Aftershock Energy Distribution by Statistical Mechanics Approach

Abstract:

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 case of aftershock sequences, it 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 the 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 combination 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 ε_i.

Therefore we can assume that the Maxwell-Boltzmann statistics can be applied to aftershock sequences with the proviso that the judgment on the validity of this hypothesis is the agreement with the data.

The aftershock energy distribution can therefore be written as follow:

 n(ε)=Ag(ε)exp(-βε)

where n(ε) is the number of aftershocks with energy, ε, A and β are constants.

Considering the above hypothesis, we can assume g(ε) is proportional to ε.

We selected and analysed different aftershock sequences (data extracted from Earthquake Catalogs of SCEC, of INGV-CNT and other institutions) with a minimum magnitude retained M_L=2 (in some cases M_L=2.6) and a time window of 35 days.

The results of our model are in agreement with the data, except in the very low energy band, where our model resulted in a moderate overestimation.