Brownian motion is the motion of a particle, such as a smoke or dust particle, in a gas, as it is buffeted by random collisions with gas molecules. Make a simple computer simulation of such a particle in two dimensions as follows. The particle is confined to a square grid or lattice L x L squares on a side, so that its position can be represented by two integers i, j = 0... L 1. It starts in the middle of the grid. In each step of the simulation, choose a random direction---up, down, left, or right---and move the particle one step in that direction. This process is called a random walk. The particle is not allowed to move outside the limits of the lattice---if it tries to do so, choose a new random direction to move in. (We choose an odd length for the side of the square so that there is one lattice site exactly in the center.) Write a program to perform a million steps of this process on a lattice with L = 101. Plot the position of the particle for different numbers of steps (100, 1000, 10000, 100000).