than the mean and asymmetry of its steady-state probability distribution, respectively, $\kappa_1(P_{yx})$ and $\kappa_3(P_{yx})$. Such behavior of athermal statistics is consistent with the observation, that the fluctuations of the nonequilibrium force become negligible in the near-equilibrium systems [@PRE2016], i.e. $\chi(P_{yx})\approx0$ and $\kappa_3(P_{yx})\approx0$, and [Eq. (\[eq:lgv2\])]{} thus can be approximated by $$\ddot\alpha(t) + a \dot\alpha(t) + b^2 \alpha(t) \approx f + A\omega(t),$$ which agrees in the macroscopic limit with [Eq. (\[eq:mac1\])]{}, cf. [Sec. \[sec:intro\]]{}.

In [Eq. (\[eq:kform\])]{}, the linear dependence of the athermal cumulant ratio on the shear rate can only arise due to the term proportional to $F$, which therefore can not vanish. In fact, [Eq. (\[eq:lgv2\])]{} admits [*an alternative form*]{}, in which one of its parameters is related linearly to the athermal cumulant ratio, as shown in Appendix \[sec:acr\].

As anticipated in [Sec. \[sec:intro\]]{}, [Eq. (\[eq:lgv2\])]{}, with $\epsilon$ represented by white exponential noise or white exponential shot noise, is accurate up to the third-order statistics. For this, three parameters, $F$, $B$, and $\tau$, are required to fit exactly three cumulants $\kappa_1$, $\kappa_2$ and $\kappa_3$; cf. Eqs. (\[eq:kappaex\])-(\[eq:kappas\]).

\[sec:mes\]Langevin Dynamics Simulation
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Once all the parameters of the stochastic differential equation (\[eq:lgv2\]) are found, it can be simulated numerically. In 