Slow dynamical modes from static averages

In recent times, efforts are being made at describing the evolution of a complex system not through the, often impossible to obtain long trajectories, but via the study of distribution evolution. This more collective approach can be made practical using the transfer operator formalism and its associated dynamics generator. Here, we reformulate in more transparent way the result of [Devergne et al https://arxiv.org/abs/2406.09028] and show that the lowest eigenfunctions and eigenvalues of the dynamics can be efficiently computed using data easily available in biased simulations. We also show explicitly that the long time dynamics can be reconstructed by using the spectral decomposition of the dynamics operator.