Physics of long-range interacting systems:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford Univ. Press
2014
|
| Ausgabe: | 1. ed. |
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | XVI, 410 S. Ill., graph. Darst. |
| ISBN: | 9780199581931 |
Internformat
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| 245 | 1 | 0 | |a Physics of long-range interacting systems |c A. Campa, T. Dauxois, D. Fanelli and S. Ruffo |
| 250 | |a 1. ed. | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | This book deals with an important class of many-body systems: those where the
interaction potential decays slowly for large inter-particle distances; in particular,
systems where the decay is slower than the inverse inter-particle distance raised to
the dimension of the embedding space. Gravitational and Coulomb interactions are
the most prominent examples, however it has become clear that long-range
interactions are more common than previously thought.
A satisfactory understanding of properties, generally considered as oddities only
a couple of decades ago, has now been reached: ensemble inequivalence, negative
specific heat, negative susceptibility, ergodicity breaking, out-of-equilibrium quasi-
stationary-states, anomalous diffusion. The book, intended for Master and PhD
students, tries to gradually acquaint the reader with the subject. The first two parts
describe the theoretical and computational instruments needed to address the study
of both equilibrium and dynamical properties of systems subject to long-range forces.
The third part of the book is devoted to applications of such techniques to the most
relevant examples of long-range systems.
A. Campa is a Senior Scientist at the Health and Technology Department of the Istituto
Superiore di Sanità, Rome.
T. Dauxois is a CNRS Research Director at the Ecole Normale Supérieure de Lyon
D. Fanelli is an Associate Professor in the Department of Physics and Astronomy
at the University of Florence.
S. Ruffo is a Full Professor in the Department of Physics and Astronomy at
the University of Florence.
The book serves as a reference volume in the field of long-range interacting
systems where the various features of these systems are presented in a coherent
way. On the one hand it makes the field easily accessible to newcomers and on the
other hand it provides a guide to the vast literature on the subject which will be
useful to researchers active in the field. It is a welcomed timely contribution.’
David Mukamel, Weizmann Institute
Cover image: Reminiscent of the core -halo picture. Untitled. earth water, willow, blades of grass,
sorb apples, privet beiries. Germany, by Nils Udo. Copyright (1999). Diasec. 1 24x124 cm. used with
the kind permission of the artist.
OXTORD
UNIVERSITY PRESS
ISBN 978-0-19-958193-1
9 780199
58193
www.oup.com
Contents
Part I Static and Equilibrium Properties £}
1 Basics of Statistical Mechanics of Short-Range Interacting Systems 3
1.1 The Microcanonical Ensemble 4
1.2 The Canonical and the Grand-Canonical Ensembles 8
1.2.1 The canonical ensemble 8
1.2.2 The grand-canonical ensemble 11
1.3 Equivalence of Ensembles for Short-Range Interactions 13
1.3.1 Concave functions 14
1.3.2 The Legendre-Fenchel transform (LFT) 15
1.3.3 Stable and tempered potentials 16
1.3.4 Ensemble equivalence 20
1.3.5 Equivalence in presence of phase transition: the Maxwell construction 23
1.4 Lattice Systems 26
1.5 Microstates and Macro states 28
1.6 A Summary of the Most Relevant Points 29
2 Equilibrium Statistical Mechanics of Long-Range Interactions 30
2.1 Non-additivity 30
2.1.1 Definition of long-range interactions 30
2.1.2 Extensivity vs additivity 31
2.1.3 Non-additivity and the lack of convexity in thermodynamic parameters 33
2.1.4 Non-additivity and the canonical ensemble 34
2.2 Ensemble Inequivalence and Negative Specific Heat 35
2.3 An Analytical Solvable Example: The Mean-Field Blume-Emery-
Griffiths (BEG) Model 38
2.3.1 Qualitative remarks 38
2.3.2 The solution in the canonical ensemble 39
2.3.3 The solution in the microcanonical ensemble 42
2.3.4 Inequivalence of ensembles 45
2.4 Entropy and Free Energy Dependence on the Order Parameter 48
2.4.1 Basic definitions 48
2.4.2 Maxwell construction in the microcanonical ensemble 51
2.4.3 Negative susceptibility 54
2.5 The Min-Max Procedure 55
xii Contents
3 The Large Deviations Method and Its Applications 61
3.1 Introduction 61
3.2 The Computation of the Entropy for Long-Range Interacting Systems 62
3.2.1 The method in three steps 62
3.2.2 The computation of the entropy of the different macrostates 63
3.3 The Three-States Potts Model: An Illustration of the Method 65
3.4 The Solution of the BEG Model Using Large Deviations 68
