  Matsubara dynamics is the classical dynamics which results when
imaginary-time path-integrals are smoothed; it conserves the quantum Boltzmann
distribution and appears in drastically approximated form in path-integral
dynamics methods such as (thermostatted) ring-polymer molecular dynamics
(T)RPMD and centroid molecular dynamics (CMD). However, it has never been
compared directly with exact quantum dynamics for non-linear operators, because
the difficulty of treating the phase has limited the number of Matsubara modes
to fewer than 10. Here, we treat up to $\sim$200 Matsubara modes in simulations
of a Morse oscillator coupled to a dissipative bath of harmonic oscillators.
This is done by expressing the Matsubara equations of motion in the form of a
generalised Langevin equation, approximating the noise to be real, and
analytically continuing the momenta to convert the Matsubara phase into
ring-polymer springs. The resulting equations of motion are stable up to a
maximum value of modes which increases with bath coupling strength and
decreases with system anharmonicity. The dynamics of the tail of highly
oscillatory Matsubara modes is found to be harmonic, and can thus be computed
efficiently. For a moderately anharmonic oscillator with a strong but
subcritical coupling to the bath, the Matsubara simulations yield non-linear
$\large\langle{\hat q^2\hat q^2(t)}\large\rangle$ time-correlation functions in
almost perfect agreement with the exact quantum results. Reasonable agreement
is also obtained for weaker coupling strengths, where errors arise because of
the real-noise approximation. These results give strong evidence that Matsubara
dynamics correctly explains how classical dynamics arises in quantum systems
which are in thermal equilibrium.
