Dispersion of the Break Energy in the GRB Internal Shock Model

In order to investigate the dispersion of the spectral break energy in gamma-ray bursts, we simulate internal shocks, including effects of shell-splitting after shell collisions and the Thomson optical depth due to electron--positron pairs produced by synchrotron photons. We produce pseudo observational data and estimate the break energy. If the distribution of initial Lorentz factors of shells has only one peak, many pulses with a break energy much lower than the typical observed value should be detected, while the effect of the Thomson optical depth reduces pulses with a break energy of $>1$ MeV. If the Lorentz factor distribution has multiple peaks, the distribution of the break energy can be consistent with observations.