  We report our investigations on the finite-size effects of the Lorenz number
in a molecular wire. Using Landauer-B\"uttiker formalism, we find that for
sufficiently long wires there are two validity regimes of the Wiedemann-Franz
(WF) law, the cotunneling and the sequential tunneling regimes, while in small
systems only the first regime survives. We compare our results with the
standard Kubo formalism and explain its failure to detect the WF law in small
systems. Furthermore, our studies on exponentially localized disordered wires
show that the Lorenz number value L_0 predicted by the WF law is obtained only
in the cotunneling regime. Also, the Lorenz number L exhibits a typical
distribution at different temperatures corresponding to different tunneling
process. In particular, first-order tunneling results in a low value of L
whereas second-order tunneling recovers the universal value L_0.
