Random Walk Distance question

I know that $s=x \sqrt {N}$ gives the distance moved by the particle after N number of collision of intervals xm between collisions.

1. Why is it the square root of N?

2. My revision guide says that it will take 10,000,000 steps to travel 1m from the starting point, assuming collisions take 1x10-7m.
however, $s=(1\times 10^-7) \times \sqrt {10,000,000} \not= 1m$

The average displacement you expect to be is zero, because the positive and negative displacements cancel out. However, the square of your displacement should increase with N. How much?

The average displacement (squared) after one step is always 1. You can then prove that the difference in the squares between D²_N and D²_N-1 is equal to 1. Induction gets you the rest of the way.

That seems a little complicated for sixth form level :/
But thank you


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The whole business dealing with squares is just to get rid of the fact that otherwise you get zero for average displacement, it's very similar to why you take a root mean square with alternating voltages.

I don't think I explained how you actually get there very well TBH. Just know that if you want to prove it, the simplest way is probably to prove that after N steps, the average of your squared distance away ought to be N as well.

The average displacement you expect to be is zero, because the positive and negative displacements cancel out. However, the square of your displacement should increase with N. How much?

The average displacement (squared) after one step is always 1. You can then prove that the difference in the squares between D²_N and D²_N-1 is equal to 1. Induction gets you the rest of the way.