  Open Quantum Walks (OQWs) are exclusively driven by dissipation and are
formulated as completely positive trace preserving (CPTP) maps on underlying
graphs. The microscopic derivation of discrete and continuous in time OQWs is
presented. It is assumed that connected nodes are weakly interacting via a
common bath. The resulting reduced master equation of the quantum walker on the
lattice is in the generalised master equation form. The time discretisation of
the generalised master equation leads to the OQWs formalism. The explicit form
of the transition operators establishes a connection between dynamical
properties of the OQWs and thermodynamical characteristics of the environment.
The derivation is demonstrated for the examples of the OQW on a circle of nodes
and on a finite chain of nodes. For both examples a transition between
diffusive and ballistic quantum trajectories is observed and found to be
related to the temperature of the bath.
