RubberBandRandomWalk

RubberBandRandomWalk - Ch1b rubber band Behavior of one...

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Ch1b rubber band 1 Behavior of one dimensional random walk - calculation of average end-to-end distance Assume each unit goes to the right or left with equal probability (1/2) - independent of what other units have done, or do. The step size length in either direction is l . Start at the origin. Let x i (n) be position of the n th unit in the i th molecule. Then: x i ( n ) = x i ( n " 1) ± l The “+” step direction applies to ~half of the units, and the “-“ applies to the other ~half. The mean displacement of the polymer is given by: x ( n ) = 1 M x i i = 1 M " ( n ) = 1 M x i i = 1 M " ( n # 1) ± l = 1 M x i i = 1 M " ( n # 1) ± 1 M l i = 1 M " last summation goes to 0 since equal numbers of + and - = x ( n # 1) on average (averaged over many molecules), the position of the n th unit is the same as the position of the n-1 th unit, . ..., is the same as the position of the first unit - ie, the average position of all units is where the molecule is centered (reflects equal numbers of right and left moves). OK - this is just like the average velocity of gas molecules along an axis is 0 - what about
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RubberBandRandomWalk - Ch1b rubber band Behavior of one...

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