This thread probably needs to be moved to Health, but that's ok.
Your chinchilla could be chewing his fur. Have you noticed your chin grooming that area a lot, picking at it or anything?
When they chew their fur it's usually due to stress, so if that's the case, start paying attention to your chins environment. Could it be bored? Are there loud noises or lots of things to disturb the chin?
Also, how does the skin look under the lost fur? Healthy? Dry, red, or scaly? Is the fur loss getting worse or is it just one area? My chin had a small patch of dry skin on his foot this winter, the fur fell off, and didn't start regrowing until I fixed the dry skin problem by cutting back on dust baths and applying neosporin to the area every day.
Another possible culprit is a fungus. There are pictures on the forums under health of what fungus looks like and what dry skin looks like. If your chin does have a fungus, all you have to do is put a little athlete's foot powder in their dust bath. After a while, that will clear it up.
The environment when she's running around shouldn't be causing this problem-at least not in any way that I can think of. As far as how long it takes to grow back, I'm not sure.