Model Hamiltonian
Reduced density operator
Kinetic equation for the momentum distribution
The collision operator can be diagonalized separetely
for each of the disjoint sets of momentum states.
an infinite number of disjoint sets of states
an infinite number of the collisional invariants
If
is a rational number, each of the disjoint sets contains
a finite number of momentum states.
If
is an irrational number, each of the disjoint sets contains
an enumerably infinite number of momentum states.
The first Brillouin zone is densely filled with the momenta
in each of the disjoint sets.
Similarity with Hofstadter butterfly?