  We present a Quantum Monte Carlo (QMC) study, based on the Langevin equation,
of a Hamiltonian describing electrons coupled to phonon degrees of freedom. The
bosonic part of the action helps control the variation of the field in
imaginary time. As a consequence, the iterative conjugate gradient solution of
the fermionic action, which depends on the boson coordinates, converges more
rapidly than in the case of electron-electron interactions, such as the Hubbard
Hamiltonian. Fourier Acceleration is shown to be a crucial ingredient in
reducing the equilibration and autocorrelation times. After describing and
benchmarking the method, we present results for the phase diagram focusing on
the range of the electron-phonon interaction. We delineate the regions of
charge density wave formation from those in which the fermion density is
inhomogeneous, caused by phase separation. We show that the Langevin approach
is more efficient than the Determinant QMC method for lattice sizes $N \gtrsim
8 \times 8$ and that it therefore opens a potential path to problems including,
for example, charge order in the 3D Holstein model.
