  We obtain thermostatted ring polymer molecular dynamics (TRPMD) from exact
quantum dynamics via Matsubara dynamics, a recently-derived form of
linearization which conserves the quantum Boltzmann distribution. Performing a
contour integral in the complex quantum Boltzmann distribution of Matsubara
dynamics, replacement of the imaginary Liouvillian which results with a
Fokker-Planck term gives TRPMD. We thereby provide error terms between TRPMD
and quantum dynamics and predict the systems in which they are likely to be
small. Using a harmonic analysis we show that careful addition of friction
causes the correct oscillation frequency of the higher ring-polymer normal
modes in a harmonic well, which we illustrate with calculation of the
position-squared autocorrelation function. However, no physical friction
parameter will produce the correct fluctuation dynamics for a parabolic
barrier. The results in this paper are consistent with previous numerical
studies and advise the use of TRPMD for the computation of spectra.
