Energy loss in Capacitors

The missing piece in the puzzle is dynamics.

The system can only settle in the stationary end configuration if some sort of damping, i.e., loss of energy, forces it to.

Without damping, the charge would actually oscillate forever between the capacitors; or, in Samuel's example, the water would keep slushing from one fish tank to the other, with energy being converted between kinetic and potential periodically, as in a pendulum.

"My question is how are we getting the energy loss without taking into consideration any terms that cause it at all? Or have we taken it and I missed it?"

You know, I was also puzzled by the very same question when I was presented with the fact that there is an energy loss in going from a single charged capacitor configuration to a configuration where the the same change is distributed among two capacitors. The basic fact is that if you assume that (1) charge is conserved and (2) the voltages across each of the two capacitors in the two-capacitor configuration are equal to each other, then the total energy of the one-capacitor configuration MUST be greater than the total energy of the two-capacitor configuration by the amount shown by the equation you presented. The only way to get the energies of the two different capacitor configurations to equal each other is to do something like add some charge to the two-capacitor configuration to raise its energy up a bit to the same energy value as the one-capacitor configuration.

It is interesting that we don't have to consider any particular energy loss mechanisms to arrive at this conclusion that some energy must be lost to heat, vibrations, sound, etc., but there are other examples that are similar to this. For example, suppose that you have two rectangular 50-gallon fish tanks, one of which is full of water and the other empty. There is a small pipe connection between the two fish tanks at near the their bottoms and a valve so that you can open this pipe connection and allow water to flow between them. You open the valve and watch as the water from the completely full fish tank quietly flows into the other fish tank until the water in the two tanks are level with each other and the flow stops.

You might think that, in principle, it's possible that no energy was lost in making the water transfer, but if you actually calculate the total gravitational potential energy of associated with the initial fish tank configuration (i.e., all the water in one fish tank) with the total gravitational potential energy of associated with the final fish tank configuration, you will find that there is an energy difference: Some amount of energy MUST have been necessarily lost to heat, sound, etc.. in transferring water from one fish tank to another in this way.