Contents 5 Random Processes 5 1 Overview 5 2 Random Processes and their Probability Distributions 5 2 1 Markov Processes 5 2 2 Gaussian Processes and the Central Limit Theorem 5 3 Correlation Functions Spectral Densities and Ergodicity 5 3 1 Correlation Functions 5 3 2 Ergodic Hypothesis 5 3 3 Fourier Transforms and Spectral Densities 5 3 4 Doob s Theorem for Gaussian Markov Processes 5 4 Noise and its Types of Spectra 5 4 1 Intuitive Meaning of Spectral Density 5 4 2 Shot Noise Flicker Noise and Random Walk Noise 5 4 3 Information Missing from Spectral Density 5 5 Filters Signal to Noise Ratio and Shot Noise 5 5 1 Filters their Kernals and the Filtered Spectral Density 5 5 2 Band Pass Filter and Signal to Noise Ratio 5 5 3 Shot Noise 5 6 Fluctuation Dissipation Theorem 5 6 1 Generalized Coordinate and its Impedance 5 6 2 Fluctuation Dissipation Theorem for Generalized Coordinate Interacting with Thermalized Heat Bath 5 6 3 Johnson Noise and Langevin Equations 5 7 Fokker Planck Equation for a Markov Processes Conditional Probability 5 7 1 Fokker Planck for a One Dimensional Markov Process 5 7 2 Fokker Planck for a Multi Dimensional Markov Process 5 7 3 Brownian Motion i 1 2 2 5 6 8 8 9 10 14 16 17 18 19 20 20 22 24 29 29 31 34 37 37 40 41 Chapter 5 Random Processes Version 0805 1 K 29 Oct 08 Please send comments suggestions and errata via email to kip tapir caltech edu or on paper to Kip Thorne 130 33 Caltech Pasadena CA 91125 Box 5 1 Reader s Guide Relativity does not enter into this chapter This chapter does not rely in any major way on previous chapters but it does make occasional reference to results from Chaps 3 and 4 about statistical equalibrium and 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 5 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 turbulence in fluids in Sec 22 2 1 in treating the quasilinear formalism for weak plasma turbulence and in Sec 27 5 7 in discussing observations of the anisotropy of the cosmic microwave background radiation The fluctuation dissipation theorem developed in Sec 5 6 will be used in Ex 10 14 for thermoelastic noise in solids and in Sec 11 5 for normal modes of an elastic body The Fokker Planck equation developed in Sec 5 7 will be referred to in Sec 19 4 3 and Ex 19 8 when discussing thermal equilibration in a plasma and thermoelectric transport coefficients and it will be used in Sec 22 3 1 in developing the quasilinear theory of wave particle interactions in a plasma 1 2 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 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 distributions that describe it and discusses two special classes of random processes Markov processes and Gaussian processes Section 5 3 introduces two powerful mathematical tools 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 characterization and to types of signal processing that can be done to extract weak signals from large noise Finally in Secs 5 6 and 5 7 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 in Sec 5 6 we develop the fluctuation dissipation theorem which characterizes the forces by which the bath interacts with the systems and in Sec 5 7 we the develop the Fokker Planck equation which describes how the probability diffuses through phase space 5 2 Random Processes and their Probability Distributions 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 3 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 this chapter we shall insist for simplicity that our random processes y take on a continuum of values ranging over some interval often but not always to The generalization to y s with discrete e g integral values is straightforward Examples of random processes are i the total energy E t in a cell of gas that is in contact with a heat bath ii the temperature T t at the corner of Main Street and Center Street in Logan Utah iii the earth longitude t of a specific oxygen molecule in the earth s atmosphere One can also deal with random processes that are vector or tensor functions of time but in this chapter s brief introduction we shall refrain from doing so except for occasional side remarks equations and brief paragraphs Ensembles of random processes Since the precise time evolution of a random process is not predictable if one wishes to make predictions one can do so only probabilistically The foundation for
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