Complexity Explorer Santa Few Institute

1. What's the justification for including the coupling, but not the  coupling or the coupling?

A. is larger than the others, so it's the dominant contribution

B. if we drop , we're back to pairwise couplings, which is closer to the original micro-level model and only includes pairwise couplings.

C. if we drop and , then we get back a lattice structure that's the same as the micro-level model.

D. all of the above.

 

Answer: (D). This is a pretty vicious approximation. We're dropping a huge amount of structure in the model just to keep it in the same model class.

 

2. What's the justification for changing the coupling to ?

A. when we ignored the next-neighbour interaction in the first attempt, we underestimated extent to which the system is coupled together. This is a heuristic that just reassigns them to nearest-neighbour, enabling us to stay within the same "pairwise square lattice" model class.

B. it's a consequence of how the next-nearest neighbour couplings interact with the synergistic coupling.

C. it's a way to approximate the synergistic couplings.

D. it's an exact compensation for neglecting the next-nearest neighbour couplings.

 

Answer (A). This is a complete hack; we just by fiat approximate a lattice with these cross-wise nearest neighbour couplings as a square one. There's no good justification other than (as we'll see) it enables us to get closer to reality. It's definitely not the case that it enables us to handle the synergistic four-way couplings -- these are much stranger, and capture something invisible to the pairwise case.

 

Don't be too hard on this approximation. It was something people did a long time ago when they were really wrestling with how to handle models like this. Today we have much neater ways to go about these approximations, in particular, by going into Fourier space -- but that's another story. What I love about this account is that you really get a sense for how people hack away when they don't have an answer yet.