A rederivation of the classical JMAK results for volume fraction transformed.

We assume that the system is an infinite line, planar surface, or a three or higher dimensional volume, in which there is no transformation at time 0, a constant (or time dependent) nucleation rate in the untransformed volume after time 0, and that each nucleus grows radially with a constant (or time dependent) rate until growth ceases at each point of impingement. The Johnson and Mehl method,[2] focusses on exact statistical estimates of the volume fraction transformed (which is , where is the fraction untransformed) by calculating the volumes transformed as if there were no impingement and no reduction in nucleation rate due to the reduction in and then making exact statistical estimates for the corrections for impingement and the reduction in nucleation rate. In this section we calculate directly the estimate that a point in the volume will be untransformed at time . This is done by computing the probability that no nuclei formed at all earlier times and positions , that could have grown to reach point at . (It has to be none; if there are some, it does not matter how many.)

We do this in three steps:

1. It will be convenient to combine a d-dimensional space with time into a single space, and define a d+1 dimensional space and a point in this space. In three dimensions, We then construct for the point a domain in this space that is the set of all points and earlier times that would have caused transformation at had nucleation occurred. This domain will be a subset of and will be designated by . Since its shape is a cone, it will be termed an ``influence cone'' or ``time cone.'' Its surface will be called an ``event horizon.''

2. Integrating the nucleation rate over the time cone to obtain the number of nuclei expected in .

3. Making use of the stochastic independence of the nucleation events to estimate

• The time cone when
• The expected number of nuclei in
• The probability, , that is untransformed at .