Liouville’s theorem is more general than conservation of energy, I think, or at least it can hold even if conservation of energy fails. You can have a system with a time-dependent Hamiltonian, for instance, and thus no energy conservation, but with phase space volume still preserved by the dynamics. So this would be a deterministic system (one where phase space trajectories don’t merge) without energy conservation.
As for the minimum necessary set of conservation laws that must be knocked out to guarantee non-determinism, I’m not sure. I can’t think of any a priori reason to suppose that determinism would crucially rely on any particular set of conservation laws, although this might be true if certain further constraints on the form of the law are specified.
If I understood the Wiki article correctly, the assumption needed to derive Liouville’s theorem is time-translation invariance; but this is the same symmetry that gives us energy conservation through Noether’s theorem. So, it is not clear to me that you can have one without the other.
Liouville’s theorem follows from the continuity of transport of some conserved quantity. If this quantity is not energy, then you don’t need time-translation invariance. For example, forced oscillations (with explicitly time-dependent force, like first pushing a child on a swing harder and harder and then letting the swing relax to a stop) still obey the theorem.
Liouville’s theorem is more general than conservation of energy, I think, or at least it can hold even if conservation of energy fails. You can have a system with a time-dependent Hamiltonian, for instance, and thus no energy conservation, but with phase space volume still preserved by the dynamics. So this would be a deterministic system (one where phase space trajectories don’t merge) without energy conservation.
As for the minimum necessary set of conservation laws that must be knocked out to guarantee non-determinism, I’m not sure. I can’t think of any a priori reason to suppose that determinism would crucially rely on any particular set of conservation laws, although this might be true if certain further constraints on the form of the law are specified.
If I understood the Wiki article correctly, the assumption needed to derive Liouville’s theorem is time-translation invariance; but this is the same symmetry that gives us energy conservation through Noether’s theorem. So, it is not clear to me that you can have one without the other.
Liouville’s theorem follows from the continuity of transport of some conserved quantity. If this quantity is not energy, then you don’t need time-translation invariance. For example, forced oscillations (with explicitly time-dependent force, like first pushing a child on a swing harder and harder and then letting the swing relax to a stop) still obey the theorem.