  We show that a single change in the derivation of the linearized
semiclassical-initial value representation (LSC-IVR or classical Wigner
approximation) results in a classical dynamics which conserves the quantum
Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing
the (exact) quantum time-correlation function in terms of the normal modes of a
free ring-polymer (i.e. a discrete imaginary-time Feynman path), taking the
limit that the number of polymer beads $N \to \infty$, such that the lowest
normal-mode frequencies take their Matsubara values. The change we propose is
to truncate the quantum Liouvillian, not explicitly in powers of $\hbar^2$ at
$\hbar^0$ (which gives back the standard LSC-IVR approximation), but in the
normal-mode derivatives corresponding to the lowest Matsubara frequencies. The
resulting Matsubara dynamics is inherently classical (since all terms
$\mathcal{O}\left(\hbar^{2}\right)$ disappear from the Matsubara Liouvillian in
the limit $N \to \infty$), and conserves the quantum Boltzmann distribution
because the Matsubara Hamiltonian is symmetric with respect to imaginary-time
translation. Numerical tests show that the Matsubara approximation to the
quantum time-correlation function converges with respect to the number of
modes, and gives better agreement than LSC- IVR with the exact quantum result.
Matsubara dynamics is too computationally expensive to be applied to complex
systems, but its further approximation may lead to practical methods.
