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E-grāmata: Thinking Probabilistically: Stochastic Processes, Disordered Systems, and Their Applications

2.75/5 (8 ratings by Goodreads)
(Harvard University, Massachusetts)
  • Formāts: EPUB+DRM
  • Izdošanas datums: 17-Dec-2020
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108802352
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Thinking Probabilistically: Stochastic Processes, Disordered Systems, and Their ApplicationsPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 17-Dec-2020
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781108802352
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Probability theory has diverse applications in a plethora of fields, including physics, engineering, computer science, chemistry, biology and economics. This book will familiarize students with various applications of probability theory, stochastic modeling and random processes, using examples from all these disciplines and more. The reader learns via case studies and begins to recognize the sort of problems that are best tackled probabilistically. The emphasis is on conceptual understanding, the development of intuition and gaining insight, keeping technicalities to a minimum. Nevertheless, a glimpse into the depth of the topics is provided, preparing students for more specialized texts while assuming only an undergraduate-level background in mathematics. The wide range of areas covered - never before discussed together in a unified fashion – includes Markov processes and random walks, Langevin and Fokker–Planck equations, noise, generalized central limit theorem and extreme values statistics, random matrix theory and percolation theory.

Assuming only an undergraduate-level background in mathematics, this book explains the power of thinking probabilistically through a diverse set of applications drawn from science. It covers a wide range of topics never before discussed together in a unified fashion while keeping technicalities to a minimum.

Recenzijas

'A remarkable demonstration that advanced topics need not be esoteric. Amir takes us through random walks on networks, extreme value statistics, Kramers theory, anomalous diffusion and other topics generally omitted from introductory texts, always rooting every discussion in applications of terrific current interest. I found the discussion of Lévy-stable distributions especially insightful as a principled approach to the nonstandard walks that abound in contexts from biophysics to finance.' Philip C. Nelson, University of Pennsylvania 'The book is a suitable springboard for self-study because it introduces a wide variety of topics and contains many references to current work. In the classroom, the book can function either as the basis for a course in special topics or as a source of material to spice up more traditional statistical-mechanics courses the methods for studying random phenomena introduced in Thinking Probabilistically will help readers understand reasoning techniques that may not be terribly familiar to physicists. Moreover, following the author's arguments is a rewarding intellectual exercise in its own right.' Rob de Ruyter, Physics Today

