  We study numerically how conservation laws affect the optical conductivity
sigma(w) of a slightly doped one-dimensional Mott insulator. We investigate a
regime where the average distance between charge excitations is large compared
to their thermal de Broglie wave length and a classical description is
possible. Due to conservation laws, the dc-conductivity is infinite and the
Drude weight D is finite even at finite temperatures. Our numerical results
test and confirm exact theoretical predictions for D both for integrable and
non-integrable models. Small deviations from integrability induce slowly
decaying modes and, consequently, low-frequency peaks in sigma(w) which can be
described by a memory matrix approach.
