Describing system in terms of 2 variables vs natural variables

[I<3NickTesla]

Realised I probably posted this in the wrong forum before, should've been here..

I often see a function's differential expressed in terms of convenient partial derivatives eg
dU=(dU/dT) dT + (dU/dV) dV

And I've seen it written that "any system is uniquely specified by two parameters, such as pressure and volume, or perhaps pressure and temperature"

But then what's the deal natural variables? What's so "natural"/good about them if any pair will do? By natural I mean that I've seen the natural variables for gibb's energy as pressure and temperature.

 

[BruceW]

right, for the Gibbs free energy, the natural variables are pressure and temperature. But there are other thermodynamic potentials than just the Gibbs free energy, and they will have some other natural variables. http://en.wikipedia.org/wiki/Thermodynamic_potential
This website is pretty good. I'll admit that I don't know as much about statistical physics as I would like to know. It is a pretty interesting topic. And then it naturally leads to things like quantum field theories. (i.e. related to renormalisation and phase transitions and symmetry and stuff).