دانلود کتاب Thinking Probabilistically: Stochastic Processes, Disordered Systems, and Their Applications

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Cover
Half-title
Title page
Copyright information
Contents
Acknowledgments
1 Introduction
1.1 Probabilistic Surprises
1.2 Summary
1.3 Exercises
2 Random Walks
2.1 Random Walks in 1D
2.2 Derivation of the Diffusion Equation for Random Walks in Arbitrary Spatial Dimension
2.3 Markov Processes and Markov Chains
2.4 Google PageRank: Random Walks on Networks as an Example of a Useful Markov Chain
2.5 Relation between Markov Chains and the Diffusion Equation
2.6 Summary
2.7 Exercises
3 Langevin and Fokker–Planck Equations and Their Applications
3.1 Application of a Discrete Langevin Equation to a Biological Problem
3.2 The Black–Scholes Equation: Pricing Options
3.3 Another Example: The “Well Function” in Hydrology
3.4 Summary
3.5 Exercises
4 Escape Over a Barrier
4.1 Setting Up the Escape-Over-a-Barrier Problem
4.2 Application to the 1D Escape Problem
4.3 Deriving Langer\'s Formula for Escape-Over-a-Barrier in Any Spatial Dimension
4.4 Summary
4.5 Exercises
5 Noise
5.1 Telegraph Noise: Power Spectrum Associated with a Two-Level-System
5.2 From Telegraph Noise to 1/f Noise via the Superposition of Many Two-Level-Systems
5.3 Power Spectrum of a Signal Generated by a Langevin Equation
5.4 Parseval’s Theorem: Relating Energy in the Time and Frequency Domain
5.5 Summary
5.6 Exercises
6 Generalized Central Limit Theorem and Extreme Value Statistics
6.1 Probability Distribution of Sums: Introducing the Characteristic Function
6.2 Approximating the Characteristic Function at Small Frequencies for Distributions with Finite Variance
6.3 Central Region of CLT: Where the Gaussian Approximation Is Valid
6.4 Sum of a Large Number of Positive Random Variables: Universal Description in Laplace Space
6.5 Application to Slow Relaxations: Stretched Exponentials
6.6 Example of a Stable Distribution: Cauchy Distribution
6.7 Self-Similarity of Running Sums
6.8 Generalized CLT via an RG-Inspired Approach
6.9 Exploring the Stable Distributions Numerically
6.10 RG-Inspired Approach for Extreme Value Distributions
6.11 Summary
6.12 Exercises
7 Anomalous Diffusion
7.1 Continuous Time Random Walks
7.2 Lévy Flights: When the Variance Diverges
7.3 Propagator for Anomalous Diffusion
7.4 Back to Normal Diffusion
7.5 Ergodicity Breaking: When the Time Average and the Ensemble Average Give Different Results
7.6 Summary
7.7 Exercises
8 Random Matrix Theory
8.1 Level Repulsion between Eigenvalues: The Birth of RMT
8.2 Wigner’s Semicircle Law for the Distribution of Eigenvalues
8.3 Joint Probability Distribution of Eigenvalues
8.4 Ensembles of Non-Hermitian Matrices and the Circular Law
8.5 Summary
8.6 Exercises
9 Percolation Theory
9.1 Percolation and Emergent Phenomena
9.2 Percolation on Trees – and the Power of Recursion
9.3 Percolation Correlation Length and the Size of the Largest Cluster
9.4 Using Percolation Theory to Study Random Resistor Networks
9.5 Summary
9.6 Exercises
Appendix A Review of Basic Probability Concepts and Common Distributions
A.1 Some Important Distributions
A.2 Central Limit Theorem
Appendix B A Brief Linear Algebra Reminder, and Some Gaussian Integrals
B.1 Basic Linear Algebra Facts
B.2 Gaussian Integrals
Appendix C Contour Integration and Fourier Transform Refresher
C.1 Contour Integrals and the Residue Theorem
C.2 Fourier Transforms
Appendix D Review of Newtonian Mechanics, Basic Statistical Mechanics, and Hessians
D.1 Basic Results in Classical Mechanics
D.2 The Boltzmann Distribution and the Partition Function
D.3 Hessians
Appendix E Minimizing Functionals, the Divergence Theorem, and Saddle-Point Approximations
E.1 Functional Derivatives
E.2 Lagrange Multipliers
E.3 The Divergence Theorem (Gauss’s Law)
E.4 Saddle-Point Approximations
Appendix F Notation, Notation…
References
Index