  We study a chaotic ratchet system under the influence of a thermal
environment. By direct integration of the Lindblad equation we are able to
analyze its behavior for a wide range of couplings with the environment, and
for different finite temperatures. We observe that the enhancement of the
classical and quantum currents due to temperature depend strongly on the
specific properties of the system. This makes difficult to extract universal
behaviors. We have also found that there is an analogy between the effects of
the classical thermal noise and those of the finite $\hbar$ size. These results
open many possibilities for their testing and implementation in kicked BECs and
cold atoms experiments.
