For a long time, it has been known that the power spectrum of Barkhausen noise had a power-law decay at high frequencies.Up to now, the theoretical predictions for this decay have been incorrect, or have only applied to a small set of models. In this paper, we describe a careful derivation of the power spectrum exponent in avalanche models, and in particular, in variations of the zerotemperature random-field Ising model. We find that the naive exponent, (3 − τ )/σνz, which has been derived in several other papers, is in general incorrect for small τ , when large avalanches are common. (τ is the exponent describing the distribution of avalanche sizes, and σνz is the exponent describing the relationship between avalanche size and avalanche duration.) We find that for a large class of avalanche models, including several models of Barkhausen noise, the correct exponent for τ < 2 is 1/σνz. We explicitly derive the mean-field exponent of 2. In the process, we calculate the average avalanche shape for avalanches of fixed duration and scaling forms for a number of physical properties. II. THE MODELSSeveral variations of the zero temperature random field Ising model have been proposed to explain the power laws in Barkhausen noise. They are differentiated on the basis of the presence of long range forces, and the details of the dynamics. In a separate paper, we examine in detail the differences between these models 14 . A general Hamiltonian for the models iswhere s i = ±1 is an Ising spin, J nn is the strength of the ferromagnetic nearest neighbor interactions, H is an external magnetic field, h i is a random local field, J inf is the strength of the infinite range demagnetizing field, 10 and J dipole is the strength of the dipole-dipole interactions. The critical exponents of the power laws are independent of the particular choice of random field distributions ρ(h i ) for a large variety of distributions. Most commonly, a Gaussian distribution of random fields is used, with a standard deviation R. (When we refer to the strength of the disorder, we are referring to the width, R, of the random field distribution.)

As computers increase in speed and memory, scientists are inevitably led to simulate more complex systems over larger time and length scales. Although a simple, straightforward algorithm is often the most efficient for small system sizes, especially when the time needed to implement the algorithm is included, the scaling of time and memory with system size becomes crucial for larger simulations.In our studies of hysteresis and avalanches in a simple model of magnetism (the random-field Ising model at zero temperature), we often have found it necessary to do very large simulations. Previous simulations were limited to relatively small systems (up to 900 2 and 128 3 [1], see however [3]). In our simulations we have found that larger systems (up to a billion spins) are crucial to extracting accurate values of the universal critical exponents and understanding important qualitative features of the physics.We have developed two efficient and relatively straightforward algorithms which allow us to simulate these large systems. The first algorithm uses sorted lists and scales as O(N log N ), and asymptotically uses N × (sizeof(double)+sizeof(int)) bytes of memory, where N is the number of spins. The second algorithm, which does not generate the random fields, also scales in time as O(N log N ), but asymptotically needs only one bit of storage per spin, about 96 times less than the first algorithm. Using the latter algorithm, simulations of a billion spins can be run on a workstation with 128MB of RAM in a few hours.In this column we discuss algorithms for simulating the zero-temperature random-field Ising model, which is defined by the energy functionwhere the spins s i = ±1 sit on a D-dimensional hypercubic lattice with periodic boundary conditions. The spins interact ferromagnetically with their z nearest neighbors ; Olga Perković is with McKinsey & Company, olga perkovic@mckinsey.com; Karin Dahmen has just joined the physics faculty of the University of Illinois at Urbana-Champaign; Bruce W. Roberts is working at Starwave Corporation in Seattle, bwr@halcyon.com; James P. Sethna is a professor of physics at Cornell University, sethna@lassp.cornell.edu. Details about his research group can be found at http://www.lassp.cornell.edu/sethna/ with strength J, and experience a uniform external field H(t) and a random local field h i . We choose units such that J = 1. The random field h i is distributed according to the Gaussian distribution ρ(h) of width R:The external field H(t) is increased arbitrarily slowly from −∞ to ∞.The dynamics of our model includes no thermal fluctuations: each spin flips deterministically when it can gain energy by doing so. That is, it flips when its local fieldchanges sign. This change can occur in two ways: a spin can be triggered when one of its neighbors flips (by participating in an avalanche), or a spin can be triggered because of an increase in the external field H(t) (starting a new avalanche). The zero-temperature random-field Ising model was introduced by Robbins and Ji [3] to study flu...

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