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 (rho)(Ic) is known to be Gaussian in the limit of large channel number, N (infinty). 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>=2) is discussed numerically, and for small critical currents analytically.