Contents6RandomProcesses 16.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Random Processes and their Probability Distributions . . . .......... 26.2.1 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.2.2 Gaussian Processes and the Central Limit Theorem . . . . . . . . . . 66.3 Correlation Functions, Sp ectral Densities, and Ergodicity . . . . . . . . . . . 86.3.1 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.3.2 Ergo dic Hyp othesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.3.3 Fourier Transforms and Sp ectral Densities . . . . . . . . . . . . . . . 106.3.4 Doob’s Theorem for Gaussian, Markov Processes . . . . . . . . . . . 146.4 Noise and its Types of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.1 Intuitive Meaning of Sp ectral Density . . . . . . . . . . . . . . . . . . 176.4.2 Shot Noise, Flicker Noise and Random-Walk Noise . . . . . . . . . . 186.4.3 Information Missing from Spectral Density . . . . . . . . . . . . . . . 196.5 Filters, Signal-to-Noise Ratio, and Shot Noise . . . . . . . . . . . . . . . . . 206.5.1 Filters, their Kernals, and the Filtered Spectral Density . . . . . . . . 206.5.2 Band-Pass Filter and Signal to Noise Ratio . . . . . . . . . . . . . . . 226.5.3 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.6 Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 296.6.1 Generalized Coordinate and its Impedance . . . . . . . . . . . . . . . 296.6.2 Fluctuation-Dissipation TheoremforGeneralizedCoordinateInteract-ing with Thermalized Heat Bath . . . . . . . . . . . . . . . . . . . . 316.6.3 Johnson Noise and Langevin Equations . . . . . . . . . . . . . . . . . 346.7 Fokker-Planck Equation for a Markov Processes Conditional Probability . . . 376.7.1 Fokker-Planck for a One-Dimensional Markov Process . . . . . . . . . 376.7.2 Fokker-Planck for a Multi-Dimensional Markov Process . . . . . . . . 406.7.3 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41iChapter 6Random ProcessesVersion 1006.1.K, 29 Oct 08Please send comments, suggestions, and errata via email to [email protected], or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 6.1Reader’s Guide• Relativity does not enter into this chapter.• This chapter does not rely in any major way on previous chapters, but it does makeoccasional reference to results from Chaps. 3 and 4 about statistical equalibriumand fluctutions in and away from statistical equilibrium. KTB??• No subsequent chapter relies in any major way on this chapter. However:– The concepts of spectral density and correlation function, developed in Sec.6.3, will be used in Ex. 8.7 in treating coherence properties of radiation, in Sec.10.5 in studying thermal noise in solids, in Sec. 14.3 in studying turbulencein fluids, in Sec. 22.2.1 in treating the quasilinear formalism for weak plasmaturbulence, and in Sec. 27.5.7 in discussing observations of the anisotropy ofthe cosmic microwave background radiation.– The fluctuation-dissipation theorem, developed in Sec. 6.6, will be used in Ex.10.14 for thermoelastic noise in solids, and in Sec. 11.5 for normal modes ofan elastic body.– The Fokker-Planc k equation, developed in Sec. 6.7, will be referred to inSec. 19.4.3 and Ex. 19.8 when discussing thermal equilibration in a plasmaand thermoelectric transport coefficients, and it will be used in Sec. 22.3.1 indeveloping the quasilinear theory of wave-particle interactions in a plasma.126.1 OverviewIn this chapter we shall analyze, among others, the following issues:• What is the time evolution of the distribution function for an ensemble of systems thatbegins out of statistical equilibrium and isbroughtintoequilibriumthroughcontactwith a heat bath?• How can one characterize the noise introduced in to experiments or observations bynoisy 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 strongnoise?• Frictional damping of a dynamical systemgenerallyarisesfromcouplingtomanyother degrees of freedom (a bath) that can sap the system’s energy. What is theconnection between the fluctuating (noise) forces that the bath exerts on the systemand its damping influence?The mathematical foundation foranalyzingsuchissuesisthetheory of random processes,and a portion of that subject is the theory of stochastic differential equations.Thefirsttwosections of this chapter constitute a quick in troduction to the theory of random processes,and subsequent sections then use that theory toanalyzetheaboveissuesandothers. Morespecifically:Section 6.2 introduces the concept of a random process and the various probability dis-tributions that describe it, and discusses two special classes of random processes: Markovprocesses and Gaussian processes. Section 6.3 introduces two powerful mathematical toolsfor the analysis of random processes: the correlation function and the spectral density. InSecs. 6.4 and 6.5 we meet the first application ofrandomprocesses: tonoise and its charac-terization, and to types of signal processing that can be done to extract weak signals fromlarge noise. Finally, in Secs. 6.6 and 6.7 we use the theory of random processes to studythe details of how an ensemble of systems, interacting with a bath, evolves into statisticalequilibrium. As we shall see, the evolution is governed by a stochastic differential equationcalled the “Langevin equation,” whose solution is described by an evolving probability distri-bution (the distribution function). As powerful tools in studying the probability’s evolution,in Sec. 6.6 we develop the fluctuation-dissipation theorem, which characterizes the forces bywhic h the bath interacts with the systems; andinSec.6.7wethedeveloptheFokker-Planckequation, which describes how the probability diffuses through phase space.6.2 Random Processes and their Probability Distribu-tionsDefinition of “random process”.A(one-dimensional)random process is a (scalar) functiony(t), where t is usually time, for which the future evolution is not determined uniquely by3any 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
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