Noise Theory and Application to Physics

1 Introduction.- 2 Random Variables.- 2.1 Random Events and Probability.- 2.2 Random Variables.- 2.3 Means and Moments.- 2.4 Median and Mode of a Probability Distribution.- 2.5 Joint Random Variables.- 2.6 Covariance.- 2.7 Change of Variables.- 2.8 Stochastic Vectors.- Exercises.- 3 Fluctuations and Covariance.- 3.1 Stochastic Processes.- 3.2 Stationarity and Ergodicity.- 3.3 Ergodicity in Statistical Physics.- 3.4 Generalization to Stochastic Fields.- 3.5 Random Sequences and Cyclostationarity.- 3.6 Ergodic and Stationary Cases.- 3.7 Application to Optical Coherence.- 3.8 Fields and Partial Differential Equations.- 3.9 Power Spectral Density.- 3.10 Filters and Fluctuations.- 3.11 Application to Optical Imaging.- 3.12 Green Functions and Fluctuations.- 3.13 Stochastic Vector Fields.- 3.14 Application to the Polarization of Light.- 3.15 Ergodicity and Polarization of Light.- 3.16 Appendix: Wiener–Khinchine Theorem.- Exercises.- 4 Limit Theorems and Fluctuations.- 4.1 Sum of Random Variables.- 4.2 Characteristic Function.- 4.3 Central Limit Theorem.- 4.4 Gaussian Noise and Stable Probability Laws.- 4.5 A Simple Model of Speckle.- 4.6 Random Walks.- 4.7 Application to Diffusion.- 4.8 Random Walks and Space Dimensions.- 4.9 Rare Events and Particle Noise.- 4.10 Low Flux Speckle.- Exercises.- 5 Information and Fluctuations.- 5.1 Shannon Information.- 5.2 Entropy.- 5.3 Kolmogorov Complexity.- 5.4 Information and Stochastic Processes.- 5.5 Maximum Entropy Principle.- 5.6 Entropy of Continuous Distributions.- 5.7 Entropy, Propagation and Diffusion.- 5.8 Multidimensional Gaussian Case.- 5.9 Kullback-Leibler Measure.- 5.10 Appendix: Lagrange Multipliers.- Exercises.- 6 Thermodynamic Fluctuations.- 6.1 Gibbs Statistics.- 6.2 Free Energy.- 6.3 Connection with Thermodynamics.- 6.4Covariance of Fluctuations.- 6.5 A Simple Example.- 6.6 Fluctuation-Dissipation Theorem.- 6.7 Noise at the Terminals of an RC Circuit.- 6.8 Phase Transitions.- 6.9 Critical Fluctuations.- Exercises.- 7 Statistical Estimation.- 7.1 The Example of Poisson Noise.- 7.2 The Language of Statistics.- 7.3 Characterizing an Estimator.- 7.4 Maximum Likelihood Estimator.- 7.5 Cramer-Rao Bound in the Scalar Case.- 7.6 Exponential Family.- 7.7 Example Applications.- 7.8 Cramer-Rao Bound in the Vectorial Case.- 7.9 Likelihood and the Exponential Family.- 7.10 Examples in the Exponential Family.- 7.11 Robustness of Estimators.- 7.12 Appendix: Scalar Cramer-Rao Bound.- 7.13 Appendix: Efficient Statistics.- 7.14 Appendix: Vectorial Cramer-Rao Bound.- Exercises.- 8 Examples of Estimation in Physics.- 8.1 Measurement of Optical Flux.- 8.2 Measurement Accuracy in the Presence of Gaussian Noise.- 8.3 Estimating a Detection Efficiency.- 8.4 Estimating the Covariance Matrix.- 8.5 Application to Coherency Matrices.- 8.6 Making Estimates in the Presence of Speckle.- 8.7 Fluctuation-Dissipation and Estimation.- Exercises.- 9 Solutions to Exercises.- 9.1 Chapter Two. Random Variables.- 9.2 Chapter Three. Fluctuations and Covariance.- 9.3 Chapter Four. Limit Theorems and Fluctuations.- 9.4 Chapter Five. Information and Fluctuations.- 9.5 Chapter Six. Statistical Physics.- 9.6 Chapter Seven. Statistical Estimation.- 9.7 Chapter Eight. Examples of Estimation in Physics.- References.