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E-raamat: Generalised Thermostatistics

  • Formaat: PDF+DRM
  • Ilmumisaeg: 07-Feb-2011
  • Kirjastus: Springer London Ltd
  • Keel: eng
  • ISBN-13: 9780857293558
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Generalised ThermostatisticsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 07-Feb-2011
  • Kirjastus: Springer London Ltd
  • Keel: eng
  • ISBN-13: 9780857293558
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The domain of non-extensive thermostatistics has been subject to intensive research over the past twenty years and has matured significantly. Generalised Thermostatistics cuts through the traditionalism of many statistical physics texts by offering a fresh perspective and seeking to remove elements of doubt and confusion surrounding the area.

The book is divided into two parts - the first covering topics from conventional statistical physics, whilst adopting the perspective that statistical physics is statistics applied to physics. The second developing the formalism of non-extensive thermostatistics, of which the central role is played by the notion of a deformed exponential family of probability distributions.

Presented in a clear, consistent, and deductive manner, the book focuses on theory, part of which is developed by the author himself, but also provides a number of references towards application-based texts.

Written by a leading contributor in the field, this book will provide a useful tool for learning about recent developments in generalized versions of statistical mechanics and thermodynamics, especially with respect to self-study. Written for researchers in theoretical physics, mathematics and statistical mechanics, as well as graduates of physics, mathematics or engineering. A prerequisite knowledge of elementary notions of statistical physics and a substantial mathematical background are required.



Presented in a clear, consistent, and deductive manner, this book offers a fresh perspective on statistical physics focused on theory, part of which is developed by the author himself, and goes on to provide numerous references to application-based texts.

Arvustused

From the reviews:

This book is written in such a manner that it presents some aspects of generalized thermostatics by a very wide context of considerations. It is introduced in a very interesting way, leading the reader to more and more general approaches. Step-by-step, the ideas for different generalizations are presented. Each chapter of the book ends with some problems intended for a better understanding of the presented topics. At the end of the book, the reader can find the solutions to these problems. (Dominik Strzaka, Zentralblatt MATH, Vol. 1231, 2012)

This book addresses fundamental issues of thermostatistics from the point of view of statistics applied to physics. Many subtle aspects of the theory that are not usually emphasised in standard textbooks are discussed, and the reader can benefit from a deeper understanding of the subject. The book can be used in an advanced course for graduate students, and it is also useful for researchers in statistical physics, since it deals with many aspects of current developments. (Ernesto P. Borges, Mathematical Reviews, Issue 2012 h)

