We develop a theory for approximating quantum time-correlation functions using the classical dynamics of coordinates subject to thermally averaged Feynman path fluctuations (“the mean field”). This theory approximates the dynamics of systems at thermal equilibrium that follow quantum Boltzmann statistics and undergo rapid quantum decoherence. As it relies on purely classical mechanics, the theory leads to simulation methods that scale favourably with the number of particles, and can be used for modelling condensed-matter systems.

We begin with the path-integral Liouvillian operator, which gives rise to exact quantum time-correlation functions but does not generate classical dynamics. By thermally averaging the fluctuations about smooth Feynman paths, we obtain an approximation in terms of purely classical trajectories. This presents an alternative derivation of Matsubara dynamics, a theory for approximating quantum time-correlation functions first developed by Althorpe and co-workers. As in the original formulation, the Matsubara thermal distribution function includes a complex phase, which gives rise to a sign problem. Unlike in the original, restricting the shapes of the smooth paths (limiting the number of Matsubara modes) only changes the accuracy of the dynamics, while the mean-field thermal distribution remains exact. This improves convergence with respect to the number of Matsubara modes, enough to achieve meaningful results for a two-dimensional model potential.

The results prompt us to apply a mean-field approximation to Matsubara dynamics itself. We show that such approximations can be made phase-free for certain choices of fluctuation coordinates. Our particular choice is based on average bond-angle coordinates, called “quasicentroids”, due to their proximity to the true centroids of the Feynman paths. This “quasicentroid molecular dynamics”, or QCMD, closely approximates the vibrational spectra of model gaseous and condensed-phase water over a broad range of temperatures, improving significantly on the established path-integral methods. We anticipate that QCMD will perform equally well for more sophisticated models and will soon be extended to general molecular systems.