Papildus informācija

An introductory text providing the reader with a thorough background to the rich world of applications of stochastic processes.
Acknowledgments viii
1 Introduction
1(14)
1.1 Probabilistic Surprises
5(7)
1.2 Summary
12(1)
1.3 Exercises
13(2)
2 Random Walks
15(24)
2.1 Random Walks in 1D
16(2)
2.2 Derivation of the Diffusion Equation for Random Walks in Arbitrary Spatial Dimension
18(6)
2.3 Markov Processes and Markov Chains
24(1)
2.4 Google Page Rank: Random Walks on Networks as an Example of a Useful Markov Chain
25(5)
2.5 Relation between Markov Chains and the Diffusion Equation
30(2)
2.6 Summary
32(1)
2.7 Exercises
32(7)
3 Langevin And Fokker--Planck Equations And Their Applications
39(28)
3.1 Application of a Discrete Langevin Equation to a Biological Problem
45(6)
3.2 The Black--Scholes Equation: Pricing Options
51(9)
3.3 Another Example: The "Well Function" in Hydrology
60(2)
3.4 Summary
62(1)
3.5 Exercises
63(4)
4 Escape Over A Barrier
67(14)
4.1 Setting Up the Escape-Over-a-Barrier Problem
70(1)
4.2 Application to the ID Escape Problem
71(2)
4.3 Deriving Langer's Formula for Escape-Over-a-Barrier in Any Spatial Dimension
73(6)
4.4 Summary
79(1)
4.5 Exercises
79(2)
5 Noise
81(15)
5.1 Telegraph Noise: Power Spectrum Associated with a Two-Level-System
83(5)
5.2 From Telegraph Noise to 1/f Noise via the Superposition of Many Two-Level-Systems
88(1)
5.3 Power Spectrum of a Signal Generated by a Langevin Equation
89(1)
5.4 Parseval's Theorem: Relating Energy in the Time and Frequency Domain
90(2)
5.5 Summary
92(1)
5.6 Exercises
92(4)
6 Generalized Central Limit Theorem And Extreme Value Statistics
96(37)
6.1 Probability Distribution of Sums: Introducing the Characteristic Function
98(1)
6.2 Approximating the Characteristic Function at Small Frequencies for Distributions with Finite Variance
99(1)
6.3 Central Region of CLT: Where the Gaussian Approximation Is Valid
100(3)
6.4 Sum of a Large Number of Positive Random Variables: Universal Description in Laplace Space
103(3)
6.5 Application to Slow Relaxations: Stretched Exponentials
106(2)
6.6 Example of a Stable Distribution: Cauchy Distribution
108(1)
6.7 Self-Similarity of Running Sums
109(1)
6.8 Generalized CLT via an RG-Inspired Approach
110(8)
6.9 Exploring the Stable Distributions Numerically
118(2)
6.10 RG-Inspired Approach for Extreme Value Distributions
120(7)
6.11 Summary
127(1)
6.12 Exercises
128(5)
7 Anomalous Diffusion
133(12)
7.1 Continuous Time Random Walks
134(3)
7.2 Levy Flights: When the Variance Diverges
137(1)
7.3 Propagator for Anomalous Diffusion
138(1)
7.4 Back to Normal Diffusion
139(1)
7.5 Ergodicity Breaking: When the Time Average and the Ensemble Average Give Different Results
139(1)
7.6 Summary
140(1)
7.7 Exercises
141(4)
8 Random Matrix Theory
145(39)
8.1 Level Repulsion between Eigenvalues: The Birth of RMT
145(4)
8.2 Wigner's Semicircle Law for the Distribution of Eigenvalues
149(6)
8.3 Joint Probability Distribution of Eigenvalues
155(7)
8.4 Ensembles of Non-Hermitian Matrices and the Circular Law
162(17)
8.5 Summary
179(1)
8.6 Exercises
180(4)
9 Percolation Theory
184(41)
9.1 Percolation and Emergent Phenomena
184(9)
9.2 Percolation on Trees - and the Power of Recursion
193(2)
9.3 Percolation Correlation Length and the Size of the Largest Cluster
195(2)
9.4 Using Percolation Theory to Study Random Resistor Networks
197(5)
9.5 Summary
202(1)
9.6 Exercises
203(4)
Appendix A Review of Basic Probability Concepts and Common Distributions
207(4)
A.1 Some Important Distributions
208(2)
A.2 Central Limit Theorem
210(1)
Appendix B A Brief Linear Algebra Reminder, and Some Gaussian Integrals
211(3)
B.1 Basic Linear Algebra Facts
211(1)
B.2 Gaussian Integrals
212(2)
Appendix C Contour Integration and Fourier Transform Refresher
214(3)
C.1 Contour Integrals and the Residue Theorem
214(1)
C.2 Fourier Transforms
214(3)
Appendix D Review of Newtonian Mechanics, Basic Statistical Mechanics, and Hessians
217(3)
D.1 Basic Results in Classical Mechanics
217(1)
D.2 The Boltzmann Distribution and the Partition Function
218(1)
D.3 Hessians
218(2)
Appendix E Minimizing Functional, the Divergence Theorem, and Saddle-Point Approximations
220(2)
E.1 Functional Derivatives
220(1)
E.2 Lagrange Multipliers
220(1)
E.3 The Divergence Theorem (Gauss's Law)
220(1)
E.4 Saddle-Point Approximations
221(1)
Appendix F Notation, Notation
222(3)
References 225(7)
Index 232
Ariel Amir is a Professor at Harvard University, Massachusetts. His research centers on the theory of complex systems.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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