Physica D 148 (2001) 1–19 Characterizing the metastable balance between chaos and diffusion Arjendu K. Pattanayak Department of Physics and Astronomy, Rice University, Houston, TX 77251-1892, USA Received 2 December 1999; received in revised form 20 June 2000; accepted 11 September 2000 Communicated by C.K.R.T. Jones Abstract We examine some new diagnostics for the behavior of a field ρ evolving in an advective–diffusive system. One of these diagnostics is approximately the Fourier second moment (denoted as χ 2 ) and the other is the linear entropy or field intensity S , the latter being significantly easier to compute or measure. We establish that as a result of chaos the increasing structure in ρ is accompanied by χ increasing exponentially rapidly in time at a rate given by ρ -dependent Lyapunov exponents Λ i and dominated by the largest one Λ max . Noise or diffusive coarse-graining of ρ causes χ to decrease as χ 2 ≈ 1 4 Dt, where D is a measure of the diffusion. When both effects are present the competition between the processes leads to metastability for χ followed by a final diffusive tail. The initial stages may be chaotic or diffusive depending upon the value of Λ −1 max 2Dχ 2 (0) but the metastable value of χ 2 is given by χ 2∗ = ∑ i Λ i /2D irrespective. Since ˙ S =−2Dχ 2 , similar analysis applies to S , and in particular there exists a metastable decay rate for S given by ˙ S ∗ = ∑ i Λ i . These arguments are verified for a simple case, the Arnol’d Cat Map with added diffusive noise. © 2001 Published by Elsevier Science B.V. PACS: 05.40.Ca; 05.45.a Keywords: Chaos; Noise; Metastability 1. Introduction The dynamics by which a passive scalar is mixed and dispersed by an incompressible fluid flow is a fundamental problem of considerable interest. There are many applications in fields such as chemical mixing and climate, for example [1–5]. These dynamics are described by the advection–diffusion equation ∂ρ ∂t + v ·  ∇ρ = D∇ 2 ρ, (1) where ρ(  x,t) is the density of the passive scalar field,  v(  x,t) the Eulerian fluid velocity, D the molecular diffusivity constant and the incompressibility is expressed by the constraint ∇· v = 0. In even dimensions, precisely the same equation with  x ⇔ (  p,  q) also describes the behavior of the Liouville probability distribution for a Hamiltonian system subject to Gaussian noise in all phase-space variables, a system with applications in the behavior of galaxies, for example [6]. The advective and the diffusive transport processes for this system, described by the  v ·  ∇ρ and the D∇ 2 ρ terms, respectively, have entirely different impacts on the field ρ . Two limits can be immediately identified, E-mail address: arjendu@rice.edu (A.K. Pattanayak). 0167-2789/01/$ – see front matter © 2001 Published by Elsevier Science B.V. PII:S0167-2789(00)00186-X