At time , there are 1000 points at the left edge of the box, representing the initial condition at . At each time step, these 1000 points will diffuse by a random walk (Monte Carlo simulation) along the box. The concentration at distance is the density (number of points) at that position.
The analytical solution of the equation is , where is the diffusion coefficient and is the initial concentration, shown as the red curve. The concentration profiles using Monte Carlo simulation are shown as blue dots.