  We investigate the continuum limit that the number of beads goes to infinity
in the ring polymer representation of thermal averages. Studying the continuum
limit of the trajectory sampling equation sheds light on possible
preconditioning techniques for sampling ring polymer configurations with large
number of beads. We propose two preconditioned Langevin sampling dynamics,
which are shown to have improved stability and sampling accuracy. We present a
careful mode analysis of the preconditioned dynamics and show their connections
to the normal mode, the staging coordinate and the Matsubara mode
representation for ring polymers. In the case where the potential is quadratic,
we show that the continuum limit of the preconditioned mass modified Langevin
dynamics converges to its equilibrium exponentially fast, which suggests that
the finite-dimensional counterpart has a dimension-independent convergence
rate. In addition, the preconditioning techniques can be naturally applied to
the multi-level quantum systems in the nonadiabatic regime, which are
compatible with various numerical approaches.
