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Contemporary Mathematics and Its Applications:
Monographs, Expositions and Lecture Notes
Print ISSN: 2591-7668
Online ISSN: 2591-7676
Series Editor
M Zuhair Nashed (University of Central Florida)
Editorial Board
Guillaume Bal Palle Jorgensen
Gang Bao Marius Mitrea
Liliana Borcea Otmar Scherzer
Raymond Chan Frederik J Simons
Adrian Constantin Edriss S Titi
Willi Freeden Luminita Vese
Charles W Groetsch Hong-Kun Xu
Mourad Ismail Masahiro Yamamoto
This series aims to inspire new curriculum and integrate current research into texts. Its aims and main
scope are to publish:
– Cutting-edge Research Monographs
– Mathematical Plums
– Innovative Textbooks for capstone (special topics) undergraduate and graduate level courses
– Surveys on recent emergence of new topics in pure and applied mathematics
– Advanced undergraduate and graduate level textbooks that may initiate new directions and new
courses within mathematics and applied mathematics curriculum
– Books emerging from important conferences and special occasions
– Lecture Notes on advanced topics
Monographs and textbooks on topics of interdisciplinary or cross-disciplinary interest are particularly
suitable for the series.
Published
Vol. 6 Generalized Radon Transforms and Imaging by Scattered Particles:
Broken Rays, Cones, and Stars in Tomography
by Gaik Ambartsoumian
Vol. 5 Tensor Algebra and Analysis for Engineers:
With Applications to Differential Geometry of Curves and Surfaces
by Paolo Vannucci
Vol. 4 Frontiers in Entropy Across the Disciplines:
Panorama of Entropy: Theory, Computation, and Applications
edited by Willi Freeden & M Zuhair Nashed
Vol. 3 Introduction to Algebraic Coding Theory
by Tzuong-Tsieng Moh
More information on this series can also be found at https://www.worldscientific.com/series/cmameln
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Published by
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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A catalogue record for this book is available from the British Library.
Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes — Vol. 4
FRONTIERS IN ENTROPY ACROSS THE DISCIPLINES
Panorama of Entropy: Theory, Computation, and Applications
Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd.
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For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
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the publisher.
ISBN 978-981-125-939-5 (hardcover)
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PREFACE
Hardlyanyterminscienceissooftenuseddeviatingfromitsactualmeaningasthat
of entropy.Nonetheless, the term“entropy”does havea narrowlydefined meaning.
However, the colloquial definition of entropy as a “measure of disorder” falls too
short. Disorder is not a defined physical concept, so it has no physical measure.
It is more correct to understand entropy as a well-defined objective measure of
the amount of information that would be required to infer from an observable
macrostate the microstate of the system that is actually present. This is what is
meantwhenentropyisalsoparaphrasedas“ameasureoftheignoranceofthestates
of all individual particles”.
A concrete definition of entropy as a physical quantity was established by the
Austrian physicist Ludwig Boltzmann (1844–1906) in the second half of the 19th
century. He focused on the microscopic behavior of a fluid, i.e., a gas or a liquid.
Ludwig Boltzmanndetermined thatentropyis proportionalto the logarithmofthe
number of microstates in a closed system with a fixed volume and fixed number of
particles. By microstates, he meant all the ways in which the molecules or atoms
of the confined fluid can arrange themselves. His formula thus defines entropy as a
measure of the “freedom of arrangement” of the molecules and atoms.
With the help of Boltzmann’s definition, one side of the term can be char-
acterized, but entropy also has another, i.e., macroscopic side, which the German
physicist Rudolf Clausius (1822–1888) had already uncovered a few years earlier.
AccordingtoClausius‘interpretation,entropyalwaysincreasesorremainsconstant.
So, an arrow of time is introduced into the physics of closed systems, because as
entropy increases, thermodynamic processes in closed systems are reversible (or
irreversible). A process would only be reversible if the entropy remained constant.
However,thisisonlytheoreticallypossible.Allrealprocessesareirreversible.Thus,
freely after Boltzmann, one can also say: The number of possible microstates
increasesatany time. This microscopicinterpretationextends the thermodynamic-
macroscopic interpretation by Rudolf Clausius.
Since the insights and concepts of Clausius and Boltzmann, entropy has also
entered other areas of science. Even outside physics, it was taken up, at least as
a mathematical concept. For example, the American mathematician and electrical
engineerClaudeShannon(1916–2001)introducedtheso-calledinformationentropy
v
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vi Frontiers in Entropy Across the Disciplines
in1948.Heusedthisquantitytocharacterizethelossofinformationintransmissions
via telephone lines.
TheKolmogoroventropyisaninvariantofmeasure-preservingmappingsinthe
mathematical subfield of dynamical systems. It generalizes the notion of entropy
known from probability theory (and originally from thermodynamics). Entropy is
meant to measure how much information is obtained with each new step of the
dynamical system. Its definition goes back to Andrei Kolmogorov (1903-1987),
but it was not until the late 1950s that Yakov Sinai succeeded in proving the
nontriviality of this invariant. Intuitively, the Kolmogorov-Sinai entropy measures
thechaoticityofdynamicalsystems.Itistrivial,i.e.,itsvalueiszero,fornon-chaotic
mappings (such as, e.g., rotations). Kolmogorov entropy is also called measure-
theoretic entropy or metric entropy, or abbreviated KS-entropy.
Entropyalsoplaysaroleinchemistryandbiology:Incertainopensystems,new
structures can form if entropy is released to the outside. These must be so-called
dissipative systems, i.e., systems in which energy is converted into thermal energy.
