Physica A 411 (2014) 138]
• $\alpha$-divergence derived as the generalized rate function in a power-law system
• Satellite image classification and segmentation using non-additive entropy
• A generalized Vitali set from nonextensive statistics
• Scaling property and the generalized entropy uniquely determined by a fundamental nonlinear differential equation
• Fisher metric from relative entropy group
• Comment on "Rényi entropy yields artifficial biases not in the data and incorrect updating due to the infinite-size data"
• Generalized Fisher information matrix in nonextensive systems with spatial correlation
• Emergent family of Tsallis entropies from the $q$-deformed combinatorics
• The ${\tau }_q$-Fourier transform: covariance and uniqueness
• Nonextensive Pythagoras' Theorem
• Group theory, entropy and the third law of thermodynamics
• Power-law entropies for continuous systems and generalized operations
• Generalized Exponential Function and some of its Applications to Complex Systems
• Complete versus incomplete definitions of the deformed logarithmic and exponential functions
• Non-Gaussian Saha's ionization in Rindler spacetime and the equivalence principle
• Generalized Statistics Framework for Rate Distortion Theory with Bregman Divergences
• Two types of $q$-Gaussian distributions used to study the diffusion in a finite region
• On Shore and Johnson properties for a Special Case of Csiszár f-divergences
• On the $q$-generalised multinomial/divergence correspondence
• New degenerated polynomials arising from non-classical Umbral Calculus
• Deformed Statistics Free Energy Model for Source Separation using Unsupervised Learning
• Analytical and numerical studies of creation probabilities of hierarchical trees
• Algebraic structures and deformed Schrödinger equations from groups entropies
• Generalized Statistics Framework for Rate Distortion Theory
• Deformed Statistics Formulation of the Information Bottleneck Method
• Probabilistic graphs using coupled random variables
• q-Calculus Revisited