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Entropy
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Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena
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Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (31 December 2018) | Viewed by 22245

Special Issue Editor

Dr. Carlo Cafaro

E-Mail Website
Guest Editor
Department of Nanoscale Science and Engineering, University at Albany-SUNY, Albany, NY 12222, USA
Interests: classical and quantum information physics; complexity; entropy; inference; information geometry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The role of geometric methods in modern physical science is very important from applied and foundational perspectives alike. The concepts of complexity, entanglement, phase transitions, and quantum algorithms are examples of physical phenomena that may be observed in cleverly prepared experimental settings whose formal description and essential conceptual understanding can be enhanced by means of geometric concepts. These geometric concepts include, among others: induced metric, curvature, isotropy, symmetries, geodesic paths, geodesic deviation, and volume growth. Explorations of the myriad connections among entropic and geometric quantities present opportunities for further lines of investigation ranging from statistical physics to network science. In this Special Issue, we propose the discussion of the following two areas of research: First, geometric descriptions of physical phenomena; second, entropic aspects of such geometrizations.

These investigations are usually undertaken by several types of scientists, including applied mathematicians, quantum physicists, and statistical physicists. The mathematical and physical tools needed to investigate such problems are quite diverse and include, in particular, inference methods, information theory, probability theory, quantum physics, Riemannian geometry, and statistical physics. More importantly, the role that the concept of entropy plays in such geometric formulations of natural phenomena is becoming increasingly important.

It is our great pleasure to welcome your contributions to this Special Issue with the aim of advancing our geometric and entropic understanding of challenging problems appearing in condensed matter physics, general relativity, network science, quantum computing, and thermodynamics, to include a few research fields. At the same time, we hope to highlight the entropic aspects uncovered by means of the geometric modeling of natural phenomena, including special scenarios covered by either classical or quantum modern theoretical physics.

Dr. Carlo Cafaro
Guest Editor

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Complexity
  • Entropy
  • Inference methods
  • Information theory
  • Phase transitions
  • Probability theory
  • Quantum algorithms
  • Quantum physics
  • Riemannian geometry
  • Statistical physics

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Published Papers (5 papers)

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Research

10 pages, 257 KiB  
Article
Probabilistic Inference for Dynamical Systems
by Sergio Davis, Diego González and Gonzalo Gutiérrez
Entropy 2018, 20(9), 696; https://doi.org/10.3390/e20090696 - 12 Sep 2018
Cited by 4 | Viewed by 3573
Abstract
A general framework for inference in dynamical systems is described, based on the language of Bayesian probability theory and making use of the maximum entropy principle. Taking the concept of a path as fundamental, the continuity equation and Cauchy’s equation for fluid dynamics [...] Read more.
A general framework for inference in dynamical systems is described, based on the language of Bayesian probability theory and making use of the maximum entropy principle. Taking the concept of a path as fundamental, the continuity equation and Cauchy’s equation for fluid dynamics arise naturally, while the specific information about the system can be included using the maximum caliber (or maximum path entropy) principle. Full article
(This article belongs to the Special Issue Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena)
26 pages, 416 KiB  
Article
Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box
by Paolo De Gregorio and Lamberto Rondoni
Entropy 2018, 20(9), 645; https://doi.org/10.3390/e20090645 - 28 Aug 2018
Cited by 1 | Viewed by 4152
Abstract
From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively [...] Read more.
From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose–Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is ‘large’ and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of ( 2 / 3 ) -rds in one dimension, and ( 3 / 5 ) -ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ. Full article
(This article belongs to the Special Issue Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena)
15 pages, 871 KiB  
Article
Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations
by Hector Zenil, Narsis A. Kiani and Jesper Tegnér
Entropy 2018, 20(7), 534; https://doi.org/10.3390/e20070534 - 18 Jul 2018
Cited by 2 | Viewed by 5118
Abstract
We introduce a definition of algorithmic symmetry in the context of geometric and spatial complexity able to capture mathematical aspects of different objects using as a case study polyominoes and polyhedral graphs. We review, study and apply a method for approximating the algorithmic [...] Read more.
We introduce a definition of algorithmic symmetry in the context of geometric and spatial complexity able to capture mathematical aspects of different objects using as a case study polyominoes and polyhedral graphs. We review, study and apply a method for approximating the algorithmic complexity (also known as Kolmogorov–Chaitin complexity) of graphs and networks based on the concept of Algorithmic Probability (AP). AP is a concept (and method) capable of recursively enumerate all properties of computable (causal) nature beyond statistical regularities. We explore the connections of algorithmic complexity—both theoretical and numerical—with geometric properties mainly symmetry and topology from an (algorithmic) information-theoretic perspective. We show that approximations to algorithmic complexity by lossless compression and an Algorithmic Probability-based method can characterize spatial, geometric, symmetric and topological properties of mathematical objects and graphs. Full article
(This article belongs to the Special Issue Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena)
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17 pages, 296 KiB  
Article
Entropy of Iterated Function Systems and Their Relations with Black Holes and Bohr-Like Black Holes Entropies
by Christian Corda, Mehdi FatehiNia, MohammadReza Molaei and Yamin Sayyari
Entropy 2018, 20(1), 56; https://doi.org/10.3390/e20010056 - 12 Jan 2018
Cited by 15 | Viewed by 4522
Abstract
In this paper we consider the metric entropies of the maps of an iterated function system deduced from a black hole which are known the Bekenstein–Hawking entropies and its subleading corrections. More precisely, we consider the recent model of a Bohr-like black hole [...] Read more.
In this paper we consider the metric entropies of the maps of an iterated function system deduced from a black hole which are known the Bekenstein–Hawking entropies and its subleading corrections. More precisely, we consider the recent model of a Bohr-like black hole that has been recently analysed in some papers in the literature, obtaining the intriguing result that the metric entropies of a black hole are created by the metric entropies of the functions, created by the black hole principal quantum numbers, i.e., by the black hole quantum levels. We present a new type of topological entropy for general iterated function systems based on a new kind of the inverse of covers. Then the notion of metric entropy for an Iterated Function System ( I F S ) is considered, and we prove that these definitions for topological entropy of IFS’s are equivalent. It is shown that this kind of topological entropy keeps some properties which are hold by the classic definition of topological entropy for a continuous map. We also consider average entropy as another type of topological entropy for an I F S which is based on the topological entropies of its elements and it is also an invariant object under topological conjugacy. The relation between Axiom A and the average entropy is investigated. Full article
(This article belongs to the Special Issue Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena)
293 KiB  
Article
A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group
by Jing Wang, Huafei Sun and Didong Li
Entropy 2017, 19(12), 698; https://doi.org/10.3390/e19120698 - 20 Dec 2017
Cited by 1 | Viewed by 4050
Abstract
In this paper, we propose an efficient algorithm to solve the averaging problem on the Lorentz group O ( n , k ) . Firstly, we introduce the geometric structures of O ( n , k ) endowed with a Riemannian metric where [...] Read more.
In this paper, we propose an efficient algorithm to solve the averaging problem on the Lorentz group O ( n , k ) . Firstly, we introduce the geometric structures of O ( n , k ) endowed with a Riemannian metric where geodesic could be written in closed form. Then, the algorithm is presented based on the Riemannian-steepest-descent approach. Finally, we compare the above algorithm with the Euclidean gradient algorithm and the extended Hamiltonian algorithm. Numerical experiments show that the geodesic-based Riemannian-steepest-descent algorithm performs the best in terms of the convergence rate. Full article
(This article belongs to the Special Issue Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena)
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