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Large Deviations in Physics: The Legacy of the Law of Large Numbers [Pehme köide]

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  • Formaat: Paperback / softback, 314 pages, kõrgus x laius: 235x155 mm, kaal: 4978 g, 18 Illustrations, color; 40 Illustrations, black and white; XIV, 314 p. 58 illus., 18 illus. in color., 1 Paperback / softback
  • Sari: Lecture Notes in Physics 885
  • Ilmumisaeg: 06-Jun-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642542506
  • ISBN-13: 9783642542503
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Large Deviations in Physics: The Legacy of the Law of Large NumbersSuurem pilt
  • Formaat: Paperback / softback, 314 pages, kõrgus x laius: 235x155 mm, kaal: 4978 g, 18 Illustrations, color; 40 Illustrations, black and white; XIV, 314 p. 58 illus., 18 illus. in color., 1 Paperback / softback
  • Sari: Lecture Notes in Physics 885
  • Ilmumisaeg: 06-Jun-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642542506
  • ISBN-13: 9783642542503
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783642542503.html
Märksõnad:
This book reviews the basic ideas of the Law of Large Numbers with its consequences to the deterministic world and the issue of ergodicity. Applications of Large Deviations and their outcomes to Physics are surveyed. The book covers topics encompassing ergodicity and its breaking and the modern applications of Large deviations to equilibrium and non-equilibrium statistical physics, disordered and chaotic systems, and turbulence.

Ergodicity A Basic Concept.- Large Deviations in Statistical Mechanics: Rigorous and Non-Rigorous.- Large Deviation Techniques for Long-Range Interactions.- Fluctuation-Dissipation and Fluctuation Relations: From Equilibrium to Nonequilibrium Phenomena and Back.- Stochastic Fluctuations in Deterministic Systems.- Large Deviation and Disordered Systems.- Large Deviations in Turbulence.- Ergodicity Breaking Challenges Monte Carlo Methods.- Anomalous Diffusion: Deterministic and Stochastic Perspectives.- The Use of Fluctuation Relations for the Analysis of Free-Energy Landscapes.
1 From the Law of Large Numbers to Large Deviation Theory in Statistical Physics: An Introduction 1(28)
Fabio Cecconi
Massimo Cencini
Andrea Puglisi
Davide Vergni
Angelo Vulpiani
1.1 Introduction
1(2)
1.2 An Informal Historical Note
3(10)
1.2.1 Law of Large Numbers and Ergodicity
4(5)
1.2.2 Central Limit Theorems
9(2)
1.2.3 Large Deviation Theory
11(2)
1.3 LDT for the Sum and Product of Random Independent Variables
13(4)
1.3.1 A Combinatorial Example
13(2)
1.3.2 Product of Random Variables
15(2)
1.4 Large Deviation Theory: Examples From Physics
17(13)
1.4.1 Energy Fluctuations in the Canonical Ensemble
17(1)
1.4.2 Multiplicative Cascade in Turbulence
18(1)
1.4.3 Chaotic Systems
19(2)
1.4.4 Disordered Systems
21(3)
1.4.5 Entropy Production in Markov Processes
24(2)
References
26(3)
2 Ergodicity: How Can It Be Broken? 29(42)
Giancarlo Benettin
Roberto Livi
Giorgio Parisi
2.1 The Ergodic Hypothesis
30(6)
2.1.1 The Fundamental Physical Ideas
30(3)
2.1.2 A Well-Posed Mathematical Setting
33(3)
2.2 Ergodicity Breaking
36(17)
2.2.1 Ergodicity Breaking in the Fermi-Pasta-Ulam Model at Low Energies
37(10)
2.2.2 Ergodicity Breaking Induced by Breather States in the Discrete Nonlinear Schr8dinger Equation
47(6)
2.3 Ergodicity Breaking at Equilibrium
53(9)
2.3.1 General Concepts
53(1)
2.3.2 The Gibbs States
54(6)
2.3.3 The Local DRL Equilibrium States
60(2)
2.4 Glassy System
62(6)
2.4.1 A Heuristic Construction: Finite Volume States
62(3)
2.4.2 The Case of Many States
65(3)
References
68(3)
3 Large Deviations in Stationary States, Especially Nonequilibrium 71(22)
Giovanni Jona-Lasinio
3.1 Introduction
71(2)
3.2 Assumptions
73(1)
3.3 The Fundamental Formula
74(4)
3.4 Time Reversal and Its Consequences
78(2)
3.5 Long Range Correlations
80(1)
3.6 Fluctuations of the Current and Dynamical Phase Transitions
81(2)
3.7 Universality in Current Fluctuations and Other Results
83(1)
3.8 Nonequilibrium Phase Transitions
84(1)
3.9 Large Deviations for Reaction-Diffusion Systems
85(1)
3.10 Thermodynamic Interpretation of the Large Deviation Functional
85(4)
3.11 Concluding Remarks and Additional References
89(1)
References
90(3)
4 Fluctuation-Dissipation and Fluctuation Relations: From Equilibrium to Nonequilibrium and Back 93(42)
Paolo Adamo
Roman Belousov
Lamberto Rondoni
4.1 Concise History
93(2)
4.2 The Brownian Motion and the Langevin Equation
95(4)
4.3 The Fluctuation-Dissipation Relation
99(4)
4.4 Evolution of Probability Distributions
