Entropy and Complexity Analyses of D-dimensional Quantum Systems

Dehesa, J. S.; López-Rosa, S.; Manzano, Daniel

doi:10.1007/978-90-481-3890-6_5

Cited by 46 publications

(85 citation statements)

References 119 publications

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“…For a revision of their properties see [38,39,40,41,42,43,44] and the reviews [45,46]. The Rényi entropies and their associated uncertainty relations have been widely used to investigate a great deal of quantum-mechanical properties and phenomena of physical systems and processes [29,45,46,44], ranging from the quantum-classical correspondence [47] and quantum entanglement [48] to pattern formation and Brown processes [49,50], fractality and chaotic systems [51,52], quantum phase transition [53] and disordered systems [54]. Moreover, the knowledge of these quantities allows us to reconstruct the corresponding probability density under certain conditions [41,55].…”

Section: Introductionmentioning

confidence: 99%

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

et al. 2016

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Abstract. The Rényi entropies R p [ρ], p > 0, 1 of the highly-excited quantum states of the D-dimensional isotropic harmonic oscillator are analytically determined by use of the strong asymptotics of the orthogonal polynomials which control the wavefunctions of these states, the Laguerre polynomials. This Rydberg energetic region is where the transition from classical to quantum correspondence takes place. We first realize that these entropies are closely connected to the entropic moments of the quantum-mechanical probability ρ n (r) density of the Rydberg wavefunctions Ψ n,l,{µ} (r); so, to the L p -norms of the associated Laguerre polynomials. Then, we determine the asymptotics n → ∞ of these norms by use of modern techniques of approximation theory based on the strong Laguerre asymptotics. Finally, we determine the dominant term of the Rényi entropies of the Rydberg states explicitly in terms of the hyperquantum numbers (n, l), the parameter order p and the universe dimensionality D for all possible cases D ≥ 1. We find that (a) the Rényi entropy power decreases monotonically as the order p is increasing and (b) the disequilibrium (closely related to the second order Rényi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of D.

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Section: Introductionmentioning

confidence: 99%

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

et al. 2016

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“…See also [10] for a recent summary. The idea that the universe is trapped on a membrane in some high-dimensional space-time may explain why gravity is so weak, and could be tested at high-energy particle accelerators [11,12] in the ongoing LHC experiments with hadronic beams colliding at 7 to 14 TeV.…”

Section: Introductionmentioning

confidence: 99%

Entropy and complexity properties of the d-dimensional blackbody radiation

Toranzo

Dehesa

2014

Eur. Phys. J. D

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Space dimensionality is a crucial variable in the analysis of the structure and dynamics of natural systems and phenomena. The dimensionality effects of the blackbody radiation has been the subject of considerable research activity in recent years. These studies are still somewhat fragmentary, posing formidable qualitative and quantitative problems for various scientific and technological areas. In this work we carry out an information-theoretical analysis of the spectral energy density of a d-dimensional blackbody at temperature T by means of various entropy-like quantities (disequilibrium, Shannon entropy, Fisher information) as well as by three (dimensionless) complexity measures (Crámer-Rao, Fisher-Shannon and LMC). All these frequency-functional quantities are calculated and discussed in terms of temperature and dimensionality. It is shown that all three measures of complexity have an universal character in the sense that they depend neither on temperature nor on the Planck and Boltzmann constants, but only on the the space dimensionality d. Moreover, they decrease when d is increasing; in particular, the values 2.28415, 1.90979 and 1.17685 are found for the Crámer-Rao, Fisher-Shannon and LMC measures of complexity of the 3-dimensional blackbody radiation, respectively. In addition, beyond the frequency at which the spectral density is maximum (which follows the well-known Wien displacement law), three further characteristic frequencies are defined in terms of the previous entropy quantities; they are shown to obey Wien-like laws. The potential usefulness of these distinctive features of the blackbody spectrum is physically discussed.

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“…Refs [6][7][8][9]16]. The study of the L p norms of orthogonal polynomials is of independent interest in the theory of general orthogonal and extremal polynomials.…”

Section: Introductionmentioning

confidence: 99%