For this actual issue I think the physical setup can most easily be thought of as a tray upon which a set of non-overlapping colored disks are randomly placed. All disks of the same color have the same diameter. An equation that calculates a type of disk density can be written as follows. $$y=10^DN$$ where $N$ is the total number of disks per unit area (not necessarily an integer) and is calculated as follows. $$N=\sum_{i=1}^k n_i$$ where $n_i$ is the number of disks per unit area for color $i$ (not necessarily an integer), and $k$ is the total number of colors.
$D$ is the root mean square diameter (or quadratic mean diameter) of all of the disks, calculated as follows. $$D=\sqrt{\left(\sum_{i=1}^k\left(n_id_i^2\right)\right)/N}$$ where $d_i$ is the diameter of disks of color $i$.
Is it possible to calculate the contribution to $y$ for disks of color $i$ (I'll refer to this as $y_i$) such that the following holds true? $$y=\sum_{i=1}^ky_i$$ If this is possible, how is it done? I searched this site but can't find anything that helps. This could be because I'm not using the right search terms, but I've tried everything I can think of. If there is an existing question/solution that will help, please let me know.
The closest thing I could find is a paper titled "Theoretical Foundation of Stage's Formulation of Stand Density Index", which is on page 64 of this document: https://www.srs.fs.usda.gov/pubs/gtr/gtr_srs224.pdf. It shows how this can be done for a similar equation for tree trunks (rather than disks). It uses first order derivatives of a homogeneous function, but that won't work in this case. Any help would be greatly appreciated.