  The dissipative dynamics of a vortex line in a superfluid is investigated
within the frame of a non-Markovian quantal Brownian motion model. Our starting
point is a recently proposed interaction Hamiltonian between the vortex and the
superfluid quasiparticle excitations, which is generalized to incorporate the
effect of scattering from fermion impurities ($^3$He atoms). Thus, a
non-Markovian equation of motion for the mean value of the vortex position
operator is derived within a weak-coupling approximation. Such an equation is
shown to yield, in the Markovian and elastic scattering limits, a $^3$He
contribution to the longitudinal friction coefficient equivalent to that
arising from the Rayfield-Reif formula. Simultaneous Markov and elastic
scattering limits are found, however, to be incompatible, since an unexpected
breakdown of the Markovian approximation is detected at low cyclotron
frequencies. Then, a non-Markovian expression for the longitudinal friction
coefficient is derived and computed as a function of temperature and $^3$He
concentration. Such calculations show that cyclotron frequencies within the
range 0.01$-$0.03 ps$^{-1}$ yield a very good agreement to the longitudinal
friction figures computed from the Iordanskii and Rayfield-Reif formulas for
pure $^4$He, up to temperatures near 1 K. A similar performance is found for
nonvanishing $^3$He concentrations, where the comparison is also shown to be
very favorable with respect to the available experimental data. Memory effects
are shown to be weak and increasing with temperature and concentration.

Statistical Mechanics;Superconductivity