Mean squared distance for (persistent) random walks

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SUMMARY

The discussion focuses on deriving the mean-squared distance (MSD) from the velocity autocorrelation function for random walks. The user references a specific page that outlines the mathematical derivation but initially struggles with the integration process involving the substitution u' = u + s. Ultimately, the user resolves the issue independently and shares a link to their solution on Mathbin.

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IttyBittyBit
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Hi, I'm looking at how to derive the mean-squared-distance from the velocity autocorrelation for a random walk. It is given on this page: http://www.compsoc.man.ac.uk/~lucky/Democritus/Theory/msd2.html

Near the middle of that page the author says 'defining u'=u+s and integrating over u, results in the following form where the ensemble average has also been taken: ', but I can't seem to figure out how that integral following that statement was derived. I know what the velocity autocorrelation is, but when I set u'=u+s and integrate, it doesn't seem to pop up. Any help would be greatly appreciated.
 
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Never mind, I figured it out. For the curious, I wrote this mathbin:

http://mathbin.net/89902
 
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