# What is Liouville's theorem all about - Thermodynamics

1. May 31, 2006

### annetjelie

I'm having difficulties understanding what liouville's theorem is all about. I was "meditating" over an adiabatic free expansion and i got stuck because of a contradiction in my reasoning so it seems i still don't have a clue what liouville actually wants me to understand :)

So imagine an ideal gas in volume V that adiabatically expands (free) to a volume 2V. This means more microstates.

Then my contradiction: liouville says that the volume of the microstates (of the macrostate) in phase space under hamiltonian flow stays the same. And as the energy of an ideal gas after free expansion also remains the same, this would mean that the volume of the microstates corresponding to the first macrostate would stay the same
So i get the same "amount" of microstates for the final macrostate which is in contradiction with the above statement and is just plain wrong

Clearly there's a fundamental flaw in my reasoning, and it would mean a lot if someone could give me a nudge in the right direction! Thanks!

*edit* okay i meditated a little more and i guess it's because you have to find a reversible path between the initial and final state because it's an irreversible process, so you can't use Q=0 (and then E won't be constant). Is this right?

Last edited: May 31, 2006
2. Jun 1, 2006

### Staff: Mentor

If the number of gas atoms/molecules N is contant, why would the number of microstates increase?

What is the definition of microstate.

3. Jun 1, 2006

### arnesmeets

Doesn't the number of macrostates increase because V increases?

4. Jun 1, 2006

### annetjelie

that's what i thought. If you expand a gas to a volume twice its initial volume than the phase space of that gas will enlarge by a factor 2^N with N the number of particles, and so there are more possible microstates. No??

5. Jan 3, 2009

### znet0201

Re: Liouville

Your doubt is correct. Liouville Theorm does not hold for irreversible process, like adiabatic expansion. Because the phase space is not conserved, as you know, the entropy change is positive.