Escort mean values and the characterization of power-law-decaying probability densities

Tsallis, Constantino; Plastino, A.; Alvarez‐Estrada, R. F.

doi:10.1063/1.3104063

Cited by 55 publications

(67 citation statements)

References 35 publications

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“…In the following, we will consider β = 1. [24] and its relevance is discussed, for instance, in [25]. S q is nonadditive for q = 1 since, for two independent systems A and B, we easily verify (assuming k = 1)…”

Section: Q-gaussiansmentioning

confidence: 99%

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

2008

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The celebrated Leibnitz triangle has a remarkable property, namely that each of its elements equals the sum of its South-West and South-East neighbors. In probabilistic terms, this corresponds to a specific form of correlation of N equally probable binary variables which satisfy scaleinvariance. Indeed, the marginal probabilities of the N -system precisely coincide with the joint probabilities of the (N − 1)-system. On the other hand, the nonadditive entropy, which grounds nonextensive statistical mechanics, is, under appropriate constraints, extremized by the). These distributions also result, as attractors, from a generalized central limit theorem for random variables which have a finite generalized variance, and are correlated in a specific way called q-independence. In order to physically enlighten this concept, we introduce here three types of asymptotically scale-invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index ν = 1, 2, 3, . . . , unifying the Leibnitz triangle (ν = 1) and the case of independent variables (ν → ∞); (ii) two slightly different discretizations of q-Gaussians; (iii) a special family, characterized by the parameter χ, which generalizes the usual case of independent variables (recovered for χ = 1/2). Models (i) and (iii) are in fact strictly scale-invariant. For models (i), we analytically show that the N → ∞ probability distribution is a q-Gaussian with q = (ν − 2)/(ν − 1). Models (ii) approach q-Gaussians by construction, and we numerically show that they do so with asymptotic scale-invariance. Models (iii), like two other strictly scale-invariant models recently discussed by Hilhorst and Schehr (2007), approach instead limiting distributions which are not q-Gaussians. The scenario which emerges is that asymptotic (or even strict) scale-invariance is not sufficient but it might be necessary for having strict (or asymptotic) q-independence, which, in turn, mandates q-Gaussian attractors.

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Section: Q-gaussiansmentioning

confidence: 99%

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

2008

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“…To check the robustness of our results with respect to q-statistcs, we have computed the q-generalized kurtosis (referred to as q-kurtosis in [15,7]) defined as follows:…”

Section: More Than One Century Ago In His Historical Book Elementarymentioning

confidence: 99%

Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

Christodoulidi¹,

Tsallis²,

Bountis³

2014

EPL

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We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) β− model. The standard quartic interaction is generalized through a coupling constant that decays as 1/r α (α ≥ 0)(with strength characterized by b > 0). In the α → ∞ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For α ≥ 1 the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators N (thus yielding ergodicity), whereas, for 0 ≤ α < 1, it asymptotically decreases as N −κ(α) (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for α large enough, whereas it is well approached by a q-Gaussian, with the index q(α) monotonically decreasing from about 1.5 to 1 (Gaussian) when α increases from zero to close to one. For α small enough, the whole picture is consistent with a crossover at time t c from q-statistics to Boltzmann-Gibbs (BG) thermostatistics. More precisely, we construct a "phase diagram" for the system in which this crossover occurs through a frontier of the form 1/N ∝ b δ /t γ c with γ > 0 and δ > 0, in such a way that the q = 1 (q > 1) behavior dominates in the lim N →∞ lim t→∞ ordering (lim t→∞ lim N →∞ ordering).

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“…However, if the orbit is trapped for a long time near islands of regular motion, the MLE does not quickly converge and when it does, one cannot tell from its value whether the dynamics can be described as weakly or strongly chaotic. Now, long-range systems are known to possess long-living quasi-stationary states (QSS) [13][14][15][16][17][18][19][20][21][22][23][24][25][26], whose statistical properties are very different from what is expected within the framework of classical BG thermostatistics [27]. More specifically, when one studies such QSS in the spirit of the central limit theorem, one finds that the pdfs of sums of their variables are well approximated by q-Gaussian functions (with 1 < q < 3) or q-statistics [28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Dynamics and statistics of the Fermi–Pasta–Ulamβ-model with different ranges of particle interactions

et al. 2016

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In the present work we study the Fermi-Pasta-Ulam (FPU) β-model involving long-range interactions (LRI) in both the quadratic and quartic potentials, by introducing two independent exponents α 1 and α 2 respectively, which make the forces decay with distance r. Our results demonstrate that weak chaos, in the sense of decreasing Lyapunov exponents, and q-Gaussian probability density functions (pdfs) of sums of the momenta, occurs only when long-range interactions are included in the quartic part. More importantly, for 0 ≤ α 2 < 1, we obtain extrapolated values for q ≡ q ∞ > 1, as N → ∞, suggesting that these pdfs persist in that limit. On the other hand, when long-range interactions are imposed only on the quadratic part, strong chaos and purely Gaussian pdfs are always obtained for the momenta. We have also focused on similar pdfs for the particle energies and have obtained q Eexponentials (with q E > 1) when the quartic-term interactions are long-ranged, otherwise we get the standard Boltzmann-Gibbs weight, with q = 1. The values of q E coincide, within small discrepancies, with the values of q obtained by the momentum distributions.

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Escort mean values and the characterization of power-law-decaying probability densities

Cited by 55 publications

References 35 publications

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

Dynamics and statistics of the Fermi–Pasta–Ulamβ-model with different ranges of particle interactions

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