  The conductance G of an interacting nano-wire containing an impurity and
coupled to non-interacting semi-infinite leads is studied using a functional
renormalization group method. We obtain results for microscopic lattice models
without any further idealizations. For an interaction which is turned on
smoothly at the contacts we show that one-parameter scaling of G holds. If
abrupt contacts are included we find power-law suppression of G with an
exponent which is twice as large as the one obtained for smooth contacts and no
one-parameter scaling. Our results show excellent agreement with the
analytically known scaling function at Luttinger liquid parameter K=1/2 and
numerical density-matrix renormalization group data.
