EXACT SOLUTION FOR THE DISTRIBUTION OF TRANSMISSION EIGENVALUES IN A DISORDERED WIRE AND COMPARISON WITH RANDOM-MATRIX THEORY

We consider the complete probability distribution P({T(n)}) of the transmission eigenvalues T1, T2, ..., T(N) of a disordered quasi-one-dimensional conductor (length L much greater than width W and mean free path 1). The Fokker-Planck equation which describes the evolution of P with increasing L is mapped onto a Schrodinger equation by a Sutherland-type transformation. In the absence of time-reversal symmetry (e.g., because of a magnetic field), the mapping is onto a free-fermion problem, which we solve exactly. The resulting distribution is compared with the predictions of random-matrix theory (RMT) in the metallic regime (L much less than Nl) and in the insulating regime (L much greater than Nl). We find that the logarithmic eigenvalue repulsion of RMT is exact for T(n)'s close to unity, but overestimates the repulsion for weakly transmitting channels. The nonlogarithmic repulsion resolves several long-standing discrepancies between RMT and microscopic theory, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.