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International Journal of Applied Mathematics & Statistics/International journal of applied mathematics and statistics

Chaos and complexity in the classical-quantum transition

A.M. Kowalski, M.T. Martín, Angelo Plastino, Osvaldo Anibal Rosso

Published Feb 2, 2012• 5 citations
OpenAlex

Abstract

By recourse to the concept of Statistical Complexity we study here the classical-quantum transition in a special system that represents the matter-field interaction. Our work con-siders two distinct disequilibrium forms based on Euclidean norm and Jensen-Shannon divergence, on the one hand, and analyzes things, on the other one, by using two differ-ent numerical approaches for probability distribution, namely, relative wavelet energy and permutation patterns.

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By recourse to the concept of Statistical Complexity we study here the classical-quantum transition in a special system that represents the matter-field interaction. Our work con-siders two distinct disequilibrium forms based on Euclidean norm and Jensen-Shannon divergence, on the one hand, and analyzes things, on the other one, by using two differ-ent numerical approaches for probability distribution, namely, relative wavelet energy and permutation patterns.
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We investigated the dynamics of a regular nonlinear semiclassical system by employing Shannon entropy together with two associated formulations of statistical complexity: the López-Ruiz–Mancini–Calbet (LMC) and Jensen–Shannon approaches. Additionally, Tsallis entropy was used as an alternative quantifier. In this context, quantum variables interacted with a classical environment, and both conservative and dissipative regimes were considered. To compute information-theoretic quantifiers, probability distributions were extracted from the system's temporal evolution using the Bandt–Pompe permutation method. The classical limit was characterized by a motion invariant linked to the uncertainty principle. Our analysis revealed three distinct zones that characterized the structure of transitions across the classical–quantum boundary. These findings confirmed earlier results obtained in markedly different dynamical systems. This consistency supported the idea of a possible generalization, which, if established, would have significant implications for both semiclassical and quantum theories.
We study the transition to chaos and the emergence of statistical relaxation in isolated dynamical quantum systems of interacting particles. Our approach is based on the concept of delocalization of the eigenstates in the energy shell, controlled by the Gaussian form of the strength function. We show that, although the fluctuations of the energy levels in integrable and nonintegrable systems are different, the global properties of the eigenstates are quite similar, provided the interaction between particles exceeds some critical value. In this case, the statistical relaxation of the systems is comparable, irrespective of whether or not they are integrable. The numerical data for the quench dynamics manifest excellent agreement with analytical predictions of the theory developed for systems of two-body interactions with a completely random character.
We consider the solution of one dimensional Schrödinger Dunkl equation for energies and eigenfunctions. Then we provide analytical expressions for various information theoretic measures. For a given density function, quantities such as position expectation value, entropic moment, disequilibrium, Rényi entropy, Shannon entropy, Tsallis entropy, Fisher information are presented. Next, a few relative information measures corresponding to two density functions, like relative entropy, relative Fisher, relative Rényi, relative Tsallis, along with their associated Jensen divergences such as Jensen–Shannon divergence, Jensen–Fisher divergence, Jensen–Rényi divergence, Jensen–Tsallis divergence are treated. Sample results are provided in graphical form. Dependence of these quantities on the Dunkl parameter μ shows distinct features for μ < 0 and μ > 0.
We investigate the classical limit of the dynamics of a semiclassical system that represents the interaction between matter and a given field.On using as a quantifier the q-Complexity, we find that it describes appropriately the quantum-classical transition, detecting the most salient details of the changeover.Additionally the q-Complexity results a better quantifier of the problem than the q-entropy, in the sense that the q-range is enlarged, describing the q-Complexity, the most important characteristics of the transition for all q-value.