One may hope, however, that the time correlation function of a first-order Langevin dynamics, with a properly designed colored noise, would match that of $\alpha(t)$ in [Eq. (\[eq:lgv2\])]{}.

\[sec:apx\]Models of athermal noise
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In [Ref. [@PRE2016II]]{} we have presented a derivation of exponential white noise for the Langevin dynamics of nonequilibrium physical systems, as an alternative to white exponential shot noise [@MorgadoQ2016]. As mentioned in [Sec. \[sec:intro\]]{}, these two models essentially account for the spontaneous variations of a nonequilibrium force, which are assumed independent from the thermal fluctuations and, therefore, called [*athermal*]{} [@MorgadoQ2016].

The derivation of exponential white noise [@PRE2016II] followed a procedure similar to that of the Gaussian white noise in Ref. [@Chandrasekhar1943 Sec. I.1], used to model thermal fluctuations in both, equilibrium and nonequilibrium, systems. This procedure suggests to design a discrete random walk, which incorporates physically relevant assumptions and properties of the fluctuating force. Then, a definition of stochastic noise, which can be used in a Langevin equation, naturally arises in a properly chosen continuous limit.

In this section we revisit the above procedure and show, that it relies on a more general mathematical structure of asymptotic approximations, which allows to derive various stochastic noises with desired 