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

Noise in physics is related to a variety of domains, such as information theory, statistical physics, probability, stochastic processes and statistics. Noise Theory and Application to Physics provides a general background on noise theory, along with techniques to describe and extract information in the presence of fluctuations, with the goal of featuring noise in the context of its connection with other domains. Readers will gain a deep understanding of noise theory, while acquiring systematic techniques for describing and extracting noise data.

From the reviews:



"Philippe Réfrégier's book is an introduction to the fundamental principles of randomness that can be encountered in a physicist's everyday life. ... Réfrégier's book is a good textbook for physics graduate students and researchers who want to go a little deeper into probability theory ... . I would also recommend it to mathematicians who want to get a different point of view on probability theory. Students will undoubtedly profit from a variety of exercises that come with ... solutions." (Achim Klenke, Zentralblatt MATH, Vol. 1092 (18), 2006)

"The book provides a good presentation of the statistical basis for theories of noise in physics. ... Noise theory and application to physics provides a precise description of the theoretical background and practical tools for noise and fluctuation analyses. It will thus be of great interest to undergraduate or postgraduate students and researchers in physics and the engineering sciences." (Stefan Adams, Mathematical Reviews, Issue 2006 e)