Random-Matrix Theory: Distribution of Mesoscopic Supercurrents through a Chaotic Cavity

We investigate the distribution of the supercurrent through a chaotic quantum dot which is strongly coupled to two superconductors when the Thouless energy is large compared to the superconducting energy gap. The distribution function of the critical currents P(I_c) is known to be Gaussian in the limit of large channel number N. For N=1, we present an analytical low-temperature expression for this distribution function, valid both in the presence and in the absence of time-reversal symmetry. It connects directly the distribution of transmission coefficients to the distribution of critical currents. The case of arbitrary channel number (N>1) is discussed numerically, and for small critical currents analytically.