  We consider a long, semiflexible polymer, with persistence length $P$ and
contour length $L$, fluctuating in a narrow cylindrical channel of diameter
$D$. In the regime $D\ll P\ll L$ the free energy of confinement $\Delta F$ and
the length of the channel $R_\parallel$ occupied by the polymer are given by
Odijk's relations $\Delta F/R_\parallel=A_\circ k_BTP^{-1/3}D^{-2/3}$ and
$R_\parallel=L[1-\alpha_\circ(D/P)^{2/3}]$, where $A_\circ$ and $\alpha_\circ$
are dimensionless amplitudes. Using a simulation algorithm inspired by PERM
(Pruned Enriched Rosenbluth Method), which yields results for very long
polymers, we determine $A_\circ$ and $\alpha_\circ$ and the analogous
amplitudes for a channel with a rectangular cross section. For a semiflexible
polymer confined to the surface of a cylinder, the corresponding amplitudes are
derived with an exact analytic approach. The results are relevant for
interpreting experiments on biopolymers in microchannels or microfluidic
devices.
