I feel like they must exist (and there may not be that many simple nice ones). I expect someone who knows more physics could design them more easily.
My best guess would be to get both properties by defining the system via some kind of discrete hamiltonian. I don’t know how that works, i.e. if there is a way of making the hamiltonian discrete (in time and in values of the CA) that still gives you both properties and is generally nice. I would guess there is and that people have written papers about it. But it also seems like that could easily fail in one way or another.
It’s surprisingly non-trivial to find that by googling though I didn’t try very hard. May look a bit more tonight (or think about it a bit since it seems fun). Finding a suitable replacement for the game of life that has good conservation laws + reversibility (while still having a similar level of richness) would be nice.
I guess the important part of the hamiltonian construction may be just having the next state depend on x(t) and x(t-1) (apparently those are called second-order cellular automata). Once you do that it’s relatively easy to make them reversible, you just need the dependence of x(t+1) on x(t-1) to be a permutation. But I don’t know whether using finite differences for the hamiltonian will easily give you conservation of momentum + energy in the same way that it would with derivatives.
I feel like they must exist (and there may not be that many simple nice ones). I expect someone who knows more physics could design them more easily.
My best guess would be to get both properties by defining the system via some kind of discrete hamiltonian. I don’t know how that works, i.e. if there is a way of making the hamiltonian discrete (in time and in values of the CA) that still gives you both properties and is generally nice. I would guess there is and that people have written papers about it. But it also seems like that could easily fail in one way or another.
It’s surprisingly non-trivial to find that by googling though I didn’t try very hard. May look a bit more tonight (or think about it a bit since it seems fun). Finding a suitable replacement for the game of life that has good conservation laws + reversibility (while still having a similar level of richness) would be nice.
I guess the important part of the hamiltonian construction may be just having the next state depend on x(t) and x(t-1) (apparently those are called second-order cellular automata). Once you do that it’s relatively easy to make them reversible, you just need the dependence of x(t+1) on x(t-1) to be a permutation. But I don’t know whether using finite differences for the hamiltonian will easily give you conservation of momentum + energy in the same way that it would with derivatives.