# Calculating Mean-Squared Displacement

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maria87 Calculating Mean-Squared Displacement

Hi,

I'm a PG student and confused. I have looked at many research articles describing mean-squared-displacement and would be grateful if anyone can see if I'm doing anything wrong in the process:

1.Run the program from t=0 to t=tend
2.Calculate the displacement, (s(t)-s(t+tau)), at each time interval and store it in a vector
3.Square the answer to number 2 and find the mean, call this answer <s^2>

Okay, so far so good?

My aim is to end up with a figure with time (t) on the x-axis and <s^2> on the y-axis (see figure attached). On the x-axis: t=100,200,300,...600. I'm not sure what to put on the y-axis.
To put it another way, for each t on the x-axis, what is the coresponding mean-squared displacement? Is it the averaged value over 100 time intervals or is it just the sum of the displacements squared over that time interval?

Any help is greatly appreciated. I hope I have explained the problem clearly, however, if anything is unclear, pleaes let me know.
Maria

maria87 Any ideas anyone?

Any ideas anyone?

Sami Tuomivaara maria87,

maria87,

I'm not quite sure what you are after, but here's something.

<rsd^2> is a function of time, it's value at any t should be the ensemble average of all the replicate simulations at that same time point. For example: You did three replicate simulations, the displacements s(t) at some time t for these three individual simulations/trajectories are, let's say, 10, 15 and 20. First, square the displacements to get the rsd^2 values 100, 225 and 400. Then, average the values to get <rsd^2> = 242. Repeat this for every different time point and you have the final vector of <rsd^2> versus t.

Does this help?

Cheers,

maria87 Thank you very much for your

Thank you very much for your reply. At the moment I'm still unsure on how to  calculate <s^2> for a single particle moving spatially over time.  I know it sounds pretty trivial but I would appreciate your help.

This is what I'm doing currently to calculate <s^2> at t=100:
- I allow a particle to move spatially over a given period of time, from t=0 to t=100.
- At t=1, I calculate the square displacement of the particle from where it was at t=0, ie. (s(1)-s(0))^2
- At t=2, I calculate the square displacement  of the particle from where it was at t=1, ie. (s(2)-s(1))^2 .... and so on until I reach t=100 ...
- At t=100, I calculate the square displacement  of the particle from where it was at t=99,  ie. (s(100)-s(99))^2 .
- I add up the result of all the displacements squared over that period and divide by 100 to get <s^2> at t=100.

So, should I divide by 100 or not to get the mean (<.>) ? Can you spot anything wrong with what I'm doing? I get a very small <s^2> compared to the data in the attached figure.

Thanks again,
Maria

PS: Also, do you know why the error bar increases for large time?

maria87 I 'm a little embarassed by

I 'm a little embarassed by the fact that this is a simple calculation and I'm not getting it. Please be kind and help me clear this up.  Anyone? There must be someone on the forum who knows how to calculate Mean-Squared Displacements.

Irtaza Did you get solution about

Did you get solution about how to calculate mean square displacement.?
If yes please share.