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From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data
One way to analyze complicated non-autonomous flows is through trying to
understand their transport behavior. In a quantitative, set-oriented approach
to transport and mixing, finite time coherent sets play an important role.
These are time-parametrized families of sets with unlikely transport to and
from their surroundings under small or vanishing random perturbations of the
dynamics. Here we propose, as a measure of transport and mixing for purely
advective (i.e., deterministic) flows, (semi)distances that arise under
vanishing perturbations in the sense of large deviations. Analogously, for
given finite Lagrangian trajectory data we derive a discrete-time and space
semidistance that comes from the "best" approximation of the randomly perturbed
process conditioned on this limited information of the deterministic flow. It
can be computed as shortest path in a graph with time-dependent weights.
Furthermore, we argue that coherent sets are regions of maximal farness in
terms of transport and mixing, hence they occur as extremal regions on a
spanning structure of the state space under this semidistance---in fact, under
any distance measure arising from the physical notion of transport. Based on
this notion we develop a tool to analyze the state space (or the finite
trajectory data at hand) and identify coherent regions. We validate our
approach on idealized prototypical examples and well-studied standard cases.Comment: J Nonlinear Sci, 201
Low cost microminiature refrigerators for large unit volume applications
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