Today, in a serious of fascinating and adrenaline-filled adventures (misadventures?), I’ve finally learned how a toilet works. To spare you the hazards of plumbing, an experimentalist’s overview (very detailed) is here:
http://home.howstuffworks.com/toilet4.htm
Of course there’s a lot of extra information in there, details of experimental apparatus and such - well experimentalists do need that kind of stuff. From a theory standpoint it’s rather superfluous, of course. Here’s how it works: there’s a single parameter, call it \phi, and it’s equation of motion (determined experimentally) appears to be:
\frac{d \phi}{\dt} = \frac{\lambda}{b-a}\left( \theta(\phi-a) - \theta(\phi-b) \right)(b-\phi) + \lambda\theta(a-\phi)
where \theta is the Heaviside step function, b-a is fairly small, and a<b. \phi is a classical parameter, not a quantum field, that experimentalists say describes the “water level” of the toilet reservoir; all we really need to know is that it’s a real-valued scalar, and that the classical approximation is okay here (no need to quantize anything). Of particular concern is the region of phase space \phi << a, wherein \dot{\phi} remains at +\lambda for a substantial interval of time and one is concerned about bathroom flooding. A proper understanding of the E.O.M. shows that such fears are groundless.
(The Hamiltonian from which this motion dervies is fairly trivial, and is left as an exercise.)
bat.JPG
