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 no ise had a power-law decay at high frequencies. Up to now, the theoretical p redictions for this decay have been incorrect, or have only applied to a sm all set of models. In this paper, we describe a careful derivation of the p ower spectrum exponent in avalanche models, and in particular, in variation s of the zero-temperature random-field Ising model. We find that the naive exponent, (3 - tau)/sigma nuz, which has been derived in several other pape rs, is in general incorrect for small tau, when large avalanches are common . (tau is the exponent describing the distribution of avalanche sizes, and sigma nuz is the exponent describing the relationship between avalanche siz e and avalanche duration.) We find that for a large class of avalanche mode ls, including several models of Barkhausen noise, the correct exponent for tau <2 is 1/<sigma>nuz. 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.