  It was shown recently that smooth and continuous "Matsubara" phase-space
loops follow a quantum-Boltzmann-conserving classical dynamics when decoupled
from non-smooth distributions, which was suggested as the reason that many
dynamical observables appear to involve a mixture of classical dynamics and
quantum Boltzmann statistics. Here we derive a mean-field version of this
"Matsubara dynamics" which sufficiently mitigates its serious phase problem to
permit numerical tests on a two-dimensional "champagne-bottle" model of a
rotating OH bond. The Matsubara-dynamics rovibrational spectra are found to
converge toward close agreement with the exact quantum results at all
temperatures tested (200-800 K), the only significant discrepancies being a
temperature-independent 22 cm$^{-1}$ blue-shift in the position of the
vibrational peak and a slight broadening in its line shape. These results are
compared with centroid molecular dynamics (CMD) to assess the importance of
non-centroid fluctuations. Above 250 K, only the lowest-frequency non-centroid
modes are needed to correct small CMD red-shifts in the vibrational peak; below
250 K, more non-centroid modes are needed to correct large CMD red-shifts and
broadening. The transition between these "shallow curvature" and "deep
curvature" regimes happens when imaginary-time Feynman paths become able to
lower their actions by cutting through the curved potential surface, giving
rise to artificial instantons in CMD.
