  Equilibrium formally can be represented as an ensemble of uncoupled systems
undergoing unbiased dynamics in which detailed balance is maintained. Many
non-equilibrium processes can be described by suitable subsets of the
equilibrium ensemble. Here, we employ the "weighted ensemble" (WE) simulation
protocol [Huber and Kim, Biophys. J., 1996] to generate equilibrium trajectory
ensembles and extract non-equilibrium subsets for computing kinetic quantities.
States do not need to be chosen in advance. The procedure formally allows
estimation of kinetic rates between arbitrary states chosen after the
simulation, along with their equilibrium populations. We also describe a
related history-dependent matrix procedure for estimating equilibrium and
non-equilibrium observables when phase space has been divided into arbitrary
non-Markovian regions, whether in WE or ordinary simulation. In this
proof-of-principle study, these methods are successfully applied and validated
on two molecular systems: explicitly solvated methane association and the
implicitly solvated Ala4 peptide. We comment on challenges remaining in WE
calculations.
