Fractals, Diffusion and Relaxation in Disordered Complex Systems, Part B

743 Pages·2006·6.842 MB·English

by  William T Coffey, Yuri P Kalmykov, John Wiley & Sons., Wiley InterScience (Online service)

FRACTALS, DIFFUSION, AND RELAXATION
IN DISORDERED COMPLEX SYSTEMS
ASPECIALVOLUMEOFADVANCES INCHEMICAL PHYSICS
VOLUME133
PARTB
EDITORIALBOARD
BRUCE J. BERNE, Department of Chemistry, Columbia University, New York,
New York, U.S.A.
KURT BINDER, Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Mainz, Mainz,
Germany
A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State
University, University Park, Pennsylvania, U.S.A.
DAVID CHANDLER, Department of Chemistry, University of California, Berkeley,
California, U.S.A.
M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford,
U.K.
WILLIAM T. COFFEY, Department of Microelectronics and Electrical Engineering,
Trinity College, University of Dublin, Dublin, Ireland
F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison,
Wisconsin, U.S.A.
ERNEST R. DAVIDSON, Department of Chemistry, Indiana University, Bloomington,
Indiana, U.S.A.
GRAHAM R. FLEMING, Department ofChemistry,Universityof California, Berkeley,
California, U.S.A.
KARL F. FREED, The James Franck Institute, The University of Chicago, Chicago,
Illinois, U.S.A.
PIERREGASPARD, CenterforNonlinearPhenomenaandComplexSystems,Brussels,
Belgium
ERIC J. HELLER, Institute for Theoretical Atomic and Molecular Physics, Harvard-
Smithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A.
ROBINM.HOCHSTRASSER, DepartmentofChemistry,TheUniversityofPennsylvania,
Philadelphia, Pennsylvania, U.S.A.
R.KOSLOFF, TheFritzHaberResearchCenterforMolecularDynamicsandDepartment
ofPhysicalChemistry,TheHebrewUniversityofJerusalem,Jerusalem,Israel
RUDOLPHA.MARCUS, DepartmentofChemistry,CaliforniaInstituteofTechnology,
Pasadena, California, U.S.A.
G. NICOLIS, Center for Nonlinear Phenomena and Complex Systems, Universite´
Libre de Bruxelles, Brussels, Belgium
THOMAS P. RUSSELL, Department of Polymer Science, University of Massachusetts,
Amherst, Massachusetts, U.S.A.
DONALD G. TRUHLAR, Department of Chemistry, University of Minnesota,
Minneapolis, Minnesota, U.S.A.
JOHN D. WEEKS, Institute for Physical Science and Technology and Department
of Chemistry, University of Maryland, College Park, Maryland, U.S.A.
PETERG.WOLYNES, DepartmentofChemistry,UniversityofCalifornia,SanDiego,
California, U.S.A.
FRACTALS, DIFFUSION,
AND RELAXATION IN
DISORDERED COMPLEX
SYSTEMS
ADVANCES INCHEMICAL PHYSICS
VOLUME133
PARTB
Edited By
WILLIAM T. COFFEYAND YURI P. KALMYKOV
Series Editor
STUARTA. RICE
DepartmentofChemistry
and
TheJamesFranckInstitute
TheUniversityofChicago
Chicago,Illinois
ANINTERSCIENCEPUBLICATION
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ISBN-13978-0-470-04607-4(Set)
ISBN-100-470-04607-4(Set)
ISBN-13978-0-471-72507-7(PartA)
ISBN-100-471-72507-2(PartA)
ISBN-13978-0-471-72508-4(PartB)
ISBN-100-471-72508-0(PartB)
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CONTRIBUTORS TO VOLUME 133
ELI BARKAI, Department of Chemistry and Biochemistry, Notre Dame
University,NotreDame,Indiana46566,USA;andDepartmentofPhysics,
Bar Ilan University, Ramat Gan 52900, Israel
ALEXANDER BRODIN, Experimentalphysik II, Universita¨t Bayreuth, D 95440
Bayreuth, Germany
THOMAS BLOCHOWICZ, Institute fur Festko¨rperphysik, Technische Universita¨t
Darmstadt, D 64289 Darmstadt, Germany
SIMONE CAPACCIOLI, Dipartimento di Fisica and INFM, Universita` di Pisa,
I-56127, Pisa, Italy; and CNR-INFM Center ‘‘SOFT: Complex Dynamics
in Structured Systems,’’ Universita` di Roma ‘‘La Sapienza,’’ I-00185
Roma, Italy
RICCARDO CASALINI, Naval Research Laboratory, Washington, DC 20375, USA;
and Chemistry Department, George Mason University, Fairfax, Virginia
