Question
Consider the random walk problem with $p=q$ snd let $m=n_{1}-n_{z}$ denote the net displacement to the right. After a total of $N$ steps, calculate the following mean values: $\bar{m}, \overline{m^{2}}, \overline{m^{3}}$, and $\overline{m^{4}}$.
Step 1
In this problem, we have a random walk where at each step, there is an equal probability \( p = q = \frac{1}{2} \) of moving right or left. After \( N \) steps, let \( n_1 \) be the number of steps taken to the right and \( n_2 \) be the number of steps taken to Show more…
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Random flights. A particle follows in space a path that consists of $N$ equal steps, each of length $s .$ The direction in space of each step is entirely random, with no relation or correlation between any two steps. The total displacement is $$ \mathbf{S}=\sum_{i=1}^{N} \mathbf{s}_{i} $$ Show that the mean square displacement between initial and final positions is $\left\langle\mathrm{S}^{2}\right\rangle=\mathrm{Ns}^{2}$, where \langle\rangle denotes mean value. [Hint: The assumption that the direction of every step is independent of the direction of every other step means that $\left\langle s_{i} \cdot \mathrm{s}_{j}\right\rangle=0$ for all $i$ and $i$, except $\left.i=j .\right]$
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