Noise in disordered systems: The power spectrum and dynamic exponents in avalanche models

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 zero-temperature random-field Ising model. We find that the naive exponent, (3-tau)/(sigma nu z), which has been derived in several other papers, is in general incorrect for small tau, when large avalanches are common. (tau is the exponent describing the distribution of avalanche sizes, and (sigma nu z) 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 tau<2 is 1/(sigma nu z). 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.