## 2 Drop Size Distribution Functions and Spray Science
### Defining Enoughness in Large Data Samples  
An experimentalist stops a measurement based of criteria related to a statistical uncertainty. The definition of statistical uncertainty contains three conditions:  
1. it is maximum for a uniform distribution where all classes have the same probability;
2. a small variation in the probability of a class generates a small variation in the uncertainty;3. and, finally, it depends on the distribution itself. The common measure used for the statistical uncertainty considers the standard deviation (\(\xi_{\overline{x}}\)) and the sample size (\(N\)) as  
\[\varepsilon=Z_{c}\frac{S_{\overline{x}}}{\sqrt{N}} \tag{4}\]  
with \(Z_{c}\) as the coefficient associated with the confidence interval considered (e.g., \(Z_{c}=1.96\) for a 95% Confidence Interval). When the mean value (\(\overline{x}\)) is different from zero, dividing \(\varepsilon\) by the mean provides the uncertainty in percentage. Since the standard deviation depends on the spray characteristics, reducing the statistical uncertainty implies adding more data to increase \(N\). However, as argued by [PERSON] [10], if the spray begins to operate in a different way and the distribution changes, the statistical uncertainty as defined in Equation (4) continues to decrease, without providing any evidence to the experimentalist about the changes occurring in the spray. In a certain sense, it fails to comply to the third condition defining a statistical uncertainty because it is more sensitive to the sample size than the distribution itself. For this reason, an approach based on information theory is a better option. In information theory, the Shannon entropy complies to all the aforementioned characteristics of a statistical uncertainty. Considering the probability values of any discrete distribution (\(p_{i}\)), the expression for calculating the Shannon entropy \(H\) is  
\[H=-\sum_{i}\left(p_{i}\ln(p_{i})\right) \tag{5}\]  
As an example, if we consider drop size, the minimum value corresponds to a monosize spray with all droplets having the same size, thus, \(p=1\), and \(H=0\). The maximum value occurs for a hypothetical spray where all droplets have the same probability of being present, corresponding to a uniform distribution with \(k\) classes, each for a different drop size. Therefore, the maximum Shannon entropy is \(max(H)=1/k\). In sprays, while measuring size and velocity, considering the Shannon entropy normalized by its maximum value, \(H_{n}=H/max(H)\), it tends to stabilize (see [PERSON] [10] for details). The meaning is that adding more data does not mean adding more information because the shape and scale of the distribution stabilized. However, as argued in [PERSON] [10], the _enoughness_ requires a criterion with interpretative value and proposes the _excess entropy_ (\(EE\)) because, as defined in [PERSON] et al. [11], it is what best captures the nature of convergence of the entropy rate as the amount of memory gained, or the cost of amnesia if all data would suddenly be lost. To evaluate the evolution of \(EE\) while measuring,  
* one calculates the entropy rate that quantifies the difference between the normalized Shannon entropy with adding one value the \(N\) samples and the Shannon entropy with \(N\) samples-\(\dot{H}_{n}=|H_{n}(N+1)-H_{n}(N)|\);
* considers the limit when \(N\rightarrow\infty\), which is \(h\);
* and the excess entropy \(EE\) is formulated as \(EE=\sum_{N=1}^{\infty}\left(\dot{H}_{n}(N)-h\right)\)  
In the case of spray data, the stabilization implies \(h=0\), thus, it simplifies \(EE\). The method proposed is setting a convergence criterion-\(\varepsilon_{EE}\) and stop measuring when \(|EE(N)-median(EE)|<\varepsilon_{EE}\), considering the number of samples corresponding to the median as the minimum required (see [PERSON] [10] for more details).