Using the Haken-Strobl-Reineker Model to Determine the Temperature Dependence of the Diffusion Coefficient

Abstract

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.