20030, USA
ALEKSEIV.CHECHKIN,InstituteforTheoreticalPhysics,NationalScienceCenter,
Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine
WILLIAM T. COFFEY, Department of Electronic and Electrical Engineering,
School of Engineering, Trinity College, Dublin 2 Ireland
YURI FELDMAN, Department of Applied Physics, The Hebrew University of
Jerusalem, Jerusalem 91904, Israel
VSEVOLD Y. GONCHAR, Institute for Theoretical Physics, National Science
Center, Kharkov Institute of Physics and Technology, Kharkov 61108,
Ukraine
PAOLO GRIGOLINI, Department of Physics, University of North Texas, Denton,
Texas, 76203 USA; and Department of Physics, University of Pisa, Pisa,
Italy
YURI P. KALMYKOV, Laboratoire de Mathe´matiques et Physique des Syste`mes,
Universite de Perpignan, 66860 Perpignan Cedex, France
JOSEPH KLAFTER, School of Chemistry, Tel Aviv University, Tel Aviv 69978,
Israel
FRIEDRICH KREMER, Universita¨t Leipzig, Fakultat fu¨r Physik und Geowis-
senschaften, 04103 Leipzig, Germany
v
vi contributors
MASARU KUNO, Department of Chemistry and Biochemistry, Notre Dame
University,NotreDame,Indiana46566,USA;andDepartmentofPhysics,
Bar Ilan University, Ramat Gan 52900, Israel
GENNADY MARGOLIN, Department of Chemistry and Biochemistry, Notre Dame
University, Notre Dame, Indiana 46556, USA
RALF METZLER, NORDITA–Nordic Institute for Theoretical Physics, DK-2100
Copenhagen Danish Denmark
KIA L. NGAI Naval Research Laboratory, Washington, DC 20375, USA
VITALY V. NOVIKOV, Odessa National Polytechnical University, 65044 Odessa,
Ukraine
MARIAN PALUCH, Institute of Physics, Silesian University, 40-007 Katowice,
Poland
NOE´LLE POTTIER, Matie`re et Syste`mes Complexes, UMR 7057 CNRS and
Universite´ Paris 7—Denis Diderot, 75251 Paris Cedex 05, France
VLADIMIRPROTASENKO,DepartmentofChemistryandBiochemistry,NotreDame
University,NotreDame,Indiana46566,USA;andDepartmentofPhysics,
Bar Ilan University, Ramat Gan 52900, Israel
ALEXANDER PUZENKO, Department of Applied Physics, The Hebrew University
of Jerusalem, Jerusalem 91904, Israel
C. M. ROLAND, Naval Research Laboratory, Washington, DC 20375, USA
ERNST A. RO¨SSLER, Experimentalphysik II, Universita¨t Bayreuth, D 95440
Bayreuth, Germany
YAROSLAV RYABOV, Department of Applied Physics, The Hebrew University of
Jerusalem, Jerusalem 91904, Israel. Maryland Center of Biomolecular
Structure and Organization, University of Maryland, College Park,
Maryland 20742-3360, USA
ANATOLI SERGHEI, Fakultat fur Physik und Geowissenschaften, Universita¨t
Leipzig, 04103 Leipzig, Germany
SERGEY V. TITOV, Institute of Radio Engineering and Electronics of the Russian
Academy of Seciences, Fryazino, Moscow Region, 141190, Russian
Federation
BRUCE J. WEST, Mathematical & Information Sciences Directorate, U.S. Army
Research Office, Research Triangle Park, North Carolina 27709, USA
INTRODUCTION
Fewofuscananylongerkeepupwiththefloodofscientificliterature,even
in specialized subfields. Any attempt to do more and be broadly educated
with respect to a large domain of science has the appearance of tilting at
windmills.Yetthesynthesisofideasdrawnfromdifferentsubjectsintonew,
powerful, general concepts is as valuable as ever, and the desire to remain
educated persists in all scientists. This series, Advances in Chemical
Physics,is devoted to helpingthereader obtain generalinformationabout a
wide variety of topics in chemical physics, a field that we interpret very
broadly. Our intent is to have experts present comprehensive analyses of
subjects of interest and to encourage the expression of individual points of
view. We hope that this approach to the presentation of an overview of a
subjectwillbothstimulatenewresearchandserveasapersonalizedlearning
text for beginners in a field.
