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From Quasicrystals to More Complex Systems : Les Houches School, February 23 - March 6, 1998 - F. Axel

From Quasicrystals to More Complex Systems

Les Houches School, February 23 - March 6, 1998

By: F. Axel (Editor), F. Denoyer (Editor), J.P. Gazeau (Editor)

Paperback Published: 12th May 2000
ISBN: 9783540674641
Number Of Pages: 375

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This book is a collection of part of the written versions of the Physics Courses given at the Winter School "Order, Chance and Risk: Aperiodic Phenomena from Solid State to Finance" held at the Les Houches Center for Physics, between February 23 and March 6, 1998. The School gathered lecturers and participants from all over the world. On a thematic level, the content of the school can be viewed both as a continuation (aperiodic phenomena in solid state physics) and an extension (mathematical aspects of fmance and economy) of the previous "Beyond Quasicrystals", also held at Les Houches, March 7-18 1994 and published in the same ·series. One of its important goals was to promote in-depth concrete scientific exchanges between theoretical physicists, experimental physicists and mathematicians on the one hand, and on the other hand practitioners of the economico-fmancial sphere and specialists of financial mathematics. Therefore, besides the mathematical tools and concepts at work in theoretical descriptions, relevant experimental data were also presented together with methods allowing their interpretation. As a result of this choice, the School was stimulated by experimentalists and fmancial market operators who joined the theoretical physicists and mathematicians at the conference. The present volume deals with the theoretical and experimental studies on aperiodic solids with long range order, incommensurate phases, quasicrystals, glasses, and more complex systems (fractal, chaotic), while a second volume to appear in the same series is devoted to the finance and economy facet.

Industry Reviews

"The deliberate pedagogical approach of the authors makes this book particularly useful for graduate students and experimentalists wishing to acquaint themselves with the fundamental concepts." (Zeitschrift fur Kristallographie, 217/10, 2002)

