Welcome to BYC. Where, in general, are you located? Climate matters.
Sterilizing the run is neither possible nor desirable -- you'd eliminate all the good bacteria along with the problems.
The first thing to do about a muddy run is to figure out why it's muddy.
Is water running into the run off the coop or other nearby buildings? If yes, you need to install gutters to direct water away from the chickens?
Is rainwater flowing through the run via a natural drainage channel in the land? If yes, you need to dig diversions ditches, create grass swales, and/or install French drain to divert the water away from the chickens.
Is water collecting because the run is in a natural low spot in the ground? If yes, you need to either move your coop and run to a place with good drainage or you need to bring in fill to raise the level up. This is best done by a professional who really understand grading so that it doesn't cause other water problems elsewhere -- especially around your house.
Is the problem your climate -- seasonal (or constant), heavy rains? Putting a roof on the run is likely to help.
Beyond this, the best thing to do about mud in the run is to put down dry organic material. Coarse wood chips, the sort you get from a tree-trimming service, are often considered the gold standard for keeping a run clean and fresh. Here in the USA's Steamy Southeast I have access to pine straw (the fallen needled of the Loblolly and Longleaf pines), which are especially good in heavy rain because they drain freely and always dry out on top quickly after the rain but they aren't available in most areas.
