445lec3 - PHYS 445 Lecture 3 Random walks at large N...

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PHYS 445 Lecture 3 - Random walks at large N 3 - 1 © 2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited. Lecture 3 - Random walks at large N What's Important: random walks in the continuum limit Text : Reif 1D Random walks at large N As the number of steps N in a walk becomes very large, the individual step sizes in n R become relatively small. Further, the difference in values of W ( n R ) between successive values of n R becomes small as well. The proof of how the difference scales with N is given in Reif. For our purposes, the important point is that we can regard n R and W ( n R ) as continuous at large N . To obtain a continuous description of W ( n R ), consider its form near its most likely value at ˜ n R ˜ n R At the peak of the distribution, its derivative with respect to ˜ n R vanishes dW dn R ˜ n R = 0 or d ln W dn R ˜ n R = 0 Let's expand W ( n R ) or its logarithm around ˜ n R using as an expansion parameter, where n R = ˜ n R + . Choosing the logarithm, for example, we have ln W ( n R ) = ln W ( ˜ n R ) + B 1 + 1 2 B 2 2 + ... (3.1) where B i = d i ln W dn R i ˜ n R At the maximum in W , the coefficient B 1 = 0 by definition. The coefficients B i can be obtained from the discrete expression for W in terms of factorials. Start with the definition ln W ( n R ) = ln N ! - ln n R ! - ln( N - n R )! + n R ln p + ( N - n R )ln q , so W ( n R ) n R
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PHYS 445 Lecture 3 - Random walks at large N 3 - 2 © 2001 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited.
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