Dynamics and Transport Properties of Aperiodic Crystals
Structure and symmetryp. 1
Phononsp. 5
Domain wall motionp. 11
Electronsp. 13
Tensorial propertiesp. 16
Surface effectsp. 17
Transport propertiesp. 19
Concluding remarksp. 20
Diffraction Experiments on Quasicrystalsand Related Phases
Introductionp. 23
Decagonal quasicrystalsp. 24
Generalitiesp. 24
X-ray structure determination of decagonal quasicrystalsp. 24
Decagonal symmetry and twinningp. 30
Tenfold twinning of Al13Fe4 and Al13Fe4-type structurep. 30
Twinning in the structure of decagonal phases and fine structure of diffraction peaksp. 32
Twinning and main characteristic features of diffraction patternsp. 32
Description of microstructures in terms of phason-strain quasicrystalsp. 35
Quasicrystal transformationsp. 41
Icosahedral short range order in glassesp. 42
Conclusionp. 45
Electronic Properties of Quasicrystals. A Comparisonwith Approximant Phases and Disordered Systems
Introductionp. 49
Quasiperiodic orderp. 50
Quasicrystals, crystals and amorphous phasesp. 52
Samples of high structural quality in ternary alloysp. 53
Unexpected physical propertiesp. 54
Conductivity and density of states in quasicrystalsp. 54
Low electrical conductivity values in i-phasesp. 54
Low electronic density of states in quasicrystalsp. 57
Comparison with other metallic alloysp. 59
Scale of conductivity in metallic alloysp. 59
Effect of diffractionp. 60
Quantum interference effects in disordered systemsp. 61
Disordered insulator and Anderson localizationp. 65
Periodicity as an approach to quasiperiodicityp. 65
Approximant phasesp. 66
Experimental electrical conductivity in approximant phasesp. 66
Calculated electronic properties in approximantsp. 68
Low dimensional perfect quasiperiodic modelsp. 69
Quasicrystals as ordered structures of high symmetryp. 72
Pseudo Brillouin zonep. 72
Role of local atomic clustersp. 74
Towards a metal-insulator transition in quasicrystals: Comparison with disordered systemsp. 75
Some experimental evidence for the approach to a metal-insulator transition in quasicrystalsp. 76
Crossing of the metal-insulator transition in i-AlPdRep. 77
Conclusionp. 79
Exact Electron States in 1D (Quasi-) Periodic Arraysof Delta-Potentials
Introduction and scopep. 85
Finite periodic strings at negative energyp. 89
Preview: An energy gauge for crystalsp. 89
Bloch and bound states in a single bandp. 90
The string Snp. 95
Rational Bloch labelsp. 96
Bound states and clusters of the string Snp. 96
Participation numberp. 98
Supercell interpretationp. 99
Large n limitp. 99
Finitequasiperiodic strings at negativeenergyp. 100
Preview: Energy gauge in Fibonacci stringsp. 100
Substitutional systems and their invariantsp. 101
Recursive calculation of the transfer matrixp. 102
Periodic strings at positiveenergyp. 106
The S-matrixp. 107
The S-matrix for the periodic string Snp. 108
Quasiperiodic strings at positive energyp. 110
The Fibonacci-atlasp. 110
Conclusionp. 112
Random Tiling Models for Quasicrystals
Introductionp. 115
Basic definitionsp. 116
Generation of quasicrystalline tilingsp. 117
Randomization of tilingsp. 120
A zoo of tiling modelsp. 121
Mathematics of random tilingsp. 123
Entropy density and phason elastic constantsp. 123
Long-wavelength behavior and stabilityp. 126
Diffractionp. 127
Random tiling resultsp. 128
Monte Carlo simulationp. 128
Combinatoricsp. 129
Transfer matrix methodp. 130
BetheAnsatz methodp. 132
Atomic models forquasicrystalsp. 132
HREM/diffraction-basedp. 133
Models based on realistic interatomic forcesp. 136
Other modelsp. 138
Quasicrystal phase transformationsp. 139
Phason unlockingp. 139
Quasicrystal ⇆ (micro)crystalp. 139
Conclusionsp. 140
Model Sets: A Survey
Introductionp. 145
Model setsp. 147
Geometric sidep. 149
Arithmetic sidep. 150
The icosian model setsp. 150
p-adic model setsp. 152
Analytic sidep. 155
Dynamical systems sidep. 157
Diffractionp. 160
Commentsp. 163
Acceptance Windows Compatiblewith a Quasicrystal Fragment
Introductionp. 167
Notation and auxiliary factsp. 171
Local invariance and the forward growthp. 174
The maximal acceptance windowp. 178
Example: Analysis of two-dimensional quasicrystal datap. 180
Comments and remarksp. 189
Counting Systems with Irrational Basis for Quasicrystals
Introductionp. 195
The set of ¿-integersp. 197
Tau-integer labelling of the Fibonacci chainp. 199
Tau-integer labelling of diffraction patternp. 202
Tau-integer labelling of two-dimensional structuresp. 205
Arithmetics and algebra of the ß-integersp. 211
Acoustic-Like Excitations in Strongly Disordered Media
Introductionp. 219
The case of mass-fractal mediap. 221
The structure of mass fractalsp. 222
The concept of mutually self-similar series of MSSSp. 226
The dynamics of mass fractalsp. 227
The case of glassesp. 234
What is already establishedp. 236
Spectroscopy of acoustic excitations in the terahertz regime - three remarksp. 240
Some studies near the end of acoustic branches in glassesp. 243
Conclusionsp. 249
Intermittent Dynamics and Ageing in Glassy Systems
Introductionp. 261
A simple model: Traps and intermittent dynamicsp. 263
Relation with mode-coupling descriptionsp. 265
Self-induced quenched disorder and open questionsp. 267
A Short Introduction to Ergodic Theoryand Its Applications
Dynamical systemsp. 273
Examplesp. 273
Recurrencep. 277
Ergodic theoremp. 278
Unique ergodicityp. 280
Expected recurrence timep. 281
Spectral properties of dynamical systemsp. 282
The spectrum of a dynamical systemp. 282
Mixingp. 283
Entropy of dynamical systemsp. 284
Isomorphismp. 286
Entropy and Hausdorff dimensionp. 287
Epiloguep. 288
Fractality and the Kinetics of Chaos
Introductionp. 291
Mapping the dynamicsp. 295
Topological zoo (singular zones)p. 297
Self-similar hierarchy of islandsp. 300
Quasi-trapsp. 301
Boundary layer as a quasi-trapp. 303
Fractal and multifractal space-time of kineticsp. 304
Dimension spectrum of the multifractal space-timep. 307
Fractional kineticsp. 309
Conclusionsp. 312
Long-Tailed Distributions in Physics
Introductionp. 315
Fractal timep. 319
Slow relaxationsp. 321
Fractal space processesp. 321
Nonlinear dynamicsp. 324
Distribution of Galaxies: Scaling vs. Fractality
Introductionp. 329
Algebra of point distributionsp. 330
Densities and correlationsp. 330
Counts in cellsp. 332
Scale invariancep. 333
Scaling ofcorrelationsp. 333
The void probabilityp. 333
Scaling of counts in cellsp. 334
Fractalityp. 336
Correlation dimensionp. 336
Hausdorff dimension for occupied cellsp. 337
Renyi indexp. 338
Multifractal dimensionp. 340
Conclusionp. 343
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540674641
ISBN-10: 3540674640
Series: Centre de Physique Des Houches
Audience: General
Format: Paperback
Language: English
Number Of Pages: 375
Published: 12th May 2000
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 2.06
Weight (kg): 0.55

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