Noise Theory and Application to Physics: From Fluctuations to Information

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Springer Science & Business Media, Apr 27, 2004 - Computers - 288 pages

In 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.

 

Selected pages

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
Index
287
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