Intrinsic Fluctuations in a Superconducting Ring Closed with a Josephson Junction

The distribution in the external flux at which a superconducting ring closed with a weak link admits a quantum of flux is determined assuming that the weak link can be treated as a Josephson junction. We find that this transition occurs at an appreciable fraction of the flux quantum from the theoretical critical external flux. To a first approximation the width of the distribution is proportional to the inductance of the ring and varies as $T^{\frac{2}{3}}{i}_c^{-\frac{1}{3}}$, where $T$ is the temperature and $i_c$ the critical current.