Exercise 11.1: Random walks in 1d

1. Implement a program to calculate the probabilities $P_N(x)$ for a given $N$.

2. Determine $P_N(x)$, $\langle x_N \rangle$, and $\langle \Delta x_N^2 \rangle$ for $p=0.5$, and $N=8,16,32,64$. Compare your results to the exact answers.

3. Plot $P_N(x)$ versus $x$ for the above values of $N$. Is $x$ a continuous function? What is the value of $P_N(x)$ at its maximum for each value of $N$? What is the approximate ``width'' of $P_N(x)$ in each case?

4. Show from your previous plot that for sufficiently large $N$, the computed distribution can be approximated by the Gaussian
   
   $$P(x)=\sqrt{\frac{1}{2\pi \sigma^2}}\exp{[-(x-\langle x \rangle )^2/2 \sigma ^2]}$$
   
   
   where $\sigma^2=\langle \Delta x^2 \rangle$. Does the fit work equally well for all $x$?

5. Suppose that $p=0.7$. Compute $\langle x_N \rangle$ and $\langle \Delta x_N^2 \rangle$ for the same values of $N$ used in part 2. What is the interpretation of $\langle x_N \rangle$ in this case?

6. Make a log-log plot of $\langle \Delta x_N^2 \rangle$ vs. $N$ and analyze the behavior for large.

Many applications of random walk models make use of asymptotic behavior for large $N$. For example in many random walk models,

$$\langle \Delta x_N^2 \rangle \sim N^{2\eta} (N \gg 1).$$

This is an example of a ``scaling law''. That is, if the number of steps is doubled, the net mean square displacement increases by a factor $2^{\eta}$. For one-dimensional random walks we find from the exact solution described above that $\eta = 1/2$. If this kind of scaling law exists, its exponent depends on the geometry of the lattice, as well as the nature of the walk.