  Stochastic quantum Liouville equations (SQLE) are widely used to model energy
and charge dynamics in molecular systems. The Haken-Strobl-Reineker (HSR) SQLE
is a particular paradigm in which the dynamical noise that destroys quantum
coherences arises from a white noise (i.e., constant-frequency) spectrum. A
system subject to the HSR SQLE thus evolves to its `high-temperature' limit,
whereby all the eigenstates are equally populated. This result would seem to
imply that the predictions of the HSR model, e.g., the temperature dependence
of the diffusion coefficient, have no validity for temperatures lower than the
particle bandwidth. The purpose of this paper is to show that this assumption
is incorrect for translationally invariant systems. In particular, provided
that the diffusion coefficient is determined via the mean-squared-displacement,
considerations about detailed-balance are irrelevant. Consequently, the
high-temperature prediction for the temperature dependence of the diffusion
coefficient may be extrapolated to lower temperatures, provided that the bath
remains classical. Thus, for diagonal dynamical disorder the long-time
diffusion coefficient, $D_{\infty}(T) = c_{1} /T$, while for both diagonal and
off-diagonal disorder, $D_{\infty}(T) = c_{1}/T + c_{2} T$, where $c_{2} \ll
c_{1}$. An appendix discusses an alternative interpretation from the HSR model
of the `quantum to classical' dynamics transition, whereby the dynamics is
described as stochastically punctuated coherent motion.
