Mean squared distance traveled by an unbiased random walker in 1-D?

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In a one-dimensional unbiased random walk, the mean position after N steps is given by <x> = Nl(2p - 1), where l is the step size and p is the probability of stepping right. The standard deviation is σ = 2l√[Np(1 - p)]. The discussion centers on finding the second moment <x^2>, with an initial assumption that it equals l√N, which is challenged by the relationship <x^2> = σ^2 + <x>^2. It is clarified that the step size l is crucial, and the second moment is proportional to l^2, serving as an intermediate for calculating variance. Understanding <x^2> accurately is essential for analyzing the behavior of random walks.
CrimsonFlash
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Hey!

I've been doing some research on random walks. From what I have gathered, a random walker in 1-D will have:
<x> = N l (2 p - 1)

σ = 2 l sqrt[N p (1 - p) ]

Here, N is the number of steps, p is the probability to take a step to the right and l is the step size.
I was wondering what <x^2> would be. From what I found, it seems to be l sqrt(N) but when I try to use <x^2> = σ^2 + <x>^2 , I don't get l sqrt(N) . I would like to know what <x^2> really is for an unbiased random walk.

Thanks
 
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In "l sqrt(N)", what is "l"?
 
mathman said:
In "l sqrt(N)", what is "l"?

l is the step size here.
 
The second moment is proportional to l^2. In general its main use is as an intermediate to get the variance.
 

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