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Models without an explicit time dependence, called singular models, arewidely used for fitting the distribution of temperatures at which waterdroplets freeze. In 1950 Levine developed the original singular model. His keyassumption was that each droplet contained many nucleation sites, and thatfreezing occurred due to the nucleation site with the highest freezingtemperature. The fact that freezing occurs due to the maximum value out oflarge number of nucleation temperatures, means that we can apply the results ofwhat is called extreme-value statistics. This is the statistics of the extreme,i.e., maximum or minimum, value of a large number of random variables. Here weuse the results of extreme-value statistics to show that we can generaliseLevine's model to produce the most general singular model possible. We showthat when a singular model is a good approximation, the distribution offreezing temperatures should always be given by what is called the generalisedextreme-value distribution. In addition, we also show that the distribution offreezing temperatures for droplets of onesize, can be used to make predictionsfor the scaling of the median nucleation temperature with droplet size, andvice versa.

Generalisation of Levine's prediction for the distribution of freezing temperatures of droplets: A general singular model for ice nucleation