Undergrad Is something wrong with my understanding of Liouville's Theorem?

[fox26]

One version of Liouville’s Theorem for non-dissipative classical systems, governed by a conserved Hamiltonian, is that the volume in phase space (position-momentum space) of an ensemble of such systems (the volume is the Lebesgue measure of the set of points where the ensemble’s density is non-zero) does not vary with time. This is equivalent to the version of the Theorem which states that the density of systems along a flow line does not vary with time.

This so-called Theorem seems to be shown to be false by the following simple counterexample.

Let each system in the ensemble consist of one point or extended spherically symmetric non-rotating classical particle with mass 1 in an otherwise empty 3-dimensional Euclidean space. This system is non-dissipative and governed by a conserved Hamiltonian. The phase space is then a 6-dimensional space with 3 dimensions of space (position) and 3 of (linear) momentum. Let the ensemble in question at time t = 0 consist of such systems distributed with non-zero density d everywhere in a 6-dimensional hypercube with sides of length 1 (3 position, 3 momentum), centered on the origin and with sides parallel to the coordinate axes, and zero elsewhere. The ensemble’s phase space volume will then be 1, and its density d inside the cube and zero elsewhere. At time t = 1 the ensemble will have evolved to have non-zero density everywhere in a 6-dimensional rectilinear parallelepiped with edges still of length 1 in the 3 momentum coordinates, but of length 2 in the 3 position coordinates. The ensemble’s phase space volume will then be 8, and its average density d/8 inside the parallelepiped and zero elsewhere.

It seems that both the ensemble’s phase space volume and its phase space density have varied with time. Comments?

 

[Haborix]

I'm not convinced you have calculated the volume of this n-parallelopiped correctly. Would you agree that there is no problem in a 2d phase space $(q,p)$ using the same setup you prescribed in 6d?

 

[fox26]

I'm not convinced you have calculated the volume of this n-parallelopiped correctly. Would you agree that there is no problem in a 2d phase space $(q,p)$ using the same setup you prescribed in 6d?

In 2d phase space the volume of the ensemble at t = 1 would be 2 and the average density d/2, still a problem. BTW, I just edited the post because I realized that at t = 1 the density, while non-zero everywhere in the parallelepiped of volume 8, won't be the same everywhere in it.

 

[Lord Jestocost]

In 2d phase space the volume of the ensemble at t = 1 would be 2 and the average density d/2, still a problem.

Did you get something like Figure 1. (a) in
[PDF]
Liouville's Theorem - Inside Mines - Colorado School of Mines

 

[fox26]

Did you get something like Figure 1. (a) in
[PDF]
Liouville's Theorem - Inside Mines - Colorado School of Mines

No, but I should have. Your linked article has cleared up my confusion. I should have thought more about this problem, and realized that there was indeed something wrong with my (mis)understanding of the situation. I carelessly failed to realize that while the total position space in each dimension occupied by some system in the ensemble at t = 1 would be [-1,1], the space occupied by systems of any momentum p would depend on p, all systems with momentum p occupying a total position space interval of length only 1, whose center varied between -1/2 and +1/2 as p varied between -1/2 and +1/2. Thanks.