Instantaneous plane, line, or point source

The mathematical translation of the plane-source problem is as follows. Initially, there is a finite amount of mass M but very high concentration at a = 0, i.e., the density or concentration at a = 0 is defined to be infinite (which is unrealistic but merely an abstraction for the case in which initially the mass is concentrated in a very small region around a = 0). The initial condition is not consistent with that required for Boltzmann transformation. Hence, other methods must be used to solve the case of plane-source diffusion. Because this is the classical random walk problem, the solution can be found by statistical treatment as the following Gaussian distribution  

If the plane source is on the surface of a semi-infinite medium, the problem is said to be a thin-film problem. The diffusion distance stays the same, but the same mass is distributed in half of the volume. Hence, the concentration must be twice that of Equation 3-45a  

The solution to a line-source (i.e., along the z-axis at a = 0 and y=0) problem with total mass M is 

These sets of equations also describe the classic case of Brownian motion or random walk. The initial condition is that all M particles were at the central point, and then spread in one dimension (along a line), two dimensions (along a 

For two-dimensional (line-source) diffusion, the mean square displacement is 4Dt. For three-dimensional (point-source) diffusion, the mean square distance is 6Dt. 

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