STUARTA. RICE
vii
PREFACE
Fractals, Diffusion, and Relaxation in Disordered Complex Systems, which
is the subject of the present anthology, may be said to have evolved in two
stages: (1) in the course of conversations with Stuart Rice during a
remarkably pleasant lunch at the University of Chicago following the
Indianapolis meeting of the American Physical Society in March 2002 and
(2) following the Royal Irish Academy Conference on Diffusion and
RelaxationinDisorderedFractalSystemsheldinDublininSeptember2002
[1]. During each of these meetings, the necessity of reviewing the progress
bothexperimentalandtheoreticalwhichhasbeenmadeinourunderstanding
of physical systems with relaxation differing substantially from exponential
behavior was recognized. Furthermore, it was considered that the Advances
in Chemical Physics, in line with its stated aspirations and with its wide
circulation, would provide an ideal means of attaining this goal.
For the best part of three centuries the fractional calculus constituted a
subject area mainly of interest to mathematicians. Indeed many great
mathematicians such as Leibniz, L’Hoˆpital, Euler, Fourier, Abel, Liouville,
Weierstrass, Riemann, Letnikov, Wiener, Le´vy, and Hardy, to name but a
few,havecontributedtoitsdevelopment(forahistoricalsurveyseeRef.2).
In contrast, applications of fractional calculus in other branches of science
have appeared only sporadically—for example, the application to the
propagation of disturbances on transmission lines in the context of
Heaviside’s operational calculus and Kohlrausch’s stretched exponential
decaylaw[2,14].However,thesituationradicallychangedtowardtheendof
the last century following the appearance of the famous books of Benoit
Mandelbrot on fractals [3]. Thus, over the past few decades, the fractional
calculushas nolongerbeenrestrictedto therealmofpuremathematics and
probability theory [2,4]. Indeed many scientists have discovered that the
behavior of a variety of complex systems (such as glasses, liquid crystals,
polymers,proteins,biopolymers,livingorganisms,orevenecosystems)may
besuccessfullydescribedbyfractionalcalculus;thusitappearsthatcomplex
systems governed by fractional differential equations play a dominant role
in both the exact and life sciences [5]. In particular in the context of
applications in physics and chemistry, the fractional calculus allows one to
describe complex systems exhibiting anomalous relaxation behavior in
much the same way as the normal relaxation of simple systems [6].
Examplesincludechargetransportinamorphoussemiconductors,thespread
ix
x preface
of contaminants in underground water, relaxation in polymer systems, and
tracer dynamics in both polymer networks and arrays of convection rolls,
and so on [6]. In general, the diffusion and relaxation processes in such
complex systems no longer follow Gaussian statistics so that the temporal
evolution of these systems deviates from the corresponding standard laws
(wherethemean-squaredisplacementofaparticleisproportionaltothetime
between observations) for normal diffusion such as exhibited by classical
Brownian particles. Furthermore, following the development for complex
systems of higher-order experimental resolutions or via a combination of
different probe techniques, the deviations from the classical diffusion and
relaxation laws have become ever more apparent. Thus the ever larger data
windows that are becoming accessible bring ever more refinement to the
experimental data [5], with the result that fractional diffusion and kinetic
equations have become extremely powerful tools for the description of
anomalous relaxation and diffusion processes in such systems. In the present
anthology we have tried to present a comprehensive account of the present
stateof the subject.Itisobvious,however, thatwe cannot surveycompletely
such an enormous area of modern research, and inevitably many important
topics will have beenomitted.Inordertoremedythisdefect,weremarkthat
the interested reader can find additional information concerning anomalous
diffusion and relaxation and applications of fractional calculus in physics,
chemistry,biology,radioengineering,andsoon,invariousreviewarticlesand
books, a selection of which is given in Refs. 5–23.
