  We study the dissipative dynamics of one-dimensional fermions, described in
terms of the sine-Gordon model in its free massive boson or semi-classical
limit, while keeping track of forward scattering processes. The system is
prepared in the gapped ground state, and then coupled to environment through
local currents within the Lindblad formalism. The heating dynamics of the
system is followed using bosonization. The single particle density matrix
exhibits correlations between the left and right moving particles. While the
density matrix of right movers and left movers is translationally invariant,
the left-right sector is not, corresponding to a translational symmetry
breaking charge density wave state. Asymptotically, the single particle density
matrix decays exponentially with exponent proportional to $-\gamma
t|x|\Delta^2$ where $\gamma$ and $\Delta$ are the dissipative coupling and the
gap, respectively. The charge density wave order parameter decays exponentially
in time with an interaction independent decay rate. The second R\'enyi entropy
grows linearly with time and is essentially insensitive to the presence of the
gap.
