[thomasahle]

Here there are N cells with N^2 couplings. So as you say, a normal computer with this many processors could also solve a linear system in O(N sqrt(kappa)) time. (Using conjugate gradient, since matrix-vector-mult on the system would be O(N) time.)

However, (1) these thermodynamic cells are much simpler than processors, and (2) it seems the overall energy required to simulate the SDE, once the couplings are initialized, only scales with O(N), not O(N^2) as in your digital case.