DIFFUSION OF MATERIAL AWAY FROM A PLANE
The concentration is a Gaussian distribution (in the field of statistics, the same curve gives the probability of an error occurring with a given deviation from the mean value). M is the total mass (at t=0 the concentration in the plane is infinite, but the total amount is not).
The profiles are drawn for times that increase by a factor of 4. At x = 0, the exponential term equals 1, and thus the height is equal to the "normalization" factor in front of it. The total mass over the square root of time appears there, so the height of the profile decreases two fold in each example as the time increases 4 fold.
In the exponential term, we see that if t increases by a factor of 4, we need to go out only twice as far (x must be twice as large) to find the same value. Thus, the profile becomes twice as wide when the time increases 4 fold.
This solution is rather special, because any initial concentration profile can be arbitrarily well approximated by the sum of a large number of planes at different positions with different amounts of material in them. Thus, all solutions of the diffusion equation can be thought of as a sum of a series of these solutions. Special solutions like this one are important in other areas of mathematical physics, and are known as Green's functions.