Data Set Models and Exponential Families in Statistical Physics and Beyond

Naudts, Jan; Anthonis, Ben

doi:10.1142/s0217984912500625

Cited by 17 publications

(39 citation statements)

References 17 publications

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“…Both Tsallis' entropy function and that of Rényi are maximised by members of the q-exponential family. However, only Tsallis' entropy function shares with the BGS entropy function the property that the heat capacity is always positive-this has been proved in a very general context in [22]. For this reason, one can say that the Tsallis' entropy function is a stable entropy function.…”

Section: Resultsmentioning

confidence: 99%

“…Recently [21], it was proved that this distribution belongs to the q-exponential family [22][23][24], with q = 1−2/(3N −2). In the thermodynamic limit N → ∞ it converges to the Boltzmann-Gibbs distribution, which corresponds with the q = 1 case.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, Rényi's entropy function is also maximized by the members of the q-exponential family. However, the corresponding free energy is not necessarily minimized, while this is necessarily so [22] in the Tsallis case. As a consequence, a description in terms of Rényi's entropy function leaves room for thermodynamic instabilities, while these are not possible when starting from the Tsallis function.…”

Section: Introductionmentioning

confidence: 99%

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On the Thermodynamics of Classical Micro-Canonical Systems

Baeten

Naudts

2011

Entropy

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We give two arguments why the thermodynamic entropy of non-extensive systems involves Rényi's entropy function rather than that of Tsallis. The first argument is that the temperature of the configurational subsystem of a mono-atomic gas is equal to that of the kinetic subsystem. The second argument is that the instability of the pendulum, which occurs for energies close to the rotation threshold, is correctly reproduced.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Thermodynamics of Classical Micro-Canonical Systems

Baeten

Naudts

2011

Entropy

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“…For more details, see [2,6], for example. Historically, Tsallis [14] introduced the notion of q-exponential function and Naudts [5] introduced the notion of q-exponential family together with a further generalization. Such a historical note is provided in [2].…”

Section: Deformed Exponential Familiesmentioning

confidence: 99%

“…Such a statistical model is described by such a deformed exponential function. In particular, the set of all q-normal distributions (or Student's t-distributions, equivalently) is a q-exponential family, which is described by a q-deformed exponential function [5] (see also [6,7]). …”

Section: Introductionmentioning

confidence: 99%

A Sequence of Escort Distributions and Generalizations of Expectations on q-Exponential Family

Matsuzoe

2016

Entropy

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Abstract:In the theory of complex systems, long tailed probability distributions are often discussed. For such a probability distribution, a deformed expectation with respect to an escort distribution is more useful than the standard expectation. In this paper, by generalizing such escort distributions, a sequence of escort distributions is introduced. As a consequence, it is shown that deformed expectations with respect to sequential escort distributions effectively work for anomalous statistics. In particular, it is shown that a Fisher metric on a q-exponential family can be obtained from the escort expectation with respect to the second escort distribution, and a cubic form (or an Amari-Chentsov tensor field, equivalently) is obtained from the escort expectation with respect to the third escort distribution.

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Gauge Freedom of Entropies on q-Gaussian Measures

Matsuzoe

Takatsu

2021

Signals and Communication Technology

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A q-Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent 1/(1 − q) (q = 1). The limit case q = 1 recovers a Gaussian measure. On the set of all q-Gaussian densities over the real line with 1 ≤ q < 3, escort expectations determine information geometric structures such as an entropy and a relative entropy. The ordinary expectation of a random variable is the integral of the random variable with respect to its law. Escort expectations admit us to replace the law by any other measures. One of the most important escort expectations on the set of all q-Gaussian densities is the q-escort expectation since this escort expectation determines the Tsallis entropy and the Tsallis relative entropy. The phenomenon gauge freedom of entropies is that different escort expectations determine the same entropy, but different relative entropies. In this chapter, we first introduce a refinement of the q-logarithmic function. Then we demonstrate the phenomenon on an open set of all q-Gaussian densities over the real line by using the refined q-logarithmic functions. We write down the corresponding Riemannian metric. Keywords Information geometry • Gauge freedom of entropies • Refined q-logarithmic function • q-Gaussian measure

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Data Set Models and Exponential Families in Statistical Physics and Beyond

Cited by 17 publications

References 17 publications

On the Thermodynamics of Classical Micro-Canonical Systems

On the Thermodynamics of Classical Micro-Canonical Systems

A Sequence of Escort Distributions and Generalizations of Expectations on q-Exponential Family

Gauge Freedom of Entropies on q-Gaussian Measures

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