ThistheoryofstructureformationoriginatesfromtheBelgianphysicalchemistIlya
Prigogine (1917–2003).
Duringthepastfivedecades,dozensofnewconceptsofentropyhavebeenintro-
ducedandstudiedinmanydisciplines.Thisvolumecapturessomeofthesignificant
developments in today’s entropy arena. It features expository, review, and research
papersbydistinguishedmathematiciansandscientistsfrommanydisciplines.Topics
include, for example, entropy and society, entropy and time, Souriau entropy on
a symplectic model of statistical physics, new definitions of entropy, geometric
theoryofheatandinformation,maximumentropyinBayesiannetworks,maximum
entropy methods, entropy analysis of biomedical signals (review and comparisonof
methods),spectralentropyanditsapplicationtovideocodingandspeechcoding,a
comprehensivereviewof50yearsofentropyindynamics,acomprehensivereviewof
entropy,entropy-likequantitiesandapplications,topologicalentropyofmultimodal
maps,entropyproductionincomplexsystems,entropyproductionandconvergence
to equilibrium, reversibility and irreversibility in entropy, nonequilibrium entropy,
index of various entropy, and entropy — the greatest blunder ever.
Concerningthevaluationoftheworkwehavetotakeintoaccounttwoessential
aspects: First, entropy is not a precise mathematical concept. It has positive and
negativefeatures.Entropybyitselfisnotasubjectoradiscipline.Yetentropycrops
up in almost every discipline, some important occurrences of entropy are featured
in this volume. And more to come. Second, the different chapters of the book
werewritteninaspecific, discipline-influencedenvironmentinwhichcontemporary
concepts ofentropygainedimportance. As a result, the originandthe genealogyof
all contributions inherently entail different mathematical approaches, expressions,
and writings regarding content presentation and methodological form, which the
editors sought to homogenize in no way.
All in all, the encyclopedia “Frontiers in Entropy Across the Disciplines“
presents a panorama of entropy emphasizing mathematical theory, physical and
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Preface vii
scientific significance, computational methods, and applications in mathematics,
physics, statistics, engineering, biomedical signals, and signal processing.
The editors of the book are very proud to have such a distinguished group of
authors.ItwasanhonorandapleasurefortheeditorstomeetMr.HongKoonChua,
Publishing Director at World Scientific Publishing Company (WSPC), Singapore,
andtoreceivehis supportforthis project.TheeditorsthankMs.RochelleKronzek
Miller, Executive Editor atWSPC, for her support in the initiation of this project.
ThanksgotoProfessorA.S.ErdoganofPalmBeachStateCollegeforhisassistance
and logistical support. Last but not least, the editors would like to thank Mr.
Rhaimie Wahap, Editor at WSPC, for doing a superb job in the production of the
volume and for his patience and understanding in handling the contributions with
the authors.
Willi Freeden
University of Kaiserslautern, Germany
M. Zuhair Nashed
University of Central Florida, USA
June 2022
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EDITOR BIOGRAPHIES
Willi Freeden received his Philosophicum 1970, Diplom in Mathematics 1971,
StaatsexameninMathematicsandGeography1972,hisPh.D.inMathematics1975,
Habilitation in Mathematics 1979, all from RWTH Aachen University, Germany.
He has served for many years as a Professor at the RWTH Aachen. He has held
various visiting professor positions, e.g., at the Ohio State University, Columbus
(Department of Geodetic Science and Surveying). He held the professorship of
Industrial Mathematics at the University of Kaiserslautern 1989 and became the
head of the Geomathematics Group 1994, the Vice-President for Research and
Technology at the University of Kaiserslautern from 2002 to 2006. He is recipient
of the RWTH Borchers Award, the 2018 IPMS (Inverse Problems: Modeling &
Simulation) Award, the Fellowship of the International Association of Geodesy
(IAG, 1996), and the 2020 Vening Meinesz Medal: Award of the European
Geosciences Union (EGU). Since 1996 he is a member of the German Geodetic
Commission of the Bavarian Academy of Sciences, Munich. He is author, editor,
and coeditor of 22 books, published more than 225 papers. He is the (founding
and executive) Editor-in-Chief of the Springer “GEM International Journal on
Geomathematics”,Editor-in-ChiefoftheSpringer“HandbookofGeomathematics”,
Editor-in-Chiefofthe Springer-Spektrum(German)“HandbuchTiefeGeothermie”
and the Springer-Spektrum (German) “Handbuch Oberfla¨chennahe Geothermie”,
Editor-in-Chief of the Springer book series “ Geosystems Mathematics”, Editor-in-
ChiefoftheSpringer“LectureNotesonGeosystemsMathematicsandComputing”,
2015Editor-in-Chiefofthe Springer“HandbookofGeomathematics”,2018Editor-
in-Chief of the Birkha¨user“ Handbook of MathematicalGeodesy”,Editor-in-Chief
of the Springer-Spektrum (German) “Handbuch der Geod¨asie”. He is member of
the editorial board of a large number of international journals. He is the organizer
of severalOberwolfachconferences,minisymposia andSpecial Sessions at meetings
of the American Mathematical Society (AMS).
M. Zuhair Nashed is a Professor of Mathematics at the University of Central
Florida, which he joined in 2002 as Chair of the department. He received his
S.B. and S.M. degrees in Electrical Engineering from MIT and his Ph.D. in
Mathematics from the University of Michigan. He has served for many years as
ix
