Iterative trajectory reweighting for estimation of equilibrium and non-equilibrium observables

We present two algorithms by which a set of short, unbiased trajectories can be iteratively reweighted to obtain various observables. The first algorithm estimates the stationary (steady state) distribution of a system by iteratively reweighting the trajectories based on the average probability in each state. The algorithm applies to equilibrium or non-equilibrium steady states, exploiting the `left' stationarity of the distribution under dynamics -- i.e., in a discrete setting, when the column vector of probabilities is multiplied by the transition matrix expressed as a left stochastic matrix. The second procedure relies on the `right' stationarity of the committor (splitting probability) expressed as a row vector. The algorithms are unbiased, do not rely on computing transition matrices, and make no Markov assumption about discretized states. Here, we apply the procedures to a one-dimensional double-well potential, and to a 208$\mu$s atomistic Trp-cage folding trajectory from D.E. Shaw Research.