  We study the Markovian process of a multi-mode open system connecting with a
non-equilibrium environment, which consists of several heat baths with
different temperatures. As an illustration, we study the steady state of three
linearly coupled harmonic oscillators in long time evolution, two of which
contact with two independent bosonic heat baths with different temperatures
respectively. We show that the inter-mode transitions mediated by the
environment is responsible for the long time behaviour of the dynamics
evolution, which is usually considered to take effect only in short time
dynamics of the system immersed in a equilibrium heat bath with a single
temperature. In non-equilibrium systems, the ignorance of the inter-mode
transitions may give rise to some counter-intuitive result, thus they cannot be
neglected.
