What is Liouville's theorem?

[Pr0x1mo]

I know this is a broad question, but can someone explain to me, in the most laymen's way, what this theorem is?

 

[Pr0x1mo]

anybody??

 

[mathman]

http://astron.berkeley.edu/~jrg/ay202/node27.html

I got the above reference using Google (Liouville's theorem). There are a lot more.

 

[Crosson]

Hmm, think of a statistical system that consists of many repetitions of the same subsystem. All of the subsystems can be in different states at the same time.

So imagine a function which maps each state a subsytem could be in (characterized by positions, momentums etc) to the number of subsystems within the statistical system which are currently in that state. What I have described is a number density in phase space, analagous to the density of a fluid p(x,y,z).

Liouville's theorem says that under certain conditions this fluid is incompressible, that is the number density in phase space is a constant (in time).

Maybe I will get in trouble with others for being too imprecise, or maybe that wasn't really very satisfying for you. You know what Feynman said, "If I'm making sense I'm lying, if I am telling the truth I'm not making sense", of people who wanted a watered down QED.

 

[SpaceTiger]

Liouville's theorem says that under certain conditions this fluid is incompressible, that is the number density in phase space is a constant (in time).

Make sure you specify that this is only true if you're following an element of the fluid. It's not true at a given point in phase space:

$$\frac{Df}{Dt}=0$$

$$\frac{\partial f}{\partial t} \ne 0$$

 

[Chronos]

It is an exercise in circular logic.. my 2 cents worth.