Accurate analytical approximation formulae for large deviation analysis of rain formation

TL;DR

This paper critiques Wilkinson's large-deviation asymptotic formulae for rain formation, demonstrating their inaccuracy and proposing a new, highly accurate analytical approximation method using Euler-Maclaurin formulae.

Contribution

It introduces a non-asymptotic analytical approximation for large deviation functions in rain formation, improving accuracy over previous asymptotic methods.

Findings

Asymptotic formulae are inaccurate even for large collision numbers.

Proposed Euler-Maclaurin based formulae are highly accurate across relevant scales.

New formulae preserve leading order power terms of Wilkinson's asymptotic expressions.

Abstract

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.