  Monte Carlo (MC) simulations are used to study the dynamics of polymer
translocation through a nanopore in the limit where the translocation rate is
sufficiently slow that the polymer maintains a state of conformational
quasi-equilibrium. The system is modeled as a flexible hard-sphere chain that
translocates through a cylindrical hole in a hard flat wall. In some
calculations, the nanopore is connected at one end to a spherical cavity.
Translocation times are measured directly using MC dynamics simulations. For
sufficiently narrow pores, translocation is sufficiently slow that the mean
translocation time scales with polymer length N according to <\tau> \propto
(N-N_p)^2, where N_p is the average number of monomers in the nanopore; this
scaling is an indication of a quasi-static regime in which polymer-nanopore
friction dominates. We use a multiple-histogram method to calculate the
variation of the free energy with Q, a coordinate used to quantify the degree
of translocation. The free energy functions are used with the Fokker-Planck
formalism to calculate translocation time distributions in the quasi-static
regime. These calculations also require a friction coefficient, characterized
by a quantity N_{eff}, the effective number of monomers whose dynamics are
affected by the confinement of the nanopore. This was determined by fixing the
mean of the theoretical distribution to that of the distribution obtained from
MC dynamics simulations. The theoretical distributions are in excellent
quantitative agreement with the distributions obtained directly by the MC
dynamics simulations for physically meaningful values of N_{eff}. The free
energy functions for narrow-pore systems exhibit oscillations with an amplitude
that is sensitive to the nanopore length. Generally, larger oscillation
amplitudes correspond to longer translocation times.
