This is the author accepted manuscript. The final version is available from AIP via http://dx.doi.org/10.1063/1.4916311 ; 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 ? ?, 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 ?2 at ?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 O(?2) disappear from the Matsubara Liouvillian in the limit N ? ?) 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. ; T.J.H.H., M.J.W., and S.C.A. acknowledge funding from the U.K. Engineering and Physical Sciences Research Council. A.M. acknowledges the European Lifelong Learning Programme (LLP) for an Erasmus student placement scholarship. T.J.H.H. also acknowledges a Research Fellowship from Jesus College, Cambridge and helpful discussions with Dr. Adam Harper.