In the appendix of the paper I reference below is a summary derivation of the relationship  between entropy and fractals (power laws). The appendix is also posted in another blog, Plate Frames. The introduction to the post reads...

In a paper published a couple of years ago (Pilger, 2012), I describe the application of a simple principle, transformed into a distinctive abstract object, to an optimization problem (within the plate tectonics paradigm): simultaneous reconstruction of lithospheric plates for a range of ages from marine geophysical data . It is rare that the relation of the principle, maximum entropy, with a particular transformation, power-series fractals, is recognized, since Pastor-Satorras and Wagensberg derived it. I'm unaware of any other application of fractal forms to optimization problems analogous to the paper. The following derivation is taken from the 2012 paper, with slight modification, in hopes that it might prove useful in other fields, not merely the earth sciences, but beyond. I'm investigating  applications in a variety of other areas, from plate tectonics, to petroleum geology, and, oddly enough, the arts.

Pilger, R. H., Jr. (2012) Fractal Plate Reconstructions with Spreading Asymmetry, Marine Geophysical  Research, Volume 33, 149-168. dx.doi.org/10.1007/s11001-012-9152-6. (rexpilger (at) gmail (dot) com.)