Complexity Explorer Santa Fe Institute

Random Walks

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4.4 Part IV » Quiz Solutions

Question 1

In a single event, the walk hops 2 steps to the left or 2 steps to the right equiprobably.  The diffusion coefficient is proportional to the average mean-square displacement in a single step, $\langle x^2\rangle$.  

For the nearest-neighbor random walk, $\langle x^2\rangle = \frac{1}{2}\times 1+\frac{1}{2}\times 1=1$. 

For a walk whose steps have length 2, this same computation gives $\langle x^2\rangle = \frac{1}{2}\times 4 +\frac{1}{2}\times 4=4$. 

Thus the diffusion coefficient is increased by a factor of 4.