Chaos, Nonlinearity, Complexity: The Dynamical Paradigm of Nature

371 Pages·2006·21.17 MB·English

by  Francisco Balibrea (auth.), A. Sengupta Professor (eds.)

A.Sengupta(Ed.)
Chaos,Nonlinearity,Complexity
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A. Sengupta
(Ed.)
Chaos, Nonlinearity,
Complexity
The Dynamical Paradigm of Nature
ABC
ProfessorA.Sengupta
DepartmentofMechanicalEngineering
NuclearEngineeringandTechnologyProgramme
IndianInstituteofTechnologyKanpur
Kanpur208016,India
E-mail:[e-mail]
LibraryofCongressControlNumber:2006927415
ISSNprintedition:1434-9922
ISSNelectronicedition:1860-0808
ISBN-10 3-540-31756-2SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-31756-2SpringerBerlinHeidelbergNewYork
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Dedicated to the
♥ synthetic cohabitation of Yang and Yin ♥
induced by
♥ Cha(os-)No(nlinearity-comple)Xity ♥
and to
♥ MPCNS-2004 ♥
that made all this possible
Preface
I think the next century will be the century of complexity. We have already
discovered the basic laws that govern matter and understand all the normal
situations. We don’t know how the laws fit together, and what happens
under extreme conditions. But I expect we will find a complete unified
theory sometime this century. There is no limit to the complexity that we
can build using those basic laws.
Stephen Hawking, January 2000.
We don’t know what we are talking about. Many of us believed that string
theory was a very dramatic break with our previous notions of quantum
theory. But now we learn that string theory, well, is not that much of a
break. The state of physics today is like it was when we were mystified by
radioactivity. They were missing something absolutely fundamental. We are
missing perhaps something as profound as they were back then.
Nobel Laureate David Gross, December 2005.
This volume is essentially a compilation of papers presented at the Inter-
national Workshop on Mathematics and Physics of Complex and Nonlinear
Systems that was held at Indian Institute of Technology Kanpur, March 14
– 26, 2004 on the theme ChaNoXity: The Nonlinear Dynamics of Nature.
ChaNoXity — symbolizing Chaos-Nonlinearity-compleXity — is an attempt
to understand and interpret the dynamical laws of Nature on a unified and
global perspective. The Workshop’s objective was to formalise the concept
of chanoxity and to get the diverse body of practitioners of its components
to interact intelligently with each other. It was aimed at a focused debate
and discussion on the mathematics and physics of chaos, nonlinearity, and
complexity in the dynamical evolution of nature. This is expected to induce
a process of reeducation and reorientation to supplement the basically linear
reductionistapproachofpresentdaysciencethatseekstobreakdownnatural
VIII Preface
systemstotheirsimpleconstituentswhosepropertiesareexpectedtocombine
in a relatively simple manner to yield the complex laws of the whole. There
were approximately 40 hours of lectures by 12 speakers; in keeping with its
aim of providing an open platform for exposition and discourse on the the-
matic topic, each of the 5-6 lectures a day were of 75 minutes duration so
as to provide an adequate and meaningful interaction, formal and informal,
between the speaker and his audience.
The goals of the workshop were to
(cid:1) Createanawarenessamongtheparticipants,drawnfromtheresearchand
educational institutions in India and abroad, of the role and significance
of nonlinearity in its various manifestations and forms.
(cid:1) Present an overview of the strong nonlinearity of chaos and complexity
in natural systems from the mathematical and physical perspectives. The
relevant mathematics were drawn from topology, measure theory, inverse
and ill-posed problems, set-valued and nonlinear functional analyses.
(cid:1) Exploretheroleofnon-extensivethermodynamicsandstatisticalmechan-
ics in open, nonlinear systems.
Therewerelivelyandanimateddiscussionsonself-organizationandemer-
gence in the attainment of steady-states of open, far-from-equilibrium, com-
plex systems, and on the mechanism of how such systems essentially cheat
the dictates of the all-pervading Second Law of Thermodynamics1: where lies
the source of Schrodinger’s negative entropy that successfully maintains life
despite the Second Law? How does Nature defeat itself in this game of the
SecondLaw,andwhatmightbethepossibleroleofgravityinthisenterprise?
