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

[CrimsonFlash]

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

 

[mathman]

In "l sqrt(N)", what is "l"?

 

[CrimsonFlash]

In "l sqrt(N)", what is "l"?

l is the step size here.

 

[mathman]

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

 

[blue_raver22]

!

Hi there!

It's great to see that you are interested in random walks and have done some research on the topic. To answer your question, the mean squared distance traveled by an unbiased random walker in 1-D is actually <x^2> = Nl^2. This can be derived from the formula you mentioned, <x> = Nl(2p-1), by substituting p=0.5 for an unbiased walk.

To clarify your confusion, the formula <x> = Nl(2p-1) represents the mean displacement of the random walker after N steps, while <x^2> represents the mean squared displacement. This means that <x^2> takes into account both the distance traveled and the direction of each step, resulting in a larger value than <x>.

I hope this helps to answer your question. Keep up the curiosity and good work in your research!