How I can't really tell you, as there is so many ways. Besides nuclear war and total destruction, it can be the government itself finally collapses on itself crossed with such distrust of it that everyone balkanize to individual city states or organizations. I am one of those who don't thing anarchy = mass chaos in the streets. I am very close to being an Anarcho-Capitalist in my politics (I fall between Minarchy and an Anarchist) and have read half of The Dispossessed, written by an Anarcho-Communist, putting it down only because I got too many other stuff I want to read. While the question of ownership of property is the biggest difference between us, Ursula Kroeber Le Guin vision of an AnCom world (actually moon) might seem like a post-app one of anarchists living peacefully, with all property shared and the concept of money does not exists.
My most ideal and peaceful effects of a big whoops:
With government weak or out of commission, people gather around in groups during (whatever big event is happening) to help survive. Once things start to calm down, and the realization anarchy now exists, those with the same beliefs band together. Like the game Alpha Centauri, who a person want to associate with deals more on ideology more than anything else. Instead of every man for himself, its every political, religious, or philosophical belief (and combination thereof) for themselves. Hopefully
something similar to this will happen. I know I am not the first with such thoughts, but don't know what the proper term for it is. Meanwhile those with knowledge of survival become the town managers of the world.
You might also want to take a peek at the following:
http://www.reddit.com/r/Apocalypse/
http://www.reddit.com/r/collapse
http://www.reddit.com/r/survivalist