Although it is widely appreciated that gravity — the only force to have suc-
cessfully resisted integration in a unified theory — is a major player in the
dynamics of life, realization of a satisfactory theory has proved to be diffi-
cult, with loop quantum cosmology holding promise in resolving the vexing
“big-bang singularity problem”. The distinctive feature of this loop quantiza-
tion is that the quantum Wheeler-DeWitt differential equation (that fails to
remove the singularity, backward evolution leading back into it), is replaced
by a difference equation, the size of the discrete steps determined by an area
gap, Riemannian geometry now being quantized with the length, area, vol-
ume operators possessing discrete eigenvalues. Discrete difference equations,
loops of one-dimensional objects (based on spin-connections rather than on
themetricsofstandardGeneralRelativity)consideredintheperiod-doubling
1 The second law of thermodynamics holds, I think, the supreme position among
the laws of Nature. If someone points out to you that your pet theory of the
universe is in disagreement with Maxwell’s equations then so much the worse
for Maxwell’s equations. If it is found to be contradicted by observation, well,
these experimentalists do bungle things sometimes. But if your theory is found
tobeagainstthe secondlawofthermodynamics I cangiveyounohope;there is
nothing for it but to collapse in deepest humiliation. A. Eddington, The Nature
of the Physical World, Macmillan, New York (1948).
Preface IX
perspective,theextremenonlinearcurvatureofbig-bangandblackholechaotic
spacetimes: do all these point to a radically different paradigm in the chaos-
nonlinearity-complexity setting of discrete dynamical systems?2 Thus is it “a
quantum foam far removed from any classical spacetime, or is there another
large, classical universe” on the “other side of the singularity responsible for a
quantumbouncefromanexpandingbranchtoacontractingbranch”?3 Could
thispossiblybetheoutcomeoftheinteractionofourclassicalrealworldwitha
negativepartneractingastheprovideroftheillusorynegativeentropy,whose
attraction manifests on us as the repulsive “quantum bounce” through the
agency of gravity? Would the complex structure of “life” and of the universe
as we know it exist without the partnership of gravity?
NotallthepaperspresentedattheWorkshopappearhere;notableexcep-
tions among those who gave three or more lectures are S. Kesavan (Institute
of Mathematical Science at Chennai, India) who spoke on Topological Degree
and Bifurcation Theory, and M. Z. Nashed (University of Central Florida,
USA) whose paper Recovery Problems from Partial or Indirect Information:
Perspectives on Inverse and Ill-Posed Problems could not be included due to
unavoidable circumstances. The volume contains three papers by Realpe and
Ordonez, Majumdar, and Johal that were not presented at the Workshop. A
brief overview of the papers appearing follows.
