On approximating a weak Markovian process as Markovian: Are we justified when discarding longtime correlations

Abstract

The effect for removing weak longtime correlation is studied using a model system that contains a driven atom at liquid density under strong thermal fluctuations. The force that drives the tagged particle is about 1% of the average random force experienced by the particle. The tagged particle is allowed to assume a range of masses from 1/8 to 80 times that of a surrounding particle to study the effects of inertia. The driving force is indefinitely correlated but much weaker than "random" fluctuations from the environment. From this study, it is shown that the environmental influence is not fully random leading to the force autocorrelation function being a poor metric for detecting the correlated driving force. Although the velocity autocorrelation function shows stronger correlation for systems with higher inertia, the velocity autocorrelation function decays to a very small value of 2.5×10 −3 even for the most massive driven particle. For systems with small inertia, our study reveals that discarding longtime correlation has negligible influence on the first passage time (FPT) estimate, whereas for particles with large inertia, the deviation can indeed be appreciable. It is interesting that the Markov State Model (MSM) still produces reasonable estimates on the FPT even when a very short lag time that clearly violates the Markovianity assumption is used. This is likely a result of favorable error cancellations when the MSM transition probability matrices were constructed using trajectories that are much longer than the lag time.