Chapter 5Random ProcessesVersion 0405.1.K, 27 Oct 04 Please send comments, suggestions, and errata via email [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 911255.1 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 is brought into equilibrium through contactwith a heat bath?• How can one characterize the noise introduced into 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 system generally arises from coupling to manyother 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 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 twosections 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. Morespecifically: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: Markovprocesses and Gaussian processes. Section 5.3 introduces two powerful mathematical toolsfor the analysis of random processes: the correlation function and the spectral density. In12Secs. 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 fromlarge noise. Finally, in Secs. 5.6 and 5.7 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. 5.6 we develop the fluctuation-dissipation theorem, which characterizes the forces bywhich the bath interacts with the systems; and in Sec. 5.7 we the develop the Fokker-Planckequation, which describes how the probability diffuses through phase space.5.2 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 byany 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”. Throughoutthis chapter we shall insist for simplicity that our random processes y take on a continuumof values ranging over some interval, often but not always −∞ to +∞. The generalizationto 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 incontact with a heat bath; (ii) the temperature T (t) at the corner of Main Street and CenterStreet in Logan, Utah; (iii) the earth-longitude φ(t) of a specific oxygen molecule in theearth’s atmosphere. One can also deal with random processes that are vector or tensorfunctions of time, but in this chapter’s brief introduction we shall refrain from doing so; thegeneralization to “multidimensional” random processes is straightforward.Ensembles of random processes. Since the precise time evolution of a random process isnot predictable, if one wishes to make predictions one can do so only probablistically. Thefoundation for probablistic predictions is an ensemble of random processes—i.e., a collectionof a huge number of random processes each of which behaves in its own, unpredictableway. In the next section we will use the ergodic hypothesis to construct, from a singlerandom process that interests us, a conceptual ensemble whose statistical properties carryinformation about the time evolution of the interesting process. However, until then we willassume that someone else has given us an ensemble; and we shall develop a probablisticcharacterization of it.Probability distributions. An ensemble of random processes is characterized completelyby a set of probability distributions p1, p2, p3, . . . defined as follows:pn(yn, tn; . . . ; y2, t2; y1, t1)dyn. . . dy2dy1(5.1)tells us the probability that a process y(t) drawn at random from the ensemble (i) will takeon a value between y1and y1+ dy1at time t1, and (ii) also will take on a value between y2and y2+dy2at time t2, and . . ., and (iii) also will take on a value between ynand yn+dynat3time tn. (Note that the subscript n on pntells us how many independent values of y appearin pn, and that earlier times are placed to the right—a practice common for physicists.) Ifwe knew the values of all of an ensemble’s probability distributions (an infinite number ofthem!) for all possible choices of their times (an infinite number of choices for each time thatappears in each probability distribution) and for all possible values of y (an infinite number ofpossible values for each time that appears in each probability distribution), then we wouldhave full information about the ensemble’s statistical properties. Not surprisingly, it willturn out that, if the ensemble in some sense is in statistical equilibrium, we can compute allits probability distributions from a very small amount of information. But that comes later;first we must develop more formalism.Ensemble averages. From the probability distributions we can compute ensemble averages(denoted by brackets). For example, the quantityhy(t1)i ≡Zy1p1(y1, t1)dy1(5.2a)is the ensemble-averaged value of y at time t1. Similarly,hy(t2)y(t1)i ≡Zy2y1p2(y2, t2; y1, t1)dy2dy1(5.2b)is the average value of the product y(t2)y(t1).Conditional probabilities. Besides the (absolute) probability distributions pn, we shallalso find useful an infinite series of conditional probability
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