103(5)
4.4.1 Ergodicity and Mixing
107(1)
4.5 Linear Response
108(4)
4.6 Onsager-Machlup: Response from Small Deviations
112(6)
4.7 Fluctuation Relations: Response from Large Deviations
118(9)
4.7.1 The Gallavotti-Cohen Approach
119(2)
4.7.2 Fluctuation Relations for the Dissipation Function
121(4)
4.7.3 Green-Kubo Relations
125(1)
4.7.4 Jarzynski Equality
126(1)
4.8 t-Mixing and General Response Theory
127(2)
4.9 Concluding Remarks
129(2)
References
131(4)
5 Large Deviations in Disordered Spin Systems 135(26)
Andrea Crisanti
Luca Leuzzi
5.1 Some General Results on Large Deviations
135(7)
5.1.1 An Example: The Mean Field Ising Model
139(3)
5.2 Sample-to-Sample Free Energy Fluctuations and Replica Trick
142(13)
5.2.1 Random Ising Chain
143(2)
5.2.2 Replica Trick
145(2)
5.2.3 Replicas in the Random Ising Chain
147(1)
5.2.4 From Small to Large Deviations
148(3)
5.2.5 Random Directed Polymer
151(1)
5.2.6 Mean-Field Spin-Glass
151(2)
5.2.7 Positive Large Deviations
153(1)
5.2.8 Spherical Spin-Glass Model and Gaussian Random Matrices
154(1)
5.3 Sample-to-Sample Fluctuation of the Overlap
155(4)
5.3.1 Back to the Replicated Random Ising Chain
156(2)
5.3.2 Random Field Ising Model
158(1)
5.4 A Final Word
159(1)
References
159(2)
6 Large Deviations in Monte Carlo Methods 161(32)
Andrea Pelissetto
Federico Ricci-Tersenghi
6.1 Introduction
161(2)
6.2 Data Reweighting
163(5)
6.3 Multiple Histogram Method
168(4)
6.4 Umbrella Sampling and Simulated Tempering
172(6)
6.4.1 Umbrella Sampling
172(1)
6.4.2 Simulated Tempering
173(1)
6.4.3 Equivalence of Simulated Tempering and Umbrella Sampling
174(4)
6.5 Generalizing the Umbrella Method: Multicanonical Sampling
178(6)
6.6 Parallel Tempering
184(6)
6.6.1 General Considerations
184(2)
6.6.2 Some General Rigorous Results
186(1)
6.6.3 Optimal Choice of Temperatures
187(2)
6.6.4 Improving Parallel Tempering
189(1)
6.7 Conclusions
190(1)
References
190(3)
7 Large Deviations Techniques for Long-Range Interactions 193(28)
Aurelio Patelli
Stefano Ruffo
7.1 Long-Range Interactions
193(5)
7.2 Some Useful Results of Large Deviations Theory
198(1)
7.3 Thermodynamic Functions From Large Deviations Theory
199(3)
7.4 Applications
202(14)
7.4.1 Three-States Potts Model
203(2)
7.4.2 The Blume-Capel Model
205(3)
7.4.3 A System with Continuous Variables: The XY Model
208(4)
7.4.4 Negative Susceptibility: 44 Model
212(3)
7.4.5 The Free Electron Laser
215(1)
7.5 The Min-Max Procedure and a Model with Short and Long-Range Interactions
216(3)
7.6 Conclusions and Perspectives
219(1)
References
219(2)
8 Large Deviations of Brownian Motors 221(22)
Alessandro Sarracino
Dario Villamaina
8.1 Introduction
221(3)
8.2 Nonequilibrium Fluctuations and Brownian Motors
224(7)
8.2.1 Kinetic Ratchets
225(2)
8.2.2 Molecular Motors
227(4)
8.3 Experiments in Granular Systems
231(7)
8.3.1 Velocity Fluctuations of a Self-Propelled Polar Particle
232(2)
8.3.2 Asymmetric Rotor in a Granular Gas
234(4)
8.4 Conclusions
238(1)
References
239(4)
9 Stochastic Fluctuations in Deterministic Systems 243(20)
Antonio Politi
9.1 Introduction
243(2)
9.2 Kolmogorov-Sinai Entropy
245(5)
9.3 Lyapunov Exponents
250(4)
9.3.1 Pesin Relation
253(1)
9.4 Non Hyperbolicity
254(3)
9.4.1 A 3d Map
256(1)
9.5 Space-Time Chaos
257(2)
9.6 Stable Chaos
259(2)
References
261(2)
10 Anomalous Diffusion: Deterministic and Stochastic Perspectives 263(32)
Roberto Artuso
Raffaella Burioni
10.1 Introduction
263(1)
10.2 Stochastic Anomalous Transport
264(17)
10.2.1 Moments and Scaling
264(3)
10.2.2 A Few Observations About Levy Stable Laws
267(2)
10.2.3 An Extension: Continuous Time Random Walks
269(3)
10.2.4 Topological Effects in Subdiffusion: Weak Anomalous Diffusion and Random Walks on Graphs
272(5)
10.2.5 Topological Effects in Superdiffusion: Strong Anomalous Diffusion and Quenched Levy Walks
277(4)
10.3 Deterministic Anomalous Transport
281(3)
10.3.1 A Brief Tour of Intermittency
281(3)
10.4 Chain of Intermittent Maps
284(5)
10.4.1 Subdiffusion
287(1)
10.4.2 Superdiffusion
287(2)
10.5 Deterministic vs Stochastic Approximation
289(2)
10.6 A Final Warning
291(1)
References
291(4)
11 Large Deviations in Turbulence 295(16)
Guido Boffetta
Andrea Mazzino
11.1 Introduction
295(1)
11.2 Global Scale Invariance and Kolmogorov Theory
296(3)
11.3 Accounting for the Fluctuations: The Multifractal Model
299(8)
11.3.1 The Statistics of Velocity Gradient
302(1)
11.3.2 The Statistics of Acceleration
303(2)
11.3.3 Multiplicative Processes for the Multifractal Model
305(2)
11.4 Fluctuations of the Energy Dissipation Rate
307(2)
11.5 Conclusions
309(1)
References
309(2)
Index 311