Exercise 11.3: A random walk in two dimensions

Consider a collection of ``bees'' which are initially localized in a circle of unit radius centered about the origin. At each time step, each bee moves at random with equal probability to one of four possible directions north, south, east, or west.

1. Implement a program to simulate the motion of the bees, and describe the qualitative nature of their motion.

2. Suppose that each bee is placed at random within a circle of unit radius and is given an initial velocity in one of the four directions. This means that at each time interval, a bee takes an additional step in the direction of the original velocity. Is the motion of the bees changes from part 1?

3. Compute $\langle x_N \rangle$, $\langle y_N \rangle$, $\langle \Delta x_N^2 \rangle$ and $\langle \Delta y_N^2 \rangle$ as a function of the number of steps $N$. The average is over all the $M$ bees. Also compute the mean square displacement defined as
   
   $$\langle \Delta R_N^2 \rangle = \langle x_N^2 \rangle + \lang... ...^2 \rangle - \langle x_N \rangle ^2 - \langle y_N \rangle ^2.$$
   
   
   What is the qualitative dependence of these quantities on the number of steps? Compute the $N$-dependence of $R_{max}^2$, the maximum displacement of the bees as step $N$. Is the behavior of $R_{max}^2$ qualitatively different that $\langle \Delta R_N^2 \rangle$ for all $N$?