Part I Parameter Estimation
1 Probability Distributions
3(18)
1.1 The Maxwell Distribution
3(1)
1.2 Probability Distributions in Phase Space
4(1)
1.3 The Boltzmann-Gibbs Distribution
5(1)
1.4 A Gas of Particles in the Canonical Ensemble
6(2)
1.5 Additional Conserved Quantities
8(2)
1.6 The Grand-Canonical Ensemble
10(2)
1.7 Quantum Statistics
12(2)
1.8 The von Neumann Density Operator
14(2)
1.9 Fermi-Dirac and Bose-Einstein Distributions
16(2)
Problems
18(1)
Notes
19(1)
Objectives
20(1)
References
20(1)
2 Statistical Models
21(16)
2.1 Parameter Estimation
21(2)
2.2 Definition of a Statistical Model
23(2)
2.3 The Exponential Family
25(2)
2.4 Curved Exponential Families
27(2)
2.5 Example: The Ising Model in d=1
29(2)
2.6 The Density of States
31(1)
2.7 The Quantum Exponential Family
32(2)
Problems
34(2)
Notes
36(1)
Objectives
36(1)
References
36(1)
3 Thermodynamic Equilibrium
37(16)
3.1 Thermodynamic Configuration Space
37(1)
3.2 Maximum Entropy Principle
38(1)
3.3 BGS Entropy Functional
39(2)
3.4 Applying the Method of Lagrange
41(1)
3.5 Thermodynamic Entropy
42(2)
3.6 Relative Entropy
44(1)
3.7 Thermodynamic Stability
45(2)
3.8 Entropy of Probability Densities
47(1)
3.9 Quantum Entropies
47(2)
Problems
49(1)
Notes
50(1)
Objectives
51(1)
References
51(2)
4 The Microcanonical Ensemble
53(16)
4.1 Introduction
53(1)
4.2 The Ergodic Theorem
54(1)
4.3 Example: The Harmonic Oscillator
55(1)
4.4 Definition
56(1)
4.5 Microcanonical Instabilities
57(5)
4.6 The Quantum Microcanonical Ensemble
62(2)
4.7 The Coherent State Ensemble
64(1)
Problems
65(1)
Notes
66(1)
Objectives
67(1)
References
67(2)
5 Hyperensembles
69(10)
5.1 Introduction
69(1)
5.2 The Canonical Ensemble
70(1)
5.3 Superstatistics
71(3)
5.4 The Hyperensemble
74(1)
5.5 Properties of the Hyperensemble
75(2)
Notes
77(1)
Objectives
78(1)
References
78(1)
6 The Mean Field Approximation
79(16)
6.1 The Ideal Paramagnet
79(2)
6.2 The Mean Field Equation
81(1)
6.3 Phase Transitions
82(1)
6.4 A Mean Field Phase Transition
83(3)
6.5 A Hyperensemble of Product States
86(1)
6.6 Generalised Mean Field Theories
86(3)
6.7 The Quantum Case
89(2)
Notes
91(1)
Objectives
91(1)
References
91(4)
Part II Deformed Exponential Families
7 q-Deformed Distributions
95(20)
7.1 q-Deformed Exponential and Logarithmic Functions
95(2)
7.2 Dual Definitions
97(1)
7.3 The q-Exponential Family
98(2)
7.4 Escort Probabilities
100(1)
7.5 q-to-1/q Duality and Escort Families
101(1)
7.6 Dual Identities
102(1)
7.7 The q-Gaussian Distribution
103(3)
7.8 Configurational Probability Distribution
106(2)
7.9 Average Kinetic Energy
108(1)
7.10 The Quantum Family
109(1)
Problems
110(2)
Notes
112(1)
Objectives
113(1)
References
113(2)
8 Tsallis' Thermostatistics
115(16)
8.1 The Tsallis Entropy Functional
115(2)
8.2 A Historical Reflection
117(1)
8.3 Maximising the Tsallis Entropy Functional
118(1)
8.4 Thermodynamic Properties
119(2)
8.5 Example: The Two-Level Atom
121(2)
8.6 Relative Entropy of the Csiszar Type
123(1)
8.7 Relative Entropy of the Bregman Type
124(3)
8.8 Quantum Expressions
127(1)
8.9 More General Entropies
127(1)
Problems
128(1)
Notes
129(1)
Objectives
129(1)
References
130(1)
9 Changes of Scale
131(18)
9.1 Kolmogorov-Nagumo Averages
131(1)
9.2 Renyi's Alpha Entropies
132(1)
9.3 Renyi or Tsallis?
133(2)
9.4 Configurational Temperature
135(2)
9.5 Fractal Dimensions
137(3)
9.6 Microcanonical Description
140(3)
9.7 Sharma-Mittal Entropy Functional
143(2)
9.8 Zipf's Law
145(1)
Problems
146(1)
Notes
147(1)
Objectives
147(1)
References
148(1)
10 General Deformations
149(16)
10.1 Deformed Exponential and Logarithmic Functions
149(1)
10.2 Dual Definitions
150(1)
10.3 Deduced Logarithms
151(2)
10.4 The Phi-Exponential Family
153(1)
10.5 Escort Probabilities
153(2)
10.6 Method to Test Phi-Exponentiality
155(2)
10.7 Example: The Site Percolation Problem
157(3)
10.8 Generalised Quantum Statistics
160(1)
Problems
161(1)
Notes
162(1)
Objectives
162(1)
References
162(3)
11 General Entropies
165(14)
11.1 The Entropy Functional
165(1)
11.2 Relative Entropies
166(1)
11.3 The Variational Principle
167(2)
11.4 Complexity
169(1)
11.5 Application to q-Deformed Distributions
170(1)
11.6 Deformed Fisher Information
171(2)
11.7 Quantum Entropies
173(1)
11.8 Quantum Stability
174(2)
Problems
176(1)
Notes
177(1)
Objectives
177(1)
References
178(1)
Solutions to the Problems 179(20)
Index 199
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