  A theory is developed to describe the coupled transport of energy and charge
in networks of electron donor-acceptor sites which are seated in a thermally
heterogeneous environment, where the transfer kinetics are dominated by
Marcus-type hopping rates. It is found that the coupling of heat and charge
transfer in such systems gives rise to exotic transport phenomena which are
absent in thermally homogeneous systems and cannot be described by standard
thermoelectric relations. Specifically, the directionality and extent of
thermal transistor amplification and cyclical electronic currents in a given
network can be controlled by tuning the underlying temperature gradient in the
system. The application of these findings toward optimal control of
multithermal currents is illustrated on a paradigmatic nanostructure.
