In dealing with random walk and diffusion if stepsize S is d

In dealing with random walk and diffusion, if step-size (S) is doubled, will the root mean square distance (RMS) double? Explain mathematically.

Solution

the root mean square distance (RMS) is,

     d_RMS = S sqrt(N)

Here, S is the step-size and N is the number of steps.

from the above relation, the root mean square distance is directly proportional to step-size.

if the step-size is doubled, the root mean square distance is doubled.