Jaime Sanudo

Statistical complexity and Fisher–Shannon information in the H-atom

The Fisher–Shannon information and a statistical measure of complexity are calculated in the position and momentum spaces for the wave functions of the H-atom. For each level of energy, it is found that these two indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum.

A generalized statistical complexity measure: Applications to quantum systems

A two-parameter family of complexity measures C(α,β) based on the Renyi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the Lopez-Ruiz–Mancini–Calbet complexity, which is recovered for α=1 and β=2. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, α or β, goes to infinity, one of the global factors becomes a local factor. For this special case, the complexity is calculated on different quantum systems: H-atom, harmonic oscillator, and square well.

Some features of the statistical complexity, Fisher–Shannon information and Bohr-like orbits in the quantum isotropic harmonic oscillator

The Fisher-Shannon information and a statistical measure of complexity are calculated in the position and momentum spaces for the wave functions of the quantum isotropic harmonic oscillator. We show that these magnitudes are independent of the strength of the harmonic potential. Moreover, for each level of energy, it is found that these two indicators take their minimum values on the orbitals that correspond to the classical (circular) orbits in the Bohr-like quantum image, just those with the highest orbital angular momentum.

A geometrical derivation of the Boltzmann factor

We show that the Boltzmann factor has a geometrical origin, which follows from the microcanonical ensemble. The Maxwell–Boltzmann distribution or the wealth distribution in human society are some direct applications of this interpretation.

Complexity Invariance by Replication in the Quantum Square Well

A new kind of invariance by replication of a statistical measure of complexity is considered. We show that the set of energy eigenstates of the quantum infinite square well displays this particular invariance. Then, this system presents a constant complexity for all the energy eigenstates.

Complexity and white-dwarf structure

Abstract From the low-mass non-relativistic case to the extreme relativistic limit, the density profile of a white dwarf is used to evaluate the C LMC complexity measure [R. Lopez-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321]. Similarly to the recently reported atomic case where, by averaging shell effects, complexity grows with the atomic number [C.P. Panos, K.Ch. Chatzisavvas, Ch.C. Moustakidis, E.G. Kyrkou, Phys. Lett. A 363 (2007) 78; A. Borgoo, F. De Proft, P. Geerlings, K.D. Sen, Chem. Phys. Lett. 444 (2007) 186; J. Sanudo, R. Lopez-Ruiz, Int. Rev. Phys. 2 (2008) 223], here complexity grows as a function of the star mass reaching a maximum finite value in the Chandrasekhar limit.

Statistical Complexity and Fisher-Shannon Information: Applications

In this chapter, a statistical measure of complexity and the Fisher-Shannon information product are introduced and their properties are discussed. These measures are based on the interplay between the Shannon information, or a function of it, and the separation of the set of accessible states to a system from the equiprobability distribution, i.e. the disequilibrium or the Fisher information, respectively. Different applications in discrete and continuous systems are shown. Some of them are concerned with quantum systems, from prototypical systems such as the H-atom, the harmonic oscillator and the square well to other ones such as He-like ions, Hooke’s atoms or just the periodic table. In all of them, these statistical indicators show an interesting behavior able to discern and highlight some conformational properties of those systems.

Evidence of Magic Numbers in Nuclei by Statistical Indicators

The calculation of a statistical measure of complexity and the Fisher-Shannon information in nuclei is carried out in this work. We use the nuclear shell model in order to obtain the fractional occupation probabilities of nuclear orbitals. The increasing of both magnitudes, the statistical complexity and the Fisher-Shannon information, with the number of nucleons is observed. The shell structure is reflected by the behavior of the statistical complexity. The magic numbers are revealed by the Fisher-Shannon information.

Statistical measures and the Klein tunneling in single-layer graphene

Abstract Statistical complexity and Fisher–Shannon information are calculated in a problem of quantum scattering, namely the Klein tunneling across a potential barrier in graphene. The treatment of electron wave functions as masless Dirac fermions allows us to compute these statistical measures. The comparison of these magnitudes with the transmission coefficient through the barrier is performed. We show that these statistical measures take their minimum values in the situations of total transparency through the barrier, a phenomenon highly anisotropic for the Klein tunneling in graphene.

Equiprobability, Entropy, Gamma Distributions and Other Geometrical Questions in Multi-Agent Systems

A set of many identical interacting agents obeying a global additive constraint is considered. Under the hypothesis of equiprobability in the high-dimensional volume delimited in phase space by the constraint, the statistical behavior of a generic agent over the ensemble is worked out. The asymptotic distribution of that statistical behavior is derived from geometrical arguments. This distribution is related with the Gamma distributions found in several multi-agent economy models. The parallelism with all these systems is established. Also, as a collateral result, a formula for the volume of high-dimensional symmetrical bodies is proposed.