Aftershock Energy Distribution by Statistical Mechanics Approach

The aim of our work is to research the most probable distribution of the energy of aftershocks. We started by applying one of the fundamental principles of statistical mechanics that, in the case of aftershock sequences, could be expressed as: the greater the number of different ways in which the energy of aftershocks can be arranged among the energy cells in phase space the more probable the distribution. We assume that each cell in phase space has the same possibility to be occupied, and that more than one cell in phase space can have the same energy. Seeing that seismic energy is proportional to products of different parameters, a number of different combinations of parameters can produce different energies (e.g. different combinations of stress drop and fault area can release the same seismic energy). Let us assume that there are g_i cells in the aftershock phase space characterised by the same energy released eps_i. Therefore we can assume that the Maxwell-Boltzmann statistics can be applied to aftershock sequences with the proviso that the judgement on the validity of this hypothesis is in agreement with the data. The aftershock energy distribution can therefore be written as follow: n(eps)=Ag(e)exp(-eps*beta) (1) (**) where n(eps) is the number of aftershocks with energy; eps, A and beta are constants. (**) Equation (1) is a well known distribution function of statistical mechanics. Throughout this presentation, for the sake of brevity we refer to it as "The Maxwell-Boltzmann model". Actually we should have said "the statistical mechanics distribution function that was adopted in the Maxwell-Boltzmann formulation".