  We study the large deviation function for the entropy production rate in two
driven one-dimensional systems: the asymmetric random walk on a discrete
lattice and Brownian motion in a continuous periodic potential. We compare two
approaches: the Donsker-Varadhan theory and the Freidlin-Wentzell theory. We
show that the wings of the large deviation function are dominated by a single
optimal trajectory: either in forward (positive rate) or in backward direction
(negative rate). The joining of both branches at zero entropy production
implies a non-differentiability and thus the appearance of a "kink". However,
around zero entropy production many trajectories contribute and thus the kink
is smeared out.
