to justify the constant term $F$ in [Eq. (\[eq:lgv2\])]{}. Hence, the second-order Langevin representation of our original model needs, in total, specification of six constants, $a$, $b$, $A$, $B$, $\tau$, and $F$. Furthermore, [Eq. (\[eq:lgv2\])]{} can be reparametrized so, that the athermal cumulant ratio is linearly related to one of the new parameters; see Appendix \[sec:acr\].

Finally, at short-time scales we observed in [Sec. \[sec:mes\]]{} some features of the LD simulations, which appear to be artifacts of the white noise approximation. As discussed there, a correlated noise could, in principle, improve this aspect of the Langevin equation. A description of noise correlations may require to add another parameter to the model. This, however, seems far from being practical, since there would be more unknowns than statistically significant measurements to fit; cf. Secs. \[sec:mic\]-\[sec:mes\]. Instead, one could try to construct a first-order Langevin equation with a correlated noise, which would reproduce the time autocorrelation function for $\alpha(t)$ in [Eq. (\[eq:lgv2\])]{}. Indeed, the term $a\dot\alpha(t)$ of the second-order dynamics is required for the accurate description of time autocorrelations in the modeled system; cf. [Ref. [@PRE2016III]]{}. This introduces essentially a nuisance parameter $a$, if we are interested merely in the behavior of $\alpha(t)$. 