q-exponential relaxation of the expected avalanche size in the coherent noise model

Recently (Sarlis and Christopoulos (2012)) the threshold distribution function pthres(k)(x) of the coherent noise model for infinite number of agents after the k-th avalanche has been studied as a function of k, and hence natural time. An analytic expression of the expectation value E(Sk+1) for the size Sk+1 of the next avalanche has been obtained in the case that the coherent stresses are exponentially distributed with an average value σ. Here, by using a statistical ensemble of initially identical systems, we investigate the relaxation of the average 〈E(Sk+1)〉 versus k. For k values smaller than kmax(σ,f), the numerical results indicate that 〈E(Sk+1)〉 collapses to the q-exponential (Tsallis (1988)) as a function of k. For larger k values, the ensemble average can be effectively described by the time average threshold distribution function obtained by Newman and Sneppen (1996). An estimate k0(σ,f)(>kmax(σ,f)) of this transition is provided. This ensemble of coherent noise models may be considered as a simple prototype following q-exponential relaxation. The resulting q-values are compatible with those reported in the literature for the coherent noise model.