Roughly speaking, the contents of the two-volume anthology may be
divided into four experimental and seven theoretical chapters that may be
described as follows.
Chapter 1, ‘‘Dielectric Relaxation Phenomena in Complex Materials,’’ by
Y. Feldman, A. Puzenko, and Y. Ryabov, concerns dielectric spectroscopy
studies of the structure, dynamics, and macroscopic behavior of materials,
which may broadly be described by the generic term complex systems.
Complex systems constitute an almost universal class of materials including
associated liquids, polymers, biomolecules, colloids, porous materials, doped
ferroelectric crystals, and so on. These systems are characterized by a new
‘‘mesoscopic’’lengthscale,intermediatebetweenmolecularandmacroscopic.
The mesoscopic structures of complex systems typically arise from
fluctuations or competing interactions and exhibit a rich variety of static
anddynamicbehavior.Thisgrowingfieldisinterdisciplinary;itcomplements
solid-stateandstatisticalphysics,anditoverlapsconsiderablywithchemistry,
chemicalengineering,materialsscience,andevenbiology.Acommontheme
in complex systems is that while such materials are disordered on the
molecular scale and homogeneous on the macroscopic scale, they usually
possessacertaindegreeoforderonaintermediate,ormesocopic,scaledueto
preface xi
the delicate balance of interaction and thermal effects. The authors
demonstrate how dielectric spectroscopy studies of complex systems can be
appliedtodetermineboththeirstructuresanddynamics,howtheybotharise,
and how both may influence the macroscopic behavior.
Theglass transition is an unsolved problem of condensed mater physics.
Thisquestionisaddressedinchapter2byT.Blochowicz,A.Brodin,andE.
Ro¨ssler, entitled ‘‘Evolution of the Dynamic Susceptibility in Supercooled
Liquids and Glasses.’’ The emergence of the mode coupling theory of the
glasstransitionhaspromptedthecompilationofalargebodyofinformation
on the glass transition phenomenon as well as on the glassy state that is
reviewed in this contribution. Thus this chapter focuses on describing the
evolution of the dynamic susceptibility; that is, its characteristic changes
while supercooling a molecular liquid. The authors provide information on
therelevantmoleculardynamics,andacomparisonbetweenexperimentand
theory is given. The phenomenon is essentially addressed from an
experimental point of view, by simultaneously discussing the results from
three different probe techniques, namely quasi-elastic light scattering,
dielectric spectroscopy, and nuclear magnetic resonance spectroscopy. The
application of each of the three methods allows one to investigate the
dynamics in the 0- to 1-THz frequency range. The crossover from liquid
dynamics at the highest temperatures to glassy dynamics at moderate
temperatures as well as the crossover to solid-state behavior at the lowest
temperatures near the glass transition temperature, is described in detail.
In addition, some remarks on the evolution of the susceptibility down to
cryogenic temperatures are given. The lesson to be drawn from this con-
tributionisthatanunderstandingofthedynamicsofdisorderedsystemscan
only be achieved by joint application of the various techniques covering a
large frequency range.
In many complex systems such as glasses, polymers, and proteins,
temporal evolutions differ as we have seen from the conventional
exponential decay laws (and are often much slower). Very slowly relaxing
systems remain out of equilibrium over very long times, and they display
aging effects so that the time scale of response and correlation functions
increases with the age of the system (i.e., the time elapsed since its
preparation):Oldersystemsrelaxmoreslowlythanyoungerones.Chapter3
by N. Pottier, entitled ‘‘Slow Relaxation, Anomalous Diffusion, and Aging
in Equilibrated or Nonequilibrated Environments,’’ describes recent
developments in the physics of slowly relaxing out of equilibrium systems.
Questions specifically related to out-of-equilibrium dynamics, such as (1)
agingeffectsand(2)theirdescriptionbymeansofaneffectivetemperature,
are discussed in the framework of a simple model. A system well adapted
to the analysis of these concepts is a diffusing particle in contact with an