of simple exponential noise, and $12\cdot10^{-5}$ for the case of exponential shot noise. These predictions of the excess kurtosis are also much less than the respective standard deviations in [Table \[tab:stat\]]{}. Therefore, within the statistical uncertainties, the excess kurtosis of our MD and LD models can effectively be considered zero, i.e. Gaussian-like.

To integrate numerically [Eq. (\[eq:lgv2\])]{}, we used the same time step, $\Delta{t} = 10^{-3}$, as in the original MD simulations; cf. [Ref. [@PRE2016III]]{}. Therefore, we can compare sample trajectories, which are traced pointwise in time for our LD models and the original MD system in [Fig. \[fig:trj\]]{}. Both, simple exponential noise and exponential shot noise, produce a qualitatively similar behavior of $\alpha(t)$. In particular, the two stochastic models match closely the amplitude of $\alpha(t)$ fluctuations, as well as the time scale of their onset and decay, observed in the MD simulations.

The trajectory of the original MD system in [Fig. \[fig:trj\]]{} mostly resembles those of the LD simulations, perhaps, except for one subtle detail. Namely, on short time intervals, the stochastic dynamics of the time derivative $\dot\alpha(t)$ may undergo quick alterations, which makes the trajectory of $\alpha(t)$ look noisy. This is especially conspicuous for $\dot\alpha(t)\approx0$, where this behavior 