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0205.1 - Chapter 5 Random Processes Version 0205.1 28 Oct...

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Chapter 5 Random Processes Version 0205.1, 28 Oct 02 Please send comments, suggestions, and errata via email to [email protected] and [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125 5.1 Overview In this chapter we shall analyze, among others, the following issues: What is the time evolution of the distribution function for an ensemble of systems that begins out of statistical equilibrium and is brought into equilibrium through contact with a heat bath? How can one characterize the noise introduced into experiments or observations by noisy devices such as resistors, amplifiers, etc.? What is the influence of such noise on one’s ability to detect weak signals? What filtering strategies will improve one’s ability to extract weak signals from strong noise? Frictional damping of a dynamical system generally arises from coupling to many other degrees of freedom (a bath) that can sap the system’s energy. What is the connection, if any, between the fluctuating (noise) forces that the bath exerts on the system and its damping influence? The mathematical foundation for analyzing such issues is the theory of random processes , and a portion of that subject is the theory of stochastic differential equations . The first two sections of this chapter constitute a quick introduction to the theory of random processes, and subsequent sections then use that theory to analyze the above issues and others. More specifically: Section 5.2 introduces the concept of a random process and the various probability dis- tributions that describe it, and discusses two special classes of random processes: Markov processes and Gaussian processes. Section 5.3 introduces two powerful mathematical tools 1
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2 for the analysis of random processes: the correlation function and the spectral density. In Secs. 5.4 and 5.5 we meet the first application of random processes: to noise and its charac- terization, and to types of signal processing that can be done to extract weak signals from large noise. Finally, in Sec. 5.6 we use the theory of random processes to study the details of how an ensemble of systems, interacting with a bath, evolves into statistical equilibrium. As we shall see, the evolution is governed by a stochastic differential equation called the “Langevin equation,” whose solution is described by an evolving probability distribution (the distribution function). As powerful tools in studying the probability’s evolution, we develop the fluctuation-dissipation theorem (which characterizes the forces by which the bath interacts with the systems), and the Fokker-Planck equation (which describes how the probability diffuses through phase space). 5.2 Random Processes and their Probability Distribu- tions Definition of “random process” . A (one-dimensional) random process is a (scalar) function y ( t ), where t is usually time, for which the future evolution is not determined uniquely by any set of initial data—or at least by any set that is knowable to you and me. In other words, “random process” is just a fancy phrase that means “unpredictable function”. Throughout
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