Why are Saturn's rings so thin? Take a look at this picture (from APOD https://apod.nasa.gov/apod/ap110308.html):

I presume that rocks within rings smash each other.  Below the picture there is a note which says that Saturn's rings are about 1 km thick.
Is it an explained phenomenon?
 A: There seems to be a known explanation. I quote from Composition, Structure, Dynamics, and Evolution of Saturn’s Rings, Larry W. Esposito (Annu. Rev. Earth Planet. Sci. 2010.38:383-410):

[The] rapid collision rate explains why each ring is a nearly flat disk. Starting with a set of particle orbits on eccentric and mutually inclined orbits (e.g., the fragments of a small, shattered moon), collisions between particles dissipate energy but also must conserve the overall angular momentum of the ensemble. Thus, the relative velocity is damped out, and the disk flattens after only a few collisions to a set of nearly coplanar, circular orbits.

I think the key is that particles in a thick ring would not move in parallel planes but would have slanted trajectories, colliding all the time and losing their energy very fast.  
A: If one could assume that the ring is a continuous distribution of mass, we could try minimizing the total energy of the system (self energy + energy of interaction with Saturn). These two conditions along with the condition that total mass of the disc is a constant, would (I think) leave us with a unique geometry (inner and outer radius, thickness).
EDIT: Some Googling gave this paper: http://dx.doi.org/10.1016/0019-1035(79)90084-8
A: The rings have formed where they are because there is greater gravity there. The reason is the shape of the gas giant. It is far from circular. It is wider at its equator. Centripetal force causes a thicker equator. Therefore because the thickness is far wider at the equator, the gravitation at the equator is more than that at the poles. There is more mass pulling you down at that point. The particles of the rings fall into a gravity well. 
This explanation is self evident because as can be seen from the above photo, the rings are so incredibly thin, and are PERFECTLY aligned with the equator.
Compare the reason why the moon only shows us one face. It must have more mass on one side. It is like a ship floating on the ocean. The ship has more mass in its hull that gravity pulls into the water. 
On the gas giants, given time every asteroid has worked their way to the equator.
I would imagine being an asteroid, it would take a very long time of maybe thousands of years and collisions to achieve such a thin presence. It is very beautiful though! 
