Noise Theory and Application to Physics: From Fluctuations to InformationIn many situations, physical quantities are perturbed or evolve in a not fully predictable way. We then speak about noise or fluctuations and we are generally faced to different questions such as: What are the correct physical models to describe them? What are the most practical mathematical tools to deal with them? How can relevant information be extracted in the presence of noise? Noise theory and application to physics provides a precise description of the theoretical background and practical tools for noise and fluctuation analyses. It not only introduces basic mathematical descriptions and properties of noise and fluctuations but also discusses the physical origin of different noise models and presents some statistical methods which optimize measurements in the presence of such fluctuations. Noise theory and application to physics investigates a number of ideas about noise and fluctuations in a single book in relation with probability and stochastic processes, information theory, statistical physics and statistical inference. The different notions are illustrated with many application examples from physics and engineering science and problems with solutions allow the reader to both check his understanding and to deepen some aspects. Indeed, the main objective of Noise theory and application to physics is to be a practical guide for the reader for going from fluctuation to information. It will thus be of great interest to undergraduate or postgraduate students and researchers in physics and engineering sciences. |
Contents
Introduction | 1 |
Random Variables | 5 |
21 Random Events and Probability | 6 |
22 Random Variables | 7 |
23 Means and Moments | 10 |
24 Median and Mode of a Probability Distribution | 12 |
25 Joint Random Variables | 13 |
26 Covariance | 16 |
Lagrange Multipliers | 133 |
Exercises | 134 |
Thermodynamic Fluctuations | 137 |
62 Free Energy | 141 |
63 Connection with Thermodynamics | 142 |
64 Covariance of Fluctuations | 143 |
65 A Simple Example | 146 |
66 FluctuationDissipation Theorem | 149 |
27 Change of Variables | 18 |
28 Stochastic Vectors | 19 |
Exercises | 22 |
Fluctuations and Covariance | 25 |
32 Stationarity and Ergodicity | 28 |
33 Ergodicity in Statistical Physics | 32 |
34 Generalization to Stochastic Fields | 34 |
35 Random Sequences and Cyclostationarity | 35 |
36 Ergodic and Stationary Cases | 40 |
37 Application to Optical Coherence | 41 |
38 Fields and Partial Differential Equations | 42 |
39 Power Spectral Density | 44 |
310 Filters and Fluctuations | 46 |
311 Application to Optical Imaging | 50 |
312 Green Functions and Fluctuations | 52 |
313 Stochastic Vector Fields | 56 |
314 Application to the Polarization of Light | 57 |
315 Ergodicity and Polarization of Light | 61 |
WienerKhinchine Theorem | 64 |
Exercises | 66 |
Limit Theorems and Fluctuations | 71 |
42 Characteristic Function | 74 |
43 Central Limit Theorem | 76 |
44 Gaussian Noise and Stable Probability Laws | 80 |
45 A Simple Model of Speckle | 81 |
46 Random Walks | 89 |
47 Application to Diffusion | 92 |
48 Random Walks and Space Dimensions | 97 |
49 Rare Events and Particle Noise | 100 |
410 Low Flux Speckle | 102 |
Exercises | 104 |
Information and Fluctuations | 109 |
52 Entropy | 111 |
53 Kolmogorov Complexity | 114 |
54 Information and Stochastic Processes | 117 |
55 Maximum Entropy Principle | 119 |
56 Entropy of Continuous Distributions | 122 |
57 Entropy Propagation and Diffusion | 124 |
58 Multidimensional Gaussian Case | 128 |
59 KullbackLeibler Measure | 130 |
67 Noise at the Terminals of an RC Circuit | 153 |
68 Phase Transitions | 158 |
69 Critical Fluctuations | 161 |
Exercises | 163 |
Statistical Estimation | 167 |
72 The Language of Statistics | 169 |
74 Maximum Likelihood Estimator | 174 |
75 CramerRao Bound in the Scalar Case | 177 |
76 Exponential Family | 179 |
77 Example Applications | 181 |
78 CramerRao Bound in the Vectorial Case | 182 |
79 Likelihood and the Exponential Family | 183 |
710 Examples in the Exponential Family | 186 |
7101 Estimating the Parameter in the Poisson Distribution | 187 |
7103 Estimating the Mean of the Gaussian Distribution | 188 |
7104 Estimating the Variance of the Gaussian Distribution | 189 |
7105 Estimating the Mean of the Weibull Distribution | 190 |
711 Robustness of Estimators | 192 |
Scalar CramerRao Bound | 196 |
Efficient Statistics | 199 |
Vectorial CramerRao Bound | 200 |
Exercises | 205 |
Examples of Estimation in Physics | 209 |
82 Measurement Accuracy in the Presence of Gaussian Noise | 212 |
83 Estimating a Detection Efficiency | 217 |
84 Estimating the Covariance Matrix | 219 |
85 Application to Coherency Matrices | 221 |
86 Making Estimates in the Presence of Speckle | 224 |
87 FluctuationDissipation and Estimation | 225 |
Exercises | 227 |
Solutions to Exercises | 231 |
92 Chapter Three Fluctuations and Covariance | 235 |
93 Chapter Four Limit Theorems and Fluctuations | 243 |
94 Chapter Five Information and Fluctuations | 250 |
95 Chapter Six Statistical Physics | 259 |
96 Chapter Seven Statistical Estimation | 266 |
97 Chapter Eight Examples of Estimation in Physics | 271 |
References | 285 |
287 | |
Other editions - View all
Noise Theory and Application to Physics: From Fluctuations to Information Philippe Réfrégier No preview available - 2011 |
Common terms and phrases
References to this book
Simultaneity: Temporal Structures and Observer Perspectives Susie Vrobel,Terry Marks-Tarlow Limited preview - 2008 |