Hysteresis, avalanches, and noise

In our studies of hysteresis and avalanches in the zero-temperature random-field Ising model, a simple model of magnetism, we often have had to do very large simulations. Previous simulations were usually limited to relatively small systems (up to 900/sup 2/ and 128/sup 3/), although there have been exceptions. In our simulations, we have found that larger systems (up to a billion spins) are crucial to extracting accurate values of the critical exponents and understanding important qualitative features of the physics. We show three algorithms for simulating these large systems. The first uses the brute-force method, which is the standard method for avalanche-propagation problems. This algorithm is simple but inefficient. We have developed two efficient and relatively straightforward algorithms that provide better results. The sorted-list algorithm decreases execution time, but requires considerable storage. The bits algorithm has an execution time that is similar to that of the sorted-list algorithm, but it requires far less storage.