4 Solutions of Mean Field Models 71
4.1 The Hamiltonian Mean-Field (HMF) Model 71
4.1.1 The canonical solution 72
4.1.2 The microcanonical solution 75
4.1.3 The min-max solution 80
4.2 The Generalized XY Model 81
4.2.1 Statistical mechanics via large deviations method 81
4.2.2 Parameter space convexity 85
4.2.3 Phase diagram in the microcanonical ensemble 88
4.2.4 Equilibrium dynamics 91
4.3 The phi-4 Model 93
4.4 The Self-Gravitating Ring (SGR) Model 97
4.4.1 Introduction of the model 97
4.4.2 Inequivalence of ensemble 100
5 Beyond Mean-Field Models 105
5.1 Ising Model 106
5.1.1 Introduction 106
5.1.2 The solution in the canonical ensemble 107
5.1.3 The solution in the microcanonical ensemble 110
5.1.4 Equilibrium dynamics: breaking of ergodicity 113
5.2 a-Ising Model 114
5.3 XY Model with Long- and Short-Range Couplings 118
5.3.1 Introduction 118
5.3.2 Solutions in the canonical and microcanonical ensembles 119
5.3.3 Ergodicity breaking 122
5.4 a-HMF Model 124
5.5 Dipolar Interactions in a Ferromagnet 128
5.5.1 Simplified Hamiltonian in elongated ferromagnets 129
5.5.2 Dynamical effects in layered ferromagnets 133
6 Quantum Long-Range Systems 139
6.1 Introduction 139
6.2 Classical Coulomb Systems 141
6.3 The Problem of Stability of Quantum Coulomb Systems 143
Contents xiii
6.3.1 Systems without exclusion principle 143
6.3.2 Systems with exclusion principle 146
6.4 The Thermodynamic Limit of Coulomb Systems 148
Part II Dynamical Properties
7 BBGKY Hierarchy, Kinetic Theories and the Boltzmann Equation 153
7.1 The BBGKY Hierarchy 154
7.2 The Boltzmann Equation and the Rapid Approach to Equilibrium
due to Collisions 160
7.2.1 The derivation of the Boltzmann equation 160
7.2.2 The H-theorem 165
7.2.3 H-theorem and irreversibility 167
8 Kinetic Theory of Long-Range Systems: Klimontovich, Vlasov
and Lenard-Balescu Equations 169
8.1 Derivation of the Klimontovich Equation 170
8.2 Vlasov Equation: Collisionless Approximation of the Klimontovich
Equation 172
8.3 The Lenard-Balescu Equation 175
8.4 The Boltzmann Entropy and the Mean-Field Approximation 182
8.5 The Kac’s Prescription in Long-Range Systems 183
9 Out-of-Equilibrium Dynamics and Slow Relaxation 185
9.1 Numerical Evidence of Quasi-stationary States 185
9.2 Fokker-Planck Equation for the Stochastic Process of a Single Particle 190
9.3 Long-Range Temporal Correlations and Diffusion 195
9.4 Lynden-Bell’s Entropy 199
9.4.1 The principle 199
9.4.2 Application to the HMF model 201
9.5 Lynden-Bell’s Entropy: Beyond the Single Water-Bag Case Study 206
9.6 The Core-Halo Solution 213
Part HI Applications
10 Gravitational Systems 219
10.1 Equilibrium Statistical Mechanics of Self-Gravitating Systems 221
10.2 Self-Gravitating Systems in Lower Dimensions 226
10.3 From General Relativity to the Newtonian Approximation 234
10.4 The Cosmological Problem 238
10.5 Particle Dynamics in Expanding Coordinates 240
10.6 The Vlasov Equation for an Expanding Universe 242
10.7 From Vlasov-Poisson Equations to the Adhesion Model 243
XIV
Contents
10.8 The One-Dimensionai Expanding Universe 245
10.9 Numerical Simulations in 3D 248
11 Two-Dimensional and Geophysical Fluid Mechanics 250
11.1 Introduction 2 51
11.1.1 Elements of fluid dynamics 251
11.1.2 Illustration of the non-additivity property 254
11.2 The Onsager Point Vortex Model 255
11.2.1 The model 255
11.2.2 Negative temperatures 256
11.2.3 The statistical mechanics approach 257
11.2.4 Deficiencies of the point vortex model 259
11.3 The Robert-Sommeria-Miller Theory for the 2D Euler Equation 260
11.3.1 Introduction 2 60
11.3.2 The two levels approximation 262
11.3.3 The generalization to the infinite number of levels 263
11.3.4 Ensemble inequivalence 2 64
11.4 The Quasi-Geostrophic (QG) Model for Geophysical Fluid Dynamics 264
11.4.1 The quasi-geostrophic model 264
11.4.2 The range of the interaction in the quasi-geostrophic model 267
11.4.3 The statistical mechanics of the quasi-geostrophic model 268
12 Cold Coulomb Systems 270
12.1 Introduction 2 70
12.2 The Main Parameters in Coulomb Systems 271
12.3 A Classification of Coulomb Systems 275
12.4 Strongly Coupled Plasmas 280
12.5 Wigner Crystals 283
13 Hot Plasma 287
13.1 Temperature, Debye Shielding and Quasi-neutrality 287
13.2 Klimontovich’s Approach for Particles and Waves: Derivation of
the Vlasov-Maxwell Equations 291
13.3 The Case of Electrostatic Waves 295
13.4 Landau Damping 297
13.5 Non-linear Landau Damping: An Heuristic Approach 300
13.6 The Asymptotic Evolution: BGK Modes 302
13.7 Case-Van Kampen Modes 306
14 Wave-Particles Interaction 308
14.1 Hamiltonian Formulation of Vlasov-Maxwell Equations 308
14.2 Interaction between a Plane Wave and a Co-propagating Beam of Particles 309
