  A 2016 paper by M Wilkinson in Physical Review Letters suggests that
large-deviation theory is a suitable framework for studying unexpectedly rapid
rain formation in collector-drop collision processes. Wilkinson derives
asymptotic approximation formulae for a set of exact large-deviation functions,
such as the cumulant generating function and the entropy function. The
asymptotic approach assumes a large number of water droplet collisions and is
motivated by the fact that the exact large-deviation functions are
prohibitively difficult to deal with directly. Wilkinson uses his asymptotic
formulae to obtain further results and also provides numerical work which
suggests that a certain log-density function for the collector-drop model
(which is a function of his asymptotic approximation formulae) is itself
approximated satisfactorily. However, the numerical work does not test the
accuracy of the individual asymptotic approximation formulae directly against
their exact large-deviation theory counterparts. When these direct checks are
carried out, they reveal that the asymptotic formulae are, in fact, rather
inaccurate, even for very large numbers of collisions. Their individual
inaccuracy is masked by their incorporation into log-density functions in
Wilkinson's numerical work. Their inaccuracy, as well as some assumptions
underlying their derivation, severely limit their applicability. The present
note points out that it is quite possible to develop accurate analytical (i.e.,
non-asymptotic) approximation formulae for the large-deviation theory functions
in the collector-drop model which also preserve the forms of the leading order
power terms in Wilkinson's asymptotic formulae. An analytical approximation
approach can be developed based on a Euler-Maclaurin formula. The resulting
analytical formulae are extremely accurate and valid for all relevant numbers
of collisions and time scales.
