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We recently obtained a quantum-Boltzmann-conserving classical dynamics bymaking a single change to the derivation of the `Classical Wigner'approximation. Here, we show that the further approximation of this `Matsubaradynamics' gives rise to two popular heuristic methods for treating quantumBoltzmann time-correlation functions: centroid molecular dynamics (CMD) andring-polymer molecular dynamics (RPMD). We show that CMD is a mean-fieldapproximation to Matsubara dynamics, obtained by discarding (classical)fluctuations around the centroid, and that RPMD is the result of discarding aterm in the Matsubara Liouvillian which shifts the frequencies of thesefluctuations. These findings are consistent with previous numerical results,and give explicit formulae for the terms that CMD and RPMD leave out.

Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics