Root mean square distance explanation

Question

We know that $D_{rms}=\sqrt N$ where $N$ is the number of steps taken by the random walker.

Now,consider a situation where a random walker walks $2$ steps in positive direction in the first two steps.

So,the $D_{rms}=\sqrt2\approx1.41$.

But the actual distance is $2$ units.

So why are the two values different? I am really a beginner in this field, so a detailed explanation is very useful.

Answer 1

There are four equally probable possibilities for the first two steps of this random walk:

• positive, positive: $D=+2$
• positive, negative: $D=0$
• negative, positive: $D=0$
• negative, negative: $D=-2$

so the square root of the mean of the squares of these is $\sqrt{\dfrac{4+0+0+4}{4}}=\sqrt{2}$ as expected. But this is before you know what actually happens.

Answer 2

Consider your two phrases "2 steps in positive direction in the first two steps", and "random walker".

How many steps does the walker need to take before it becomes "random"? Maybe 1000 random steps? This is more a problem of statistics/probability: how many samples do you need to get a certainty of a specified amount.

In your case, the answer 1.41 is correct, within the probability function. I.e. it will never be exactly 1.41, but it is likely that it is 1.41 ± X. (EDIT: For two steps, X = some number.) As the number of random steps increase, X decreases.