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445lec2 - PHYS 445 Lecture 2 Random walks II Lecture 2...

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PHYS 445 Lecture 2 - Random walks - II 2 - 1 © 2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited. Lecture 2 - Random walks - II What's Important: expectations tip-to-tail distributions Text : Reif "Tip-to-tail" displacement In the first lecture, we determined the probability W N ( n R , n L ) for a random walk with N steps to have n R steps to the right and n L steps to the left: W N ( n R , n L ) = N ! n R ! n L ! p n R q n L . (2.1) In this lecture, we show how to use this distribution to evaluate observables such as ensemble averages. In the random walk, the "tip-to-tail" distance, or equivalently the displacement of the walker, is one observable of interest. For a walk with equal length steps a , the displacement x is x = ma = ( n R - n L ) a where m = n R - n L = ( n R - [ N - n R ] ) a = (2 n R - N ) a . (2.2) Eq. (2.2) demonstrates that m changes by 2 units as a function of n R , and can range from - N to + N . Quick review of mean values The function W N ( n R ) tells us the probability that, in an ensemble of walks, there are walks with n R steps to the right. From this distribution, we can extract quantities such as the average number of steps to the right, or to the left etc . Let's find a formal
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