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.Comment: The research is supported in part by the grant from U.S. National
Science Foundation, DMS-9971322 and Lech Wolowsk