Francisco Balibrea (Universidad de Murcia, Spain) provides a compre-
hensive review of the complicated dynamics of discrete dynamical systems in
a compact metric space using the notions of Li-Yorke and Devaney chaos,
sensitive dependence of initial conditions, transitivity, Lyapunov exponents,
and the Kolmogorov-Sinai and topological entropies. Sumiyoshi Abe (Uni-
versity of Tsukuba, Japan) surveys the fundamental aspects of nonextensive
statistical mechanics based on the Tsallis entropy, and demonstrates how the
methodofsteepestdescents,thecountingalgorithmandtheevaluationofthe
densityofstatescanappropriatelybegeneralizedfordescribingthepower-law
distributions. Alberto Robledo (Universidad Nacional Autonoma de Mex-
ico, Mexico) gives an account of the dynamics at critical attractors of simple
one-dimensional nonlinear maps relevant to the applicability of the Tsallis
generalization of canonical statistical mechanics. Continuing in this spirit of
non-extensivity, A. G. Bashkirov (R.A.S. Moscow, Russia) considers the
RenyientropyasacumulantaverageoftheBoltzmannentropy,andfindsthat
the thermodynamic entropy in Renyi thermostatistics increases with system
complexity,withtheRenyidistributionbecomingapurepower-lawunderap-
propriate conditions. He concludes that “because a power-law distribution is
characteristicforself-organizingsystems,theRenyientropycanbeconsidered
as a potential that drives the system to a self-organized state”. Karmeshu
2 Gerard’tHooft,QuantumGravityasaDissipativeDeterministicSystem,Class.
Quantum Grav., 16, 3263-3279 (1999)
3 AbhayAshtekar,TomaszPawlowski,andParampreetSingh,QuantumNatureof
the Big Bang, ArXiv: gr-qc/0602086
X Preface
and Sachi Sharma (J.N.U., India) proposes a theoretical framework based
onnon-extensiveTsallisentropytostudytheimplicationoflong-rangedepen-
dence in traffic process on network performance. John Realpe (Universidad
del Valle, Colombia) and Gonzalo Ordonez (Butler University, Indianapo-
lis and The University of Texas at Austin, USA) study two points of view
on the origin of irreversible processes. While the “chaotic hypothesis” holds
that irreversible processes originate in the randomness generated by chaotic
dynamics, the approach of the Prigogine school maintains that irreversibility
is rooted in Poincare non-integrability associated with resonances. Consider-
ing the simple model of Brownian motion of a harmonic oscillator coupled
to lattice vibration modes, the authors show that Brownian trajectories re-
quire resonance between the particle and the lattice, with chaos playing only
a secondary role for random initial conditions. If the initial conditions are
not random however, chaos is the dominant player leading to thermalization
of the lattice and consequent appearance of Brownian resonance character-
istics. R. S. Johal (Lyallpur Khalsa College, India) considers the approach
to equilibrium of a system in contact with a heat bath and concludes, in the
context of non-extensivity, that differing bath properties yield differing equi-
librium distributions of the system. Parthasarathi Majumdar (S.I.N.P.,
India) reviews black hole thermodynamics for non-experts, underlining the
need for considerations beyond classical general relativity. The origin of the
microcanonical entropy of isolated, non-radiant, non-rotating black holes is
tracedinthisperspectiveintheLoopQuantumGravity formulationofquan-
tumspacetime.Russ Marion(ClemsonUniversity,USA)appliescomplexity
theorytoorganizationalsciencesandfindsthat“theimplicationsaresosignif-
icantthattheysignalaparadigmshiftinthewayweunderstandorganization
and leadership”. Complexity theory, in his view, alters our perceptions about
the logic of organizational behavior which rediscovers the significant impor-
tance of firms’ informal social dynamics that have long been “suppressed or
channeled”. He feels that a complexity appraoch to organizations is particu-
larly relevant in view of the recent emphasis in industrialized nations toward
knowledge-based, rather than production-based, economies. A. Sengupta
(I.I.T. Kanpur, India) employs the topological-multifunctional mathematical
language and techniques of non-injective illposedness to formulate the notion
of chanoxity in describing the specifically nonlinear dynamical evolutionary
processesofNature.Non-bijectiveill-posednessisthenaturalmodeofexpres-
sion for chanoxity that aims to focus on the nonlinear interactions generating
dynamical evolution of real irreversible processes. The basic dynamics is con-
sidered to take place in a matter-negmatter kitchen space of Nature which
is inaccessible to both the matter and negmatter components, distinguished
byopposingevolutionarydirectionalarrows.Dynamicalequilibriumisconsid-
eredtoberepresentedbysuchcompetitivelycollaboratinghomeostaticstates
of the matter-negmatter constituents of Nature, modelled as a self-organizing
engine-pump system.
Preface XI
Acknowledgement A project of this magnitude would not have succeeded
without the help and assistance of many individuals and organizations. Fi-
nancial support was provided by Department of Science and Technology and
All India Council for Technical Education, New Delhi, National Board for
Higher Mathematics, Mumbai, and the Department of Mechanical Engineer-
ing IIT Kanpur. It is a great pleasure to acknowledge the very meaningful
participation of Professor Brahma Deo in the organization and conduct of
theWorkshop,theadviceandsuggestionsofProfessorsN.Sathyamurthyand
N. N. Kishore, the infrastructural support provided by IIT Kanpur, and the
assistance of Professor Pradip Sinha. I am also grateful to Professors A. R.
Thakur,Vice-Chancellor,WestBengalUniversityofTechnologyKolkata,and
A. B. Roy of Jadavpur University for their continued support to ChaNoXity
during and after the Workshop.
April 30, 2006 A. Sengupta
Kanpur