Table Of ContentNon-Linear Dynamics and
Fundamental Interactions
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Series II:Mathematics,Physics and Chemistry –Vol.213
Non-Linear Dynamics and
Fundamental Interactions
edited by
Faqir Khanna
University of Alberta,
Edmonton, AB, Canada
and
Davron Matrasulov
Uzbek Academy of Sciences,
Tashkent, Uzbekistan
Published in cooperation with NATO Public Diplomacy Division
Proceedings of the NATO Advanced Research Workshop on
Non-Linear Dynamics and Fundamental Interactions
Tashkent, Uzbekistan
October 10-16, 2004
A C.I.P.Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-3948-4 (PB)
ISBN-13 978-1-4020-3948-5 (PB)
ISBN-10 1-4020-3947-6 (HB)
ISBN-13 978-1-4020-3947-8 (HB)
ISBN-10 1-4020-3949-2 (e-book)
ISBN-13 978-1-4020-3949-2 (e-book)
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PREFACE
This volume contains the papers presented at the NATO Advanced Research
Institute on "Non-Linear Dynamics and Fundamental Interactions" held in Tashkent,
Uzbekistan, from Oct.10-16,2004.
The main objective of the Workshop was to bring together people working in areas
of Fundamental physics relating to Quantum Field Theory, Finite Temperature Field
theory and their applications to problems in particle physics, phase transitions and
overlap regions with the areas of Quantum Chaos. The other important area is
related to aspects of Non-Linear Dynamics which has been considered with the topic
of chaology. The applications of such techniques are to mesoscopic systems,
nanostructures, quantum information, particle physics and cosmology. All this forms
a very rich area to review critically and then find aspects that still need careful
consideration with possible new developments to find appropriate solutions.
There were 29 one-hour talks and a total of seven half-hour talks, mostly by the
students. In addition two round table discussions were organised to bring the
important topics that still need careful consideration. One was devoted to questions
and unsolved problems in Chaos, in particular Quantum Chaos. The other round
table discussion considered the outstanding problems in Fundamental Interactions.
There were extensive discussions during the two hours devoted to each area.
Applications and development of new and diverse techniques was the real focus of
these discussions.
The conference was ably organised by the local committee consisting of D.U.
Matrasulov, M.M. Musakhanov, Kh.Yu. Rakhimov, A.A. Saidov, B.S. Yuldashev
with P.K. Khabibullaev as the chair. A whole cadre of students helped with many
detailed jobs. Our thanks go to Olga V. Karpova for help with the secretarial duties
before, during and after the conference. The task of getting the proceedings ready for
publication was carried out by Olga Karpova and Lee Grimard (University of
Alberta). We thank them for their help.
The conference was held at the Grand Mir hotel in Tashkent. The hotel management
provided every assistance possible to make the conference run smoothly.
Finally we wish to thank NATO Science Program for their financial assistance.
Additional assistance was provided by Science and Technology center of
Uzbekistan, Heat Physics department of Uzbek Academy of Sciences.
Faqir Khanna
Davron Matrasulov
v
CONTENTS
Preface v
Nonlinear Dynamics
1. Heat conduction in one dimensional systems: Fourier law,
chaos, and heat control
1
Guilio Casati and Baowen Li
2. Quantum graphs which sound the same
Talia Shapira and Uzy Smilansky 17
3. Scarred states in strongly coupled quantum systems
31
Bala Sundaram
4. Nonlinear quantum dynamics
Salman Habib 43
5. Signatures of quantum chaos in open chaotic billiards
Almas F. Sadreev and Karl-Fredrik Berggren 57
6. From quantum graphs to quantum random walks
Gregor K. Tanner 69
7. Semiclassics and propagation of coherent states for spin-orbit
coupling problems
89
Jens Bolte
8. Lass of inhomogeneously driven dynamical systems: general
theory, regular and chaotic properties, applications
101
Vladimir N. Damgov
9. Quantum chaos in floppy molecular systems: the licn molecule
Florentino Borondo and R.M. Benito 115
10. Modern aspects of the three-body Coulomb problem
Javier Madroñero and Andreas Buchleitner 129
11. Concerning regularities of particlés motion in the electric and
thermoelectric fields with distributed potential
Victor I. Fedulov 141
vii
12. Theory of relativistic ideal gas for quasi and ordinary particles
Mustakim R. Jumaev 155
13. Finite-temperature quantum billiards
Uktam R. Salomov, D.U. Matrasulov, F.C. Khanna and G.M.
