  We investigate the behavior in time of the energy current between a quantum
spin chain and its surrounding non-Markovian, finite temperature baths,
together with its relationship to the coherence dynamics of the system. To be
specific, both the system and the baths are assumed to be initially in thermal
equilibrium at temperature $T_s$ and $T_b$, respectively. This model plays a
fundamental role for the study of quantum system evolution towards thermal
equilibrium in an open system. The non-Markovian quantum state diffusion
(NMQSD) equation approach is used to calculate the dynamics of the spin chain.
The effects of bath non-Markovinity, temperature difference and system-bath
interaction strength on the energy current and the coherence in warm and cold
baths are analyzed, respectively. For both cases, our calculation results show
that strong non-Markovianity, weak system-bath interaction and low temperature
difference will be helpful to maintain the coherence of the system and
correspond to a small energy current. Interestingly, the warm baths destroy the
coherence while the cold baths help to generate coherence. Furthermore, the
effects of the Dzyaloshinskii-Moriya ($DM$) interaction and the external
magnetic field on the energy current and coherence are analyzed. Both energy
current and coherence will change due to the improvement of the system energy
induced by the $DM$ interaction and magnetic field. Significantly, the lowest
coherence corresponds to the critical magnetic field which causes the first
order phase transition.
