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

Models without an explicit time dependence, called singular models, are widely used for fitting the distribution of temperatures at which water droplets freeze. In 1950 Levine developed the original singular model. His key assumption was that each droplet contained many nucleation sites, and that freezing occurred due to the nucleation site with the highest freezing temperature. The fact that freezing occurs due to the maximum value out of a large number of nucleation temperatures, means that we can apply the results of what 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 we use the results of extreme-value statistics to show that we can generalise Levine's model to produce the most general singular model possible. We show that when a singular model is a good approximation, the distribution of freezing temperatures should always be given by what is called the generalised extreme-value distribution. In addition, we also show that the distribution of freezing temperatures for droplets of one size, can be used to make predictions for the scaling of the median nucleation temperature with droplet size, and vice versa.