167
Milibaeva
14. Chaotic dynamics of the relativistic kicked rotor
Guljamal M. Milibaeva, D.U. Matrasulov, U.R. Salomov and Bala
173
Sundaram
15. The largest Lyapunov exponents for the relativistic hydrogen-
like atom in a uniform magnetic field
Sanat K. Avazbaev, D.U. Matrasulov and P.K. Khabibullaev 179
Fundamental Interactions
16. Thermal field theory: Algebraic aspects and applications to
confined Systems
Ademir E. Santana, J.M.C. Malbouisson, A.P.C. Malbouisson and
187
F.C. Khanna
17. Thermofield dynamics: Generalized bogoliubov transormations
and applications to Casimir effect
Faqir C. Khanna, J.M.C. Malbouisson and A.E. Santana 215
18. A force from nothing onto nothing: Casimir effect between
bubbles in the Fermi sea
Andreas Wirzba 229
19. Quantum chaos in QCD and hadrons
Harald Markum, Willibald Plessas, Rainer Pullirsch, Bianka Sengl,
and Robert F. Wagenbrunn 243
20. QCD instanton vacuum effective action
Mirzayousuf M. Musakhanov 255
21. New Casimir energy calculations for single cavities
H. Ahmedov, Ismail H. Duru 269
22. Quantum Field Theory for Nonequilibrium Phase Transitions
Sang Pyo Kim 277
23. The(cid:73)(cid:314)(cid:90)(cid:83)0 decay in the Chiral Model
Kamil Nasriddinov, Boris Kuranov, and Nurmukhammad
293
Iskandarov
viii
24. Ginzburg Landau theory of superconductivity: Beyond the post
Gaussian approximation
Abdulla Rakhimov, Jae Hyung Yee and Chul Koo Kim 303
25. Thes-wave(cid:83)d scattering length from (cid:83)d atom using effective
field theory
Bahodir F. Irgaziev and B.A. Fayzullaev 317
26. Spectra of Quarkonia at finite-temperature
Khakim T. Butanov, D.U. Matrasulov, Kh.Yu. Rakhimov and F.C. 329
Khanna
27. Chaotization of the periodically driven quarkonia
Davron Otajanov, Faqir Khanna and Davron Matrasulov 335
28. Quantum chaos at finite temperature
Davron U. Matrasulov, U.R. Salomov, F.C. Khanna and A. Santana 341
Index 347
ix
Nonlinear Dynamics
Heat conduction in one dimensional systems: Fourier
law, chaos, and heat control
Giulio Casati(a,b) and Baowen Li (b)
a) Center for Nonlinear and Complex Systems,
Universita’ degli studi dell’Insubria, Como
Istituto Nazionale di Fisica della Materia,
Unita’ di Como, and Istituto Nazionale di Fisica Nucleare,
sezione di Milano, Milano, Italy
b) Department of Physics, National University of Singapore,
Singapore 117542, Republic of Singapore
Keywords: chaos, heat control
1. Introduction
“It seems there is no problem in modern physics for which there are on
record as many false starts, and as many theories which overlook some
essential feature, as in the problem of the thermal conductivity of non-
conducting crystals” (Peierls, 1961). This statement by R. Peierls goes
back to almost 50 years ago, yet it appears to be still valid. Compared
with charge flow (electric current), much less is known about the heat
flow.
Nevertheless,encouragingresultshavebeenobtainedinrecentyears
(seereviews(Bonettoetal,1961)andthereferencestherein).Forexam-
pleitisnowknownthatinonedimensionalsystemsoftheFermi-Pasta-
Ulam(FPU)type(Leprietal,1998),heatconductionisanomalousand
the coefficient of thermal conductivity κ diverges with the system size
L as κ ∼ L2/5 (when the transverse motion is considered κ ∼ L1/3
(?)). The connection between anomalous conductivity and anomalous
diffusion has been also established(Li and Wang, 2003; Li et al, 2005),
which implies in particular that a subdiffusive system is an insulator
in the thermodynamic limit and a ballistic system is a perfect thermal
conductor, the Fourier law being therefore valid only when phonons
undergo a normal diffusive motion. More profoundly, it has been clar-
ified that exponential dynamical instability is a sufficient(Casati et al,
2005; Alonso et al, 2005) but not a necessary condition for the validity
of Fourier law (Li et al, 2005; Alonso et al, 2002; Li et al, 2003; Li
et al, 2004). These basic studies not only enrich our knowledge of the
fundamental transport laws in statistical mechanics, but also open the
way for applications such as designing novel thermal materials and/or
1
F. Khanna and D. Matrasulov (eds.), Non-Linear Dynamics and Fundamental Interactions, 1–16.
© 2006 Springer. Printed in the Netherlands.
