Nearest-neighbor Diffusion Research Articles

A global study of enhanced stretching and diffusion in chaotic tangles

A global, finite-time study is made of stretching and diffusion in a class of chaotic tangles associated with fluids described by periodically forced two-dimensional dynamical systems. Invariant lobe structures formed by intersecting global stable and unstable manifolds of persisting invariant hyperbolic sets provide the geometrical framework for studying stretching of interfaces and diffusion of passive scalars across these interfaces. In particular, the present study focuses on the material curve that initially lies on the unstable manifold segment of the boundary of the entraining turnstile lobe. A knowledge of the stretch profile of a corresponding curve that evolves according to the unperturbed flow, coupled with an appreciation of a symbolic dynamics that applies to the entire original material curve in the perturbed flow, provides the framework for understanding the mechanism for, and topology of, enhanced stretching in chaotic tangles. Secondary intersection points (SIP’s) of the stable and unstable manifolds are particularly relevant to the topology, and the perturbed stretch profile is understood in terms of the unperturbed stretch profile approximately repeating itself on smaller and smaller scales. For sufficiently thin diffusion zones, diffusion of passive scalars across interfaces can be treated as a one-dimensional process, and diffusion rates across interfaces are directly related to the stretch history of the interface. An understanding of interface stretching thus directly translates to an understanding of diffusion across interfaces. However, a notable exception to the thin diffusion zone approximation occurs when an interface folds on top of itself so that neighboring diffusion zones overlap. An analysis which takes into account the overlap of nearest neighbor diffusion zones is presented, which is sufficient to capture new phenomena relevant to efficiency of mixing. The analysis adds to the concentration profile a saturation term that depends on the distance between neighboring segments of the interface. Efficiency of diffusion thus depends not only on efficiency of stretching along the interface, but on how this stretching is distributed relative to the distance between neighboring segments of the interface.

Domain growth and nucleation in a discrete bistable system.

A cubic map lattice, which consists of an array of cubic discrete-time maps coupled through nearest-neighbor diffusion interactions, is studied in a parameter region where the isolated maps possess bistable steady states. Within this parameter region, the spatio-temporal evolution of this deterministic dynamical system exhibits the phenomena of phase separation and nucleation. When the two coexisting states have the same stability, the domain growth is like that for continuous systems with a nonconserved order parameter, and follows Allen-Cahn scaling on long distance and time scales. When the states have different stabilities, growth typically occurs by a nucleation process. In addition to exploring these phenomena, which have their analogs in continuous systems, features of the inhomogeneous states peculiar to the discrete model are also investigated.