### Abstract

A variant of the Kac-Zwanzig model is used to test the prediction of transition state theory (TST) and variational transition state theory (VTST). The model describes the evolution of a distinguished particle moving in a double-well external potential and coupled to N free particles through linear springs. While the Kac-Zwanzig model is deterministic, under appropriate choice of the model parameters the evolution of the distinguished particle can be approximated by a two-state Markov chain whose transition rate constants can be computed exactly in suitable limit. Here, these transition rate constants are compared with the predictions of TST and VTST. It is shown that the application of TST with a naive (albeit natural) choice of dividing surface leads to the wrong prediction of the transition rate constants. This is due to crossings of the dividing surface that do not correspond to actual transition events. However, optimizing over the dividing surface within VTST allows one to eliminate completely these spurious crossings, and therefore derive the correct transition rate constants for the model. The reasons why VTST is successful in this model are discussed, which allows one to speculate on the reliability of VTST in more complicated systems.

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

Pages (from-to) | 43-73 |

Number of pages | 31 |

Journal | Journal of Statistical Physics |

Volume | 126 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2007 |

### Fingerprint

### Keywords

- Effective dynamics
- Harmonic oscillators
- Heat bath
- Metastability
- Stochastic equation
- Transition rates
- Transition state theory

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*126*(1), 43-73. https://doi.org/10.1007/s10955-006-9165-0

**Testing transition state theory on kac-Zwanzig model.** / Ariel, G.; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 126, no. 1, pp. 43-73. https://doi.org/10.1007/s10955-006-9165-0

}

TY - JOUR

T1 - Testing transition state theory on kac-Zwanzig model

AU - Ariel, G.

AU - Vanden Eijnden, Eric

PY - 2007/1

Y1 - 2007/1

N2 - A variant of the Kac-Zwanzig model is used to test the prediction of transition state theory (TST) and variational transition state theory (VTST). The model describes the evolution of a distinguished particle moving in a double-well external potential and coupled to N free particles through linear springs. While the Kac-Zwanzig model is deterministic, under appropriate choice of the model parameters the evolution of the distinguished particle can be approximated by a two-state Markov chain whose transition rate constants can be computed exactly in suitable limit. Here, these transition rate constants are compared with the predictions of TST and VTST. It is shown that the application of TST with a naive (albeit natural) choice of dividing surface leads to the wrong prediction of the transition rate constants. This is due to crossings of the dividing surface that do not correspond to actual transition events. However, optimizing over the dividing surface within VTST allows one to eliminate completely these spurious crossings, and therefore derive the correct transition rate constants for the model. The reasons why VTST is successful in this model are discussed, which allows one to speculate on the reliability of VTST in more complicated systems.

AB - A variant of the Kac-Zwanzig model is used to test the prediction of transition state theory (TST) and variational transition state theory (VTST). The model describes the evolution of a distinguished particle moving in a double-well external potential and coupled to N free particles through linear springs. While the Kac-Zwanzig model is deterministic, under appropriate choice of the model parameters the evolution of the distinguished particle can be approximated by a two-state Markov chain whose transition rate constants can be computed exactly in suitable limit. Here, these transition rate constants are compared with the predictions of TST and VTST. It is shown that the application of TST with a naive (albeit natural) choice of dividing surface leads to the wrong prediction of the transition rate constants. This is due to crossings of the dividing surface that do not correspond to actual transition events. However, optimizing over the dividing surface within VTST allows one to eliminate completely these spurious crossings, and therefore derive the correct transition rate constants for the model. The reasons why VTST is successful in this model are discussed, which allows one to speculate on the reliability of VTST in more complicated systems.

KW - Effective dynamics

KW - Harmonic oscillators

KW - Heat bath

KW - Metastability

KW - Stochastic equation

KW - Transition rates

KW - Transition state theory

UR - http://www.scopus.com/inward/record.url?scp=33846588411&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846588411&partnerID=8YFLogxK

U2 - 10.1007/s10955-006-9165-0

DO - 10.1007/s10955-006-9165-0

M3 - Article

VL - 126

SP - 43

EP - 73

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -