Upper bounds on the entropy of radiation systems

The upper bounds on the entropy of a radiation system confined to a spherical box are calculated in six cases by using the equation of state of radiation in flat spacetime and the equation of state of radiation near black-hole horizon, which was derived by Li and Liu (hereafter the Li-Liu equation). It turns out that the Li-Liu equation does have unique advantage in dealing with the entropy bound of critical self-gravitating radiation systems, while the usual equation of state will result in entropy divergence. In the case of non-self-gravitating radiation systems and non-critical self-gravitating radiation systems, there is no difference in the entropy bounds derived by these two equations of state.