  We study the fluctuation entropy for two-dimensional matter systems with an
internal symmetry coupled to Jackiw--Teitelboim(JT) gravity joined to a
Minkowski region. The fluctuation entropy is the Shannon entropy associated to
probabilities of finding particular charge for a region. We first consider a
case where the matter has a global symmetry. We find that the fluctuation
entropy of Hawking radiation shows an unbounded growth and exceeds the
entanglement entropy in presence of islands. This indicates that the global
symmetry is violated. We then discuss the fluctuation entropy for matter
coupled to a two-dimensional gauge field. We find a lower bound on the gauge
coupling $g_0$ in order to avoid a similar issue. Also, we point out a few
puzzles related to the island prescription in presence of a gauge symmetry.

High Energy Physics - Theory;Statistical Mechanics;General Relativity and Quantum Cosmology