Mean squared distance for (persistent) random walks

In summary, the conversation is about deriving the mean-squared-distance from the velocity autocorrelation for a random walk. The author suggests defining u'=u+s and integrating over u to obtain the desired form, but the speaker is having trouble understanding how the integral was derived. Later, the speaker figures it out and provides a link for those interested in the calculation.
  • #1
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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  • #2
Never mind, I figured it out. For the curious, I wrote this mathbin:

http://mathbin.net/89902
 
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FAQ: Mean squared distance for (persistent) random walks

1. What is the meaning of "mean squared distance" in the context of random walks?

The mean squared distance refers to the average of the squared distances between the starting point and each step taken in a random walk. It is a measure of how far the walker has traveled from their starting point, taking into account the direction and length of each step.

2. How is the mean squared distance calculated for a random walk?

The mean squared distance is calculated by taking the sum of the squared distances from the starting point to each step, and then dividing by the total number of steps taken. This gives an overall average of how far the walker has traveled from their starting point.

3. What is the significance of the mean squared distance in analyzing random walks?

The mean squared distance is an important measure in analyzing random walks because it provides insight into the overall behavior of the walker. It can help determine whether the walker is moving towards or away from their starting point, and can also provide information about the randomness of the walk.

4. How does the mean squared distance differ for persistent random walks compared to regular random walks?

The mean squared distance for persistent random walks is typically larger than that of regular random walks. This is because persistent random walks tend to have a more directional or biased movement, resulting in longer distances traveled from the starting point.

5. Can the mean squared distance be used to predict the future behavior of a random walk?

No, the mean squared distance cannot be used to predict the future behavior of a random walk. It is a measure of past behavior and does not take into account any external factors or changes in the random walk's path. Each step in a random walk is independent and cannot be predicted based on previous steps.

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