Characterizing the metastable balance between chaos and diffusion

We examine some new diagnostics for the behavior of a field rho evolving in an advective-diffusive system. One of these diagnostics is approximately t he Fourier second moment (denoted as chi (2)) and the other is: the linear entropy or field intensity S, the latter being significantly easier to comp ute or measure. We establish that as a result of chaos the increasing struc ture in rho is accompanied by chi increasing exponentially rapidly in time at a rate given by rho -dependent Lyapunov exponents Lambda (i) and dominat ed by the largest one Lambda (max). Noise or diffusive coarse-graining of r ho causes chi to decrease as chi (2) approximate to 1/4Dt, where D is a mea sure of the diffusion. When both effects are present the competition betwee n the processes leads to metastability for chi followed by a final diffusiv e tail. The initial stages may be chaotic or diffusive depending upon the v alue of Lambda (-1)(max)2D chi (2)(0) but the metastable value of chi (2) i s given by chi (2)* = Sigma (i)A (i)/2D irrespective. Since (S)over dot = - 2D chi (2), similar analysis applies to S. and in particular there exists a metastable decay rate for S given by (S)over dot * = Sigma (i)Lambda (i). These arguments are verified for a simple case, the Amol'd Cat Map with add ed diffusive noise. (C) 2001 Published by Elsevier Science B.V.