We study a cellular automaton derived from the phenomenon of magnetic flux creep in two-dimensional granular superconductors. We model the superconductor as an array of inductively coupled Josephson junctions evolving according to a set of coupled ordinary differential equations. In the limit of slowly increasing magnetic field, we reduce these equations to a simple cellular automaton.
The Flux Creep Automaton is the two-dimensional generalization of the one-dimensional Pendulum Automaton. The flux creep dynamics, derived from Kirchoff's laws and the Josephson relations, reduce to a Gradient Sand Pile Automaton with an unusual non-local seeding, wherein all sites except the boundaries are seeded concurrently. Loop and line currents in the Flux Creep Automaton correspond, respectively, to heights and gradients in the Sand Pile Automaton.
We implement the Flux Creep Automaton on lattices of very different rotational symmetries, including periodic lattices that are 3-fold, 4-fold, and 6-fold symmetric as well as aperiodic lattices that have 5-fold, 7-fold and higher symmetries. In each case, the automaton evolves to a state of constant flux gradient, as described by the Bean model. The intimate connection with the Gradient Sand Pile Automaton reinforces the idea that in the Bean state superconducting vortices exhibit avalanching dynamics.