Statistical mechanics based on Renyi entropy

Lenzi, E. K.; Mendes, R. S.; Silva, L.R. da

doi:10.1016/s0378-4371(00)00007-8

Cited by 125 publications

(107 citation statements)

References 34 publications

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“…(2) is recovered, proving that it is a particular case of (6). We notice that a similar generalization exists for the Tsallis entropy [18,19,20], which is widely used in statistical physics and related to the Rényi entropy by an invertible transformation [10,11,12,21,22]. In this paper we will focus on the Rényi entropy since the resulting form of the uncertainty relation is very similar to the B.B.M.I.…”

Section: The Rényi Entropy Uncertainty Relationmentioning

confidence: 71%

“…When λ tends to 1, by l'Hospital's rule, H λ converges to the Shannon entropy that will thus be denoted by continuity H 1 = H. The Rényi entropy is widely used, not only in physics (e.g. statistical mechanics, physics of turbulence, cosmology, see [10,11,12] and references therein), but in various other areas such as in signal processing (time scale analysis, decision problems, machine learning, see [4,13,14,15] and references therein), or image processing (image matching, image registration see [16,17] and references therein). …”

Section: The Rényi Entropy Uncertainty Relationmentioning

confidence: 99%

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On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Zozor

Vignat

2007

Physica A: Statistical Mechanics and its Applications

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In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle [1] and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Rényi entropies. We recall that in both Shannon and Rényi cases, and for a given dimension n, the only case of equality occurs for Gaussian random vectors. We show that as n grows, however, the bound is also asymptotically attained in the cases of n-dimensional Student-t and Student-r distributions. A complete analytical study is performed in a special case of a Student-t distribution. We also show numerically that this effect exists for the particular case of a n-dimensional Cauchy variable, whatever the Rényi entropy considered, extending the results of Abe [2] and illustrating the analytical asymptotic study of the student-t case. In the Student-r case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the vector when the dimension increases.

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Section: The Rényi Entropy Uncertainty Relationmentioning

confidence: 71%

Section: The Rényi Entropy Uncertainty Relationmentioning

confidence: 99%

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Zozor

Vignat

2007

Physica A: Statistical Mechanics and its Applications

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“…Now, it has been shown in Ref. [10,11] that in the TMP instance on also finds the requirement q > 0.…”

Section: Connections Between De 1988 and The Tmp Formalismmentioning

confidence: 89%

Equivalence of the four versions of Tsallis’s statistics

Ferri¹,

Martı́nez²,

Plastino³

2005

J. Stat. Mech.

106

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In spite of its undeniable success, there are still open questions regarding Tsallis' non-extensive statistical formalism, whose founding stone was laid in 1988 in JSTAT. Some of them are concerned with the so-called normalization problem of just how to evaluate expectation values. The Jaynes' MaxEnt approach for deriving statistical mechanics is based on the adoption of (1) a specific entropic functional form S and (2) physically appropriate constraints. The literature on non-extensive thermostatistics has considered, in its historical evolution, four possible choices for the evaluation of expectation values: (i) 1988 Tsallis-original (TO), (ii) Curado-Tsallis (CT), (iii) Tsallis-MendesPlastino (TMP), and (iv) the same as (iii), but using centered operators as constraints (OLM).The 1988 was promptly abandoned and replaced, mostly with versions ii) and iii). We will here (a) show that the 1988 is as good as any of the others, (b) demonstrate that the four cases can be easily derived from just one (any) of them, i.e., the probability distribution function in each of these four instances may be evaluated with a unique formula, and (c) numerically analyze some consequences that emerge from these four choices.PACS: 05.30.Jp

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“…Although Rényi entropy has been introduced by A. Rényi as early as 1961 [1], only recently some authors have studied Rényi entropy in the framework of a generalized thermostatistics [2,3]. Lenzi et al [2] and Bashkirov [3] for example showed that it results in a power-law equilibrium distribution by maximization.…”

Section: Introductionmentioning

confidence: 99%

The Physical Meaning of the Rényi Relative Entropy

Bagci

2017

SDÜ Fen Bil Enst Der

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The Boltzmann-Gibbs relative entropy provides the difference between the offequilibrium and equilibrium free energy terms associated with Boltzmann-Gibbs entropy. In this work, we studied whether this physical meaning can be given to Rényi relative entropy definition. We find that this is possible only in the limit as q approaches to 1. This shows that Rényi relative entropy has a physical meaning only when the system can already be explained by ordinary Boltzmann-Gibbs entropy. We also note that this result is independent of the internal energy constraint employed. Rényi Göreceli Entropisinin Fiziksel AnlamıAnahtar Kelimeler Rényi göreceli entropisi, Serbest enerji, Entropi maksimizasyonu Özet: Boltzmann-Gibbs göreceli entropisi, Boltzmann-Gibbs entropisine ilişkin denge dışı ve denge durumları arasındaki serbest enerji farkını vermektedir. Bu çalışmada, böyle bir fiziksel anlamın Rényi göreceli entropi tanımına verilip verilemeyecegini araştırdık. Bu türden bir fiziksel anlamın sadece q degeri 1'e yaklaştıgında mümkün oldugunu bulduk. Bu sonuç göstermektedir ki, Rényi göreceli entropisi sadece sistem zaten BoltzmannGibbs entropisi tarafından açıklanabildiginde bir fiziksel anlama sahip olmaktadır. Ayrıca bulunan bu sonucun kullanılan iç enerji kısıtından da bagımsız oldugu gösterilmiştir.

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Statistical mechanics based on Renyi entropy

Cited by 125 publications

References 34 publications

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

Equivalence of the four versions of Tsallis’s statistics

The Physical Meaning of the Rényi Relative Entropy

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