Energy dissipation statistics in the random fuse model

We study the statistics of the dissipated energy in the two-dimensional random fuse model for fracture under different imposed strain conditions. By means of extensive numerical simulations we compare different ways to compute the dissipated energy. In the case of a infinitely slow driving rate (quasi-static model) we find that the probability distribution of the released energy shows two different scaling regions separated by a sharp energy crossover. At low energies, the probability of having an event of energy $E$ decays as $\sim E^{-1/2}$, which is robust and independent of the energy quantifier used (or lattice type). At high energies fluctuations dominate the energy distribution leading to a crossover to a different scaling regime, $\sim E^{-2.75}$, whenever the released energy is computed over the whole system. On the contrary, strong finite-size effects are observed if we only consider the energy dissipated at microfractures. In a different numerical experiment the quasi-static dynamics condition is relaxed, so that the system is driven at finite strain load rates, and we find that the energy distribution decays as $\mathcal{P} (E) \sim E^{-1}$ for all the energy range.