English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

A 2016 paper by M Wilkinson in Physical Review Letters suggests thatlarge-deviation theory is a suitable framework for studying unexpectedly rapidrain formation in collector-drop collision processes. Wilkinson derivesasymptotic approximation formulae for a set of exact large-deviation functions,such as the cumulant generating function and the entropy function. Theasymptotic approach assumes a large number of water droplet collisions and ismotivated by the fact that the exact large-deviation functions areprohibitively difficult to deal with directly. Wilkinson uses his asymptoticformulae to obtain further results and also provides numerical work whichsuggests that a certain log-density function for the collector-drop model(which is a function of his asymptotic approximation formulae) is itselfapproximated satisfactorily. However, the numerical work does not test theaccuracy of the individual asymptotic approximation formulae directly againsttheir exact large-deviation theory counterparts. When these direct checks arecarried out, they reveal that the asymptotic formulae are, in fact, ratherinaccurate, even for very large numbers of collisions. Their individualinaccuracy is masked by their incorporation into log-density functions inWilkinson's numerical work. Their inaccuracy, as well as some assumptionsunderlying their derivation, severely limit their applicability. The presentnote points out that it is quite possible to develop accurate analytical (i.e.,non-asymptotic) approximation formulae for the large-deviation theory functionsin the collector-drop model which also preserve the forms of the leading orderpower terms in Wilkinson's asymptotic formulae. An analytical approximationapproach can be developed based on a Euler-Maclaurin formula. The resultinganalytical formulae are extremely accurate and valid for all relevant numbersof collisions and time scales.

Accurate analytical approximation formulae for large deviation analysis of rain formation