- #1
jordi
- 197
- 14
I am trying to understand ergodic theory, i.e. how simple systems reach equilibrium.
I consider a classical particle in a 2D (or 3D) box. Funnily, I have never seen this example in books (probably due to lack of knowledge). Instead, in QM, the particle in a box is a prototypical example.
My question is: is a classical particle in a 2D box ergodic? In other words: given any point in the 4D phase space (position and momentum), will the particle visit any other point in the 4D phase space, and when doing so, with the same probability for all states?
From inspection, it seems clear there are some points in the phase space for which ergodicity fails: for example, when the particle moves parallel to any of the walls (the particle keeps bouncing up and down through the same line). Also, there are some other "cycles", such as four rebounds, for example hitting with 45 degrees, which results in a cycle parallel and perpendicular to the diagonals of the box.
However, it could be that these cycles are a "measure zero" of all points in the phase space, so ergodicity could still hold "almost always" (in probabilistic terms).
Or maybe not? Could it be that for any starting point in the 4D phase space, eventually there is a cycle, that returns to the original point in the phase space after a finite amount of rebounds on the wall?
Is this classical example, apparently so simple (no interactions!) been solved?
I consider a classical particle in a 2D (or 3D) box. Funnily, I have never seen this example in books (probably due to lack of knowledge). Instead, in QM, the particle in a box is a prototypical example.
My question is: is a classical particle in a 2D box ergodic? In other words: given any point in the 4D phase space (position and momentum), will the particle visit any other point in the 4D phase space, and when doing so, with the same probability for all states?
From inspection, it seems clear there are some points in the phase space for which ergodicity fails: for example, when the particle moves parallel to any of the walls (the particle keeps bouncing up and down through the same line). Also, there are some other "cycles", such as four rebounds, for example hitting with 45 degrees, which results in a cycle parallel and perpendicular to the diagonals of the box.
However, it could be that these cycles are a "measure zero" of all points in the phase space, so ergodicity could still hold "almost always" (in probabilistic terms).
Or maybe not? Could it be that for any starting point in the 4D phase space, eventually there is a cycle, that returns to the original point in the phase space after a finite amount of rebounds on the wall?
Is this classical example, apparently so simple (no interactions!) been solved?