1. Constructing and exploring wells of energy landscapes.
- Author
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Aubin, Jean-Pierre and Lesne, Annick
- Subjects
- *
MATHEMATICAL models , *LANDSCAPES , *RELIEF models , *TOPOGRAPHICAL drawing , *ALGORITHMS , *MATHEMATICS , *STOCHASTIC approximation - Abstract
Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features “wells of duration T and altitude y.” We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells, and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) that have been developed during the last two decades; we therefore propose a minimal Appendix exposing them at the end of this paper to bridge the possible gap. [ABSTRACT FROM AUTHOR]
- Published
- 2005
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