I Query about statistical ensemble and Liouville's Theorem

[Shan K]

Hi,
I was studying about the statistical ensemble theory and facing some problem to understand these concepts ,
I have understood that the ensemble is a collection of systems which are macroscopically identical but microscopically different . In some books they are called as systems with different initial conditions .
Now when they are trying to prove the Louville's Equation concerning these elements in ensemble, they are saying that these phase points are moving . My question is what kind of movement is this ? Are they moving as a bulk ? I know that these systems can replace their position in the phase space as the time goes on . Are they assuming this movement ?
My second question is, in the proof of the conservation of extension in phase space they are saying that two curves which are followed by two different elements in the ensemble can't intersect each other . Why?

 

[Timo]

I'm not exactly sure what you read. So here's some comments of mine to your questions according to my interpretation of what you wrote:
1) The phase space of a single point-particle in 1D is the (p,x) plane, where p is the particle's momentum and x its position. At each time, the system consisting of this particle will take one point in this phase space. At another time the particle will likely be at another position, and d(x,p)/dt is continuous. In that sense, the system moves in phase space. The same holds true for more complex systems, and the same holds true for each system in an ensemble of systems. So the answer to your question probably is "yes".
2) If your system is deterministic and there is no explicit time dependence of the equation of motion then for each current state (x,p) there must be a unique state (x', p') that the system assumes some time later. If the equation of motion can be reversed in time then there also cannot be two different states (x1, p1) and (x2, p2) that develop into the same state (x', p') some identical time later. I think these criteria are sufficient to claim that trajectories in phase space cannot intersect (but note that they can be identical).

 

[Shan K]

I'm not exactly sure what you read. So here's some comments of mine to your questions according to my interpretation of what you wrote:
1) The phase space of a single point-particle in 1D is the (p,x) plane, where p is the particle's momentum and x its position. At each time, the system consisting of this particle will take one point in this phase space. At another time the particle will likely be at another position, and d(x,p)/dt is continuous. In that sense, the system moves in phase space. The same holds true for more complex systems, and the same holds true for each system in an ensemble of systems. So the answer to your question probably is "yes".

Thanks for the reply.
But what I am saying is that for an ensemble, it consists of all the possible accessible microstate of a system. Then the time evolution of those elements will totally lie in that ensemble, isn't it right ? So in equilibrium how the elements can flow through the phase space ?

 

[Timo]

It is a bit a question about the point of view. Consider a constant stream of water running through a pipe section (or running around in a frictionless torus if you also want the molecules to be the same). In a sense, there is motion. On the other hand the density in the pipe is constant with time at each location.

Similarly, you can see a point in phase space moving away with time evolution and being replaced by another one. And in fact without having thought it through I think that Liouville's theorem directly leads to statements such that the microcanonical ensemble, in with all possible states are being weighted equally, in a finite phase space is a stationary distribution: Any subset of states remains its volume and any phase-space covering sets of disjoint subsets of states is still disjoint after time evolution (because trajectories may not intersect) and still covers the full phase-space (because they are disjoint and the sum of their volumes equals the total volume) => stationary distribution in the "constant density of weights at each location"-sense.