### Abstract

This paper is devoted to the theoretical analysis of the zero-temperature string method, a scheme for identifying minimum energy paths (MEPs) on a given energy landscape. By definition, MEPs are curves connecting critical points on the energy landscape which are everywhere tangent to the gradient of the potential except possibly at critical points. In practice, MEPs are mountain pass curves that play a special role, e.g., in the context of rare reactive events that occur when one considers a steepest descent dynamics on the potential perturbed by a small random noise. The string method aims to identify MEPs by moving each point of the curve by steepest descent on the energy landscape. Here we address the question of whether such a curve evolution necessarily converges to an MEP. Surprisingly, the answer is no, for an interesting reason: MEPs may not be isolated, in the sense that there may be families of them that can be continuously deformed into one another. This degeneracy is related to the presence of critical points of Morse index 2 or higher along the MEP. In this paper, we elucidate this issue and completely characterize the limit set of a curve evolving by the string method. We establish rigorously that the limit set of such a curve is again a curve when the MEPs are isolated. We also show under the same hypothesis that the string evolution converges to an MEP. However, we identify and classify situations where the limit set is not a curve and may contain higher dimensional parts.We present a collection of examples where the limit set of a path contains a 2D region, a 2D surface, or a region of an arbitrary dimension up to the dimension of the space. In some of our examples the evolving path wanders around without converging to its limit set. In other examples it fills a region, converging to its limit set, which is not an MEP.

Original language | English (US) |
---|---|

Pages (from-to) | 193-230 |

Number of pages | 38 |

Journal | Journal of Nonlinear Science |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2011 |

### Fingerprint

### Keywords

- Convergence
- Critical point
- Curve evolutio
- Heteroclinic trajectory
- Limit set
- Morse index
- String method

### ASJC Scopus subject areas

- Applied Mathematics
- Modeling and Simulation
- Engineering(all)

### Cite this

*Journal of Nonlinear Science*,

*21*(2), 193-230. https://doi.org/10.1007/s00332-010-9081-y

**The string method as a dynamical system.** / Cameron, Maria; Kohn, Robert V.; Vanden-Eijnden, Eric.

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 21, no. 2, pp. 193-230. https://doi.org/10.1007/s00332-010-9081-y

}

TY - JOUR

T1 - The string method as a dynamical system

AU - Cameron, Maria

AU - Kohn, Robert V.

AU - Vanden-Eijnden, Eric

PY - 2011/4

Y1 - 2011/4

N2 - This paper is devoted to the theoretical analysis of the zero-temperature string method, a scheme for identifying minimum energy paths (MEPs) on a given energy landscape. By definition, MEPs are curves connecting critical points on the energy landscape which are everywhere tangent to the gradient of the potential except possibly at critical points. In practice, MEPs are mountain pass curves that play a special role, e.g., in the context of rare reactive events that occur when one considers a steepest descent dynamics on the potential perturbed by a small random noise. The string method aims to identify MEPs by moving each point of the curve by steepest descent on the energy landscape. Here we address the question of whether such a curve evolution necessarily converges to an MEP. Surprisingly, the answer is no, for an interesting reason: MEPs may not be isolated, in the sense that there may be families of them that can be continuously deformed into one another. This degeneracy is related to the presence of critical points of Morse index 2 or higher along the MEP. In this paper, we elucidate this issue and completely characterize the limit set of a curve evolving by the string method. We establish rigorously that the limit set of such a curve is again a curve when the MEPs are isolated. We also show under the same hypothesis that the string evolution converges to an MEP. However, we identify and classify situations where the limit set is not a curve and may contain higher dimensional parts.We present a collection of examples where the limit set of a path contains a 2D region, a 2D surface, or a region of an arbitrary dimension up to the dimension of the space. In some of our examples the evolving path wanders around without converging to its limit set. In other examples it fills a region, converging to its limit set, which is not an MEP.

AB - This paper is devoted to the theoretical analysis of the zero-temperature string method, a scheme for identifying minimum energy paths (MEPs) on a given energy landscape. By definition, MEPs are curves connecting critical points on the energy landscape which are everywhere tangent to the gradient of the potential except possibly at critical points. In practice, MEPs are mountain pass curves that play a special role, e.g., in the context of rare reactive events that occur when one considers a steepest descent dynamics on the potential perturbed by a small random noise. The string method aims to identify MEPs by moving each point of the curve by steepest descent on the energy landscape. Here we address the question of whether such a curve evolution necessarily converges to an MEP. Surprisingly, the answer is no, for an interesting reason: MEPs may not be isolated, in the sense that there may be families of them that can be continuously deformed into one another. This degeneracy is related to the presence of critical points of Morse index 2 or higher along the MEP. In this paper, we elucidate this issue and completely characterize the limit set of a curve evolving by the string method. We establish rigorously that the limit set of such a curve is again a curve when the MEPs are isolated. We also show under the same hypothesis that the string evolution converges to an MEP. However, we identify and classify situations where the limit set is not a curve and may contain higher dimensional parts.We present a collection of examples where the limit set of a path contains a 2D region, a 2D surface, or a region of an arbitrary dimension up to the dimension of the space. In some of our examples the evolving path wanders around without converging to its limit set. In other examples it fills a region, converging to its limit set, which is not an MEP.

KW - Convergence

KW - Critical point

KW - Curve evolutio

KW - Heteroclinic trajectory

KW - Limit set

KW - Morse index

KW - String method

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U2 - 10.1007/s00332-010-9081-y

DO - 10.1007/s00332-010-9081-y

M3 - Article

VL - 21

SP - 193

EP - 230

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 2

ER -