  We derive an effective 1D theory from the Hamiltonian of the 3D system which
consists of a mesoscopic conductor and reservoirs. We assume that the many-body
interaction have the same magnitude in the conductor as that in the reservoirs,
in contrast to the previous theories which made the ad hoc assumption that the
many-body interaction were absent in the reservoirs. We show the following: (i)
The effective potentials of impurities and two-body interaction for the 1D
modes become weaker as $x$ goes away from the conductor. (ii) On the other
hand, the interaction between the 1D and the reservoir modes is important in
the reservoir regions, where the reservoir modes excite and attenuate the 1D
modes through the interaction. (iii) As a result, the current $\hat I_1$ of the
1D modes is not conserved, whereas the total current $\hat I$ is of course
conserved. (iv) For any steady state the total current $\bra I \ket$, its
equilibrium fluctuation $\bra \delta I^2 \ket^{eq}$ at low frequency, and
non-equilibrium fluctuation $\bra \delta I^2 \ket^{noneq}$ at low frequency, of
the original system are independent of $x$, whereas $\bra \delta I^2 \ket^{eq}$
and $\bra \delta I^2 \ket^{noneq}$ at higher frequencies may depend on $x$. (v)
Utilizing this property, we can evaluate $\bra I \ket$, $\bra \delta I^2
\ket^{eq}$, and $\bra \delta I^2 \ket^{noneq}$ at low frequency from those of
the 1D current $\hat I_1$. (vi) In general, the transmittance $T$ in the
Landauer formula should be evaluated from a single-body Hamiltonian which
includes a Hartree potential created by the density deformation which is caused
by the external bias.
