Fluctuations of time averages for Langevin dynamics in a binding force field

We derive a simple formula for the fluctuations of the time average x̄(t) around the thermal mean x _eq for overdamped Brownian motion in a binding potential U(x). Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large nonergodic fluctuations and show how they are related to a superaging correlation function.