I am a CS undergraduate. I understand how Turing came up with his abstract machine (modeling a person doing a computation), but it seems to me to be an awkward, inelegant abstraction. Why do we consider a "tape", and a machine head writing symbols, changing state, shifting the tape back and forth? 

What is the underlying significance? A DFA is elegant - it seems to capture precisely what is necessary to recognize the regular languages. But the Turing machine, to my novice judgement, is just a clunky abstract contraption.

After thinking about it, I think the most idealized model of computation would be to say that some physical system corresponding to the input string, after being set into motion, would reach a static equilibrium which, upon interpretation equivalent to the the one used to form the system from the original string, would correspond to the correct output string. This captures the notion of "automation", since the system would change deterministically based solely on the original state.

Edit: 

After reading a few responses, I've realized that what confuses me about the Turing machine is that it does not seem minimal. Shouldn't the canonical model of computation obviously convey the essence of computability?

Also, in case it wasn't clear I know that DFAs are not complete models of computation.

Thank you for the replies.

Well, a DFA is just a Turing machine that's only allowed to move to the right and that must accept or reject as soon as it runs out of input characters. So I'm not sure one can really say that a DFA is natural but a Turing machine isn't.

Critique of the question aside, remember that Turing was working before computers existed. As such, he wasn't trying to codify what electronic computers do but, rather, computation in general. My parents have a dictionary from the 1930s that defines computer as "someone who computes" and this is basically where Turing was coming from: for him, at that time, computation was about slide rules, log tables, pencils and pieces of paper.  In that mind-set, rewriting symbols on a paper tape doesn't seem like a bad abstraction.

OK, fine, you're saying (I hope!) but we're not in the 1930s any more so why do we still use this? Here, I don't think there's any one specific reason. The advantage of Turing machines is that they're reasonably simple and we're decently good at proving things about them. Although formally specifying a Turing machine program to do some particular task is very tedious, once you've done it a few times, you have a reasonable intuition about what they can do and you don't need to write the formal specifications any more. The model is also easily extended to include other natural features, such as random access to the tape.  So they're a pretty useful model that we understand well and we also have a pretty good understanding of how they relate to actual computers. 

One could use other models but one would then have to do a huge amount of translation between results for the new model and the vast body of existing work on what Turing machines can do. Nobody has come up with a replacement for Turing machines that have had big enough advantages to make that look like a good idea.
