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

Abstract

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, $(3\ensuremath{-}\ensuremath{\tau})/\ensuremath{\sigma}\ensuremath{\nu}z,$ which has been derived in several other papers, is in general incorrect for small $\ensuremath{\tau},$ when large avalanches are common. $(\ensuremath{\tau}$ is the exponent describing the distribution of avalanche sizes, and $\ensuremath{\sigma}\ensuremath{\nu}z$ 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 $\ensuremath{\tau}<2$ is $1/\ensuremath{\sigma}\ensuremath{\nu}z.$ 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.