We consider dissipative systems resulting from the Gaussian and $alpha$-stable
noise perturbations of measure-preserving maps on the $d$ dimensional torus. We study the
dissipation time scale and its physical implications as the noise level $\vep$ vanishes. We
show that nonergodic maps give rise to an $O(1/\vep)$ dissipation time whereas ergodic
toral automorphisms, including cat maps and their $d$-dimensional generalizations, have an
$O(\ln{(1/\vep)})$ dissipation time with a constant related to the minimal, {\em
dimensionally averaged entropy} among the automorphism's irreducible blocks. Our approach
reduces the calculation of the dissipation time to a nonlinear, arithmetic optimization
problem which is solved asymptotically by means of some fundamental theorems in theories of
convexity, Diophantine approximation and arithmetic progression. We show that the same
asymptotic can be reproduced by degenerate noises as well as mere coarse-graining. We also
discuss the implication of the dissipation time in kinematic dynamo.