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The Flux Creep Automaton

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David Jonathan Miller

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.