Dynamics of non-Markovian systems: Markovian embedding versus effective mass approach

Dynamics of non-Markovian systems is a classic problem yet it attracts everlasting activity in physics and beyond. A powerful tool for modeling such setups is the generalized Langevin equation, however, its analysis typically poses a major challenge even for numerical means. For this reason, various approximations have been proposed over the years that simplify the original model. In this paper, we compare two methods allowing us to tackle this great challenge: (i) the well-known and successful Markovian embedding technique and (ii) the recently developed effective mass approach. We discuss their scope of applicability, numerical accuracy, and computational efficiency. In doing so, we consider a paradigmatic model of a free Brownian particle subjected to power-law correlated thermal noise. We show that when the memory time is short, the effective mass approach offers satisfying precision and typically is much faster than the Markovian embedding. Moreover, the concept of effective mass can be used to find optimal parameters allowing us to reach supreme accuracy and minimal computational cost within the embedding. Our paper therefore provides a blueprint for investigating the dynamics of non-Markovian systems.