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2015 | OriginalPaper | Chapter

1. Introduction

Author : Matthias Heymann

Published in: Minimum Action Curves in Degenerate Finsler Metrics

Publisher: Springer International Publishing

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Abstract

In this chapter we introduce the reader to the problem addressed by this monograph. First we explain the main question at hand and its motivation in the context of the Wentzell-Freidlin theory of rare transition paths. We then summarize the main features of our existence theory, and the various approaches used in the literature. Finally, we explain the structure of this monograph and introduce some notation.

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next chapter Geometric Action Functionals
Footnotes
1
The reader with no background in probability theory should not feel discouraged here: No knowledge in that field will be required to understand the results or proofs in this monograph.
 
2
That is, the definition of the dynamics via its drift and noise covariance matrix in the case of an SDE, or via its jump rates in the case of a jump process, cannot explicitly depend on time.
 
3
See, however, [10, Sect. 2.4] for how the optimal time parameterization can be recovered from the minimum action curve γ ⋆.
 
4
Note that we may occasionally reuse the letter n of our state space dimension also for other purposes, e.g., as an index for sequences such as \((\gamma _{n})_{n\in \mathbb{N}}\).
 
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Bao, D.D.W., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000)
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Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften, vol. 260, 2nd edn. Springer, New York (1998)
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Heymann, M.: The geometric minimum action method: A least action principle on the space of curves. Ph.D. thesis, New York University (2007)
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Heymann, M., Vanden-Eijnden, E.: The geometric minimum action method: A least action principle on the space of curves. Commun. Pure Appl. Math. 61(8), 1052–1117 (2008)CrossRefMATHMathSciNet
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Heymann, M., Vanden-Eijnden, E.: Pathways of maximum likelihood for rare events in nonequilibrium systems: Application to nucleation in the presence of shear. Phys. Rev. Lett. 100, 140601 (2008)CrossRefMATH
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Vanden-Eijnden, E., Heymann, M.: The geometric minimum action method for computing minimum energy paths. J. Chem. Phys. 128(6), 061103 (2008)CrossRef
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Metadata
Title
Introduction
Author
Matthias Heymann
Copyright Year
2015
Book
Minimum Action Curves in Degenerate Finsler Metrics

Print ISBN: 978-3-319-17752-6

Electronic ISBN: 978-3-319-17753-3

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-17753-3_1