0205.1

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.7)since it does not depend on the time t1.We shall also denoteP2(y2, t|y1) a" P2(y2, t|y1, 0) (5.8)for the probability that, if a random process begins with the value y1, then after the lapseof a time t it has the value y2.Markov process.A random process y(t) is said to be Markov (also sometimes calledMarkovian) if and only if all of its future probabilities are determined by its most recentlyknown value:Pn(yn, tn|yn-1, tn-1;.; y1, t1) = P2(yn, tn|yn-1, tn-1) for all tn e".e" t2 e" t1.(5.9)5This relation guarantees that any Markov process (which, of course, we require to be sta-tionary without saying so) is completely characterized by the probabilitiesp2(y2, t; y1, 0)p1(y) and P2(y2, t|y1) a" ; (5.10)p1(y1)i.e., by one function of one variable and one function of three variables.From these p1(y)and P2(y2, t|y1) one can reconstruct, using the Markovian relation (5.9) and the generalrelation (5.5) between conditional and absolute probabilities, all of the process s distributionfunctions.As an example, the x-component of velocity vx(t) of a dust particle in a room filled withconstant-temperature air is Markov (if we ignore the effects of the floor, ceiling, and walls bymaking the room be arbitrarily large).By contrast, the position x(t) of the particle is notMarkov because the probabilities of future values of x depend not just on the initial value ofx, but also on the initial velocity vx or, equivalently, the probabilities depend on the valuesof x at two initial, closely spaced times.The pair {x(t), vx(t)} is a two-dimensional Markovprocess.We shall consider multidimensional random processes in Exercises 5.1 and 5.9, andin Chap.8 (especially Ex.8.7).The Smoluchowski equation.Choose three (arbitrary) times t1, t2, and t3 that are ordered,so t1 [ Pobierz całość w formacie PDF ]

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