A.M. Kowalski

Distances in Probability Space and the Statistical Complexity Setup

Statistical complexity measures (SCM) are the composition of two ingredients: (i) entropies and (ii) distances in probability-space. In consequence, SCMs provide a simultaneous quantification of the randomness and the correlational structures present in the system under study. We address in this review important topics underlying the SCM structure, viz., (a) a good choice of probability metric space and (b) how to assess the best distance-choice, which in this context is called a “disequilibrium” and is denoted with the letter Q. Q, indeed the crucial SCM ingredient, is cast in terms of an associated distance D. Since out input data consists of time-series, we also discuss the best way of extracting from the time series a probability distribution P. As an illustration, we show just how these issues affect the description of the classical limit of quantum mechanics.

ENTROPIC NON-TRIVIALITY, THE CLASSICAL LIMIT AND GEOMETRY-DYNAMICS CORRELATIONS

The quantum-classical limit together with the associated onset of chaos is employed here in order to illustrate the importance of a proper choice of distance in probability space if one wishes to describe dynamical properties from the information theory viewpoint.

On extracting probability distribution information from time series

Time-series (TS) are employed in a variety of academic disciplines. In this paper we focus on extracting probability density functions (PDFs) from TS to gain an insight into the underlying dynamic processes. On discussing this “extraction” problem, we consider two popular approaches that we identify as histograms and Bandt–Pompe. We use an information-theoretic method to objectively compare the information content of the concomitant PDFs.

Tsallis’ deformation parameter q quantifies the classical–quantum transition

We investigate the classical limit of a type of semiclassical evolution, the pertinent system representing the interaction between matter and a given field. On using Tsallis q-entropy as a quantifier of the ensuing dynamics, we find that it not only appropriately describes the quantum–classical transition, but that the associated deformation-parameter q itself characterizes the different regimes involved in the process, detecting the most salient fine details of the changeover.

Generalized Complexity and Classical-Quantum Transition

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 theq-Complexity results a better quan- tifier 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.

Information theory and chaotic motion

We present a general method to study the dynamics of quantum-classical systems. The emergency of chaotic motion in the classical limit together with the transition between regimes are also described.

Relative Entropies and Jensen Divergences in the Classical Limit

Metrics and distances in probability spaces have shown to be useful tools for physical purposes. Here we use this idea, with emphasis on Jensen Divergences and relative entropies, to investigate features of the road towards the classical limit. A well-known semiclassical model is used and recourse is made to numerical techniques, via the well-known Bandt and Pompe methodology, to extract probability distributions from the pertinent time-series associated with dynamical data.

Kullback-Leibler Approach to Chaotic Time Series

We focus discussion on extracting probability distribution functions (PDFs) from semi-chaotic time series (TS). We wish to ascertain what is the best extraction approach and to such an end we use an extremely well known semiclassical system in its classical limit [1, 2]. Since this systems possesses a very rich dynamics, it can safely be regarded as representative of many other physical scenarios. In discussing this “extraction” problem, we consider the two most natural approaches, namely, i) histograms and ii) the Bandt‐Pompe technique. We use the Kullback-Leibler relative entropy to compare the information content of the concomitant PDFs.

Dissipative behaviour and the interface between quantum and classical dynamical systems

Abstract We consider the interaction between a two level system and a classical harmonic oscillator. The dynamical evolution is described via a convenient generalization of Ehrenfests theorem, which leads to a set of Bloch-like equations. This model is able to mimic a dissipating temporal evolution, without violating any quantum rule.