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

• The expected avalanche size in the coherent noise model is studied in natural time.
• After the kk-th avalanche, we evaluate the expected size E(Sk+1)E(Sk+1) of the next avalanche.
• In a statistical ensemble of initially identical systems, the average expected avalanche size 〈E(Sk+1)〉〈E(Sk+1)〉 is numerically studied.
• 〈E(Sk+1)〉〈E(Sk+1)〉 relaxes as a qq-exponential function of kk and the reasons behind this behavior are investigated.
• The non-extensivity parameter qq found from studying this relaxation is compatible with those found by other methods.

Recently (Sarlis and Christopoulos (2012)) the threshold distribution function pthres(k)(x) of the coherent noise model for infinite number of agents after the kk-th avalanche has been studied as a function of kk, and hence natural time. An analytic expression of the expectation value E(Sk+1)E(Sk+1) for the size Sk+1Sk+1 of the next avalanche has been obtained in the case that the coherent stresses are exponentially distributed with an average value σσ. Here, by using a statistical ensemble of initially identical systems, we investigate the relaxation of the average 〈E(Sk+1)〉〈E(Sk+1)〉 versus kk. For kk values smaller than kmax(σ,f), the numerical results indicate that 〈E(Sk+1)〉〈E(Sk+1)〉 collapses to the qq-exponential (Tsallis (1988)) as a function of kk. For larger kk 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 qq-exponential relaxation. The resulting qq-values are compatible with those reported in the literature for the coherent noise model.