14.2.1 Canonical formulation of the Hamiltonian and reduction
to a one-dimensional system 309
Contents XV
14.2.2 Studying the dynamics in the particles-field phase frame 312
14.2.3 Resonance condition and high-gain amplification 314
14.3 Alternative Derivation of the Wave-Particles Hamiltonian from
the Microscopic Equations 316
14.4 Free Electron Lasers (FEL) 31S
14.4.1 Introduction 318
14.4.2 Storage ring FEL 318
14.4.3 Single-pass FEL 322
14.4.4 On the dynamical evolution of the single-pass FEL 326
14.5 Large Deviations Method Applied to the Colson-Bonifacio Model 329
14.5.1 Equilibrium solution of the Colson-Bonifacio model 329
14.5.2 Mapping the Colson-Bonifacio model onto HMF 332
14.6 Derivation of the Lynden-Bell Solution 333
14.7 Comparison with Numerical Results 336
14.7.1 The single-wave model 336
14.7.2 The case of two harmonics 339
14.8 Analogies with the Traveling Wave Tube 342
14.9 Collective Atomic Recoil Laser (CARL) 344
15 Dipolar Systems 349
15.1 Introduction 349
15.2 The Demagnetizing Field 353
15.2.1 Uniformly magnetized bodies 354
15.2.2 The magnetostatic energy 358
15.3 The Thermodynamic Limit for Dipolar Media 359
15.3.1 Some useful relations 360
15.3.2 The lower bound 362
15.3.3 The upper bound 363
15.3.4 The thermodynamic limit 366
15.3.5 Further remarks 368
15.4 The Physical Consequences of the Existence
of the Thermodynamic Limit 369
15.4.1 The computation of the dipolar energy 370
15.4.2 The large-scale structure of the magnetization profile:
domains and curling 373
15.4.3 Isotropic and anisotropic ferromagnets 374
15.5 Experimental Studies of Dipolar Interactions 376
15.5.1 Spin ice systems 376
15.5.2 2D optical lattices 377
15.5.3 Bose-Einstein condensates 378
Appendix A Features of the Main Models Studied throughout the Book 381
Appendix В Evaluation of the Laplace Integral Outside the
Analyticity Strip 382
xvi Contents
Appendix C The Equilibrium Form of the One-Particle
Distribution Function in Short-Range Interacting Systems 384
Appendix D The Differential Cross-Section of a Binary Collision 387
Appendix E Autocorrelation of the Fluctuations of the One-
Particle Density 390
Appendix F Derivation of the Fokker-Planck Coefficients 392
References 397
Index 407
|
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| id | DE-604.BV041978683 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:09:45Z |
| institution | BVB |
| isbn | 9780199581931 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027421186 |
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| spelling | Physics of long-range interacting systems A. Campa, T. Dauxois, D. Fanelli and S. Ruffo 1. ed. Oxford Oxford Univ. Press 2014 XVI, 410 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Statistische Physik (DE-588)4057000-9 gnd rswk-swf Vielteilchensystem (DE-588)4063491-7 gnd rswk-swf Hydrodynamik (DE-588)4026302-2 gnd rswk-swf Kinetische Theorie (DE-588)4030669-0 gnd rswk-swf Vielteilchensystem (DE-588)4063491-7 s Statistische Physik (DE-588)4057000-9 s Kinetische Theorie (DE-588)4030669-0 s Hydrodynamik (DE-588)4026302-2 s DE-604 Campa, Alessandro Sonstige oth Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027421186&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027421186&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Physics of long-range interacting systems Statistische Physik (DE-588)4057000-9 gnd Vielteilchensystem (DE-588)4063491-7 gnd Hydrodynamik (DE-588)4026302-2 gnd Kinetische Theorie (DE-588)4030669-0 gnd |
| subject_GND | (DE-588)4057000-9 (DE-588)4063491-7 (DE-588)4026302-2 (DE-588)4030669-0 |
| title | Physics of long-range interacting systems |
| title_auth | Physics of long-range interacting systems |
| title_exact_search | Physics of long-range interacting systems |
| title_full | Physics of long-range interacting systems A. Campa, T. Dauxois, D. Fanelli and S. Ruffo |
| title_fullStr | Physics of long-range interacting systems A. Campa, T. Dauxois, D. Fanelli and S. Ruffo |
| title_full_unstemmed | Physics of long-range interacting systems A. Campa, T. Dauxois, D. Fanelli and S. Ruffo |
| title_short | Physics of long-range interacting systems |
| title_sort | physics of long range interacting systems |
| topic | Statistische Physik (DE-588)4057000-9 gnd Vielteilchensystem (DE-588)4063491-7 gnd Hydrodynamik (DE-588)4026302-2 gnd Kinetische Theorie (DE-588)4030669-0 gnd |
| topic_facet | Statistische Physik Vielteilchensystem Hydrodynamik Kinetische Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027421186&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027421186&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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