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

Recently (Sarlis and Christopoulos (2012)) the threshold distribution function p t h r e s ( 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 ( S k + 1 ) for the size S k + 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 ( S k + 1 ) 〉 versus k . For k values smaller than k max ( σ , f ) , the numerical results indicate that 〈 E ( S k + 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 k 0 ( σ , f ) ( > k max ( 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.