The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics

Abstract

As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) standard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistics.

References in corpus (5)

• Thermostatistics of overdamped motion of interacting particles
• Experimental validation of nonextensive scaling law in confined granular media
• Central limit behavior of deterministic dynamical systems
• Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics
• Nonextensive Statistical Analysis of Meteor Showers and Lunar Flashes

Cited by in corpus (47)

• Analytic approaches of the anomalous diffusion: a review
• Role of dimensionality in complex networks: Connection with nonextensive statistics
• Validity and Failure of the Boltzmann Weight
• Sensitivity to initial conditions of a $d$-dimensional long-range-interacting quartic Fermi-Pasta-Ulam model: Universal scaling
• Analytic results for the Tsallis thermodynamic variables
• Cauchy distributions for the integrable standard map
• Approach of complexity in nature: Entropic nonuniqueness
• Avalanches and Generalized Memory Associativity in a Network Model for Conscious and Unconscious Mental Functioning
• Fermi--Pasta--Ulam--Tsingou problems: Passage from Boltzmann to $q$-statistics
• Statistical characterization of the standard map
• Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
• von Neumann entropy and the entropy production of a damped harmonic oscillator
• Möbius transforms, cycles and q-triplets in statistical mechanics
• A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas
• Statistical characterization of discrete conservative systems: The web map
• The limit distribution in the $q$-CLT for $q\ge 1$ is unique and can not have a compact support
• Statistical Entropy of Open Quantum Systems
• Scaling properties of d-dimensional complex networks
• QCD phase diagram and critical exponents within the nonextensive Polyakov-Nambu-Jona-Lasinio model
• Thermal conductance of the coupled-rotator chain: Influence of temperature and size
• On the robustness of the $q$-Gaussian family
• Nonextensive effects on QCD chiral phase diagram and baryon-number fluctuations within Polyakov-Nambu-Jona-Lasinio model
• Approaching a large deviation theory for complex systems
• Chaotic behavior in Lotka-Volterra and May-Leonard models of biodiversity
• An Extensive Study of Bose-Einstein Condensation in Liquid Helium using Tsallis Statistics
• Generalization of the Gauss Map: A jump into chaos with universal features
• Dynamical robustness of discrete conservative systems: Harper and generalized standard maps
• Composing $\alpha$-Gauss and logistic maps: Gradual and sudden transitions to chaos
• On the connection between linear combination of entropies and linear combination of extremizing distributions
• Stochastic differential equations driven by deterministic chaotic maps: analytic solutions of the Perron-Frobenius equation
• A unified time scale for quantum chaotic regimes
• Fisher metric from relative entropy group
• Independent Approximates enable closed-form estimation of heavy-tailed distributions
• Is the public goods game a chaotic system?
• Strong correlations between the exponent $\alpha$ and the particle number for a Renyi-monoatomic gas in Gibbs' statistical mechanics
• Distinguishability notion based on Wootters statistical distance: application to discrete maps
• Non-Gaussian Saha's ionization in Rindler spacetime and the equivalence principle
• A unified framework for divergences, free energies, and Fokker-Planck equations
• A possible correction of the Saha curve for non-equilibrium states
• Entanglement can preserve the compact nature of the phase-space occupancy
• Generalized Algebra Grounded on Nonadditive Entropies
• A closer look at coupled logistic maps at the edge of chaos
• Statistics of a Family of Piecewise Linear Maps
• Features of constrained entropic functional variational problems
• Algebraic structures and deformed Schrödinger equations from groups entropies
• Reciprocity relations and generalized entropic quantifiers that lack trace-form
• The statistical mechanics and machine learning of the $\alpha$-Rényi ensemble