# Generalized Complexity and Classical-Quantum Transition

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## Abstract

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**PACS:**89.70.Cf. 03.65.Sq, 05.45.Mt

## 1. Introduction

**Statistical complexity**. In [3], López-Ruiz, Mancini and Calbet (LMC) advanced a statistical complexity measure (SCM) based on the notion of “disequilibrium" as a quantifier of the degree of physical structure in a time series. Given a probability distribution associated with a system’s state, the LMC measure is the product of an normalized entropy H times a distance to the uniform-equilibrium state Q. It vanishes for a totally random process and for a periodic one. Martín et al. [10] improved on this measure by modifying the distance-component (in the concomitant probability space). In Ref. [10], Q is built-up using Wootters’ statistical distance while H is a normalized Shannon-entropy. Regrettably enough, the ensuing statistical complexity measure is neither an intensive nor an extensive quantity, although it does yield useful results. A reasonable complexity measure should be able to distinguish among different degrees of periodicity and it should vanish only for periodicity unity. In order to attain such goals it would seem desirable to give this statistical measure an intensive character. This was achieved in Ref. [4] obtaining a SCM that is (i) able to grasp essential details of the dynamics, (ii) an intensive quantity, and (iii) capable of discerning among different degrees of periodicity and chaos.

**Deformed q-statistics**. It is a well-known fact that physical systems that are characterized by either long-range interactions, long-term memories, or multi-fractal nature, are best described by a generalized statistical mechanics’ formalism [11] that was proposed 20 years ago: the so-called q-statistics. More precisely, Tsallis [12] advanced in 1988 the idea of using in a thermodynamics’ scenario an entropic form, the Harvda-Chavrat one, characterized by the entropic index $q\in \mathcal{R}$ ($q=1$ yields the orthodox Shannon measure):

**Quantum-classical frontier**. The classical limit of quantum mechanics (CLQM) continues attracting the attention of many theorists and is the source of much exciting discussion (see, for instance, Refs. [17,18] and references therein). In particular, the investigation of “quantum" chaotic motion is considered important in this limit. Recent literature provides us with many examples, although the adequate definition of the underlying phenomena is understood in diverse fashion according to the different authors (see Ref. [19] and references therein).

## 2. A semi-classical model and the CLQM

#### 2.1. Previous q-entropy Results

## 3. Present results

#### 3.1. Introducing the q-statistical complexity

## 4. Numerical results

**Figure 1.**$q-$Statistical Complexity ${\mathcal{C}}_{q,J}$ vs. ${E}_{r}$ for $q\le 0.4$ (Fig. 1a) and $0.5\le q<1$ (Fig. 1b). Shannon’s complexity are also displayed. Three zones are to be differentiated. They are delimited by special ${E}_{r}-$values, namely, ${{E}_{r}}^{\mathcal{P}}=3.3282$ and ${{E}_{r}}^{cl}=21,55264$. Notice the local complexity maximum at ${E}_{r}^{M}\simeq 6,8155$.

## 5. Conclusions

## Acknowledgments

## A Normalized Tsallis wavelet entropy

## References and Notes

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**Figure 2.**$q-$Statistical Complexity ${\mathcal{C}}_{q,J}$ vs ${E}_{r}$ for $1<q<2$. The three zones and the point ${E}_{r}^{M}$ of Fig. 1 are also seen here.

**Figure 3.**$q-$Statistical Complexity ${\mathcal{C}}_{q,J}$ vs. ${E}_{r}$ for $2\le q\le 5$. No great changes are observed.

**Figure 4.**$q-$Statistical Complexity ${\mathcal{C}}_{q,J}$ vs. ${E}_{r}$ for $10\le q\le 20$. The local maximum at ${E}_{r}^{M}$ disappears.

**Figure 5.**$q-$Statistical Complexity ${\mathcal{C}}_{q,J}$ for different ${E}_{r}$-values. Quantal (Figs. 5a - 5b), transitional (Figs. 5c, 5d, 5e and 5f) and classic (5g - 5h). The curves corresponding to the quantal zone resemble each other and exhibit a different aspect compared to those pertaining to the classical region.

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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Kowalski, A.M.; Plastino, A.; Casas, M.
Generalized Complexity and Classical-Quantum Transition. *Entropy* **2009**, *11*, 111-123.
https://doi.org/10.3390/e11010111

**AMA Style**

Kowalski AM, Plastino A, Casas M.
Generalized Complexity and Classical-Quantum Transition. *Entropy*. 2009; 11(1):111-123.
https://doi.org/10.3390/e11010111

**Chicago/Turabian Style**

Kowalski, A. M., Angelo Plastino, and Montserrat Casas.
2009. "Generalized Complexity and Classical-Quantum Transition" *Entropy* 11, no. 1: 111-123.
https://doi.org/10.3390/e11010111