## 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 |
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Pages (from-to) | 43-73 |

Number of pages | 31 |

Journal | Journal of Statistical Physics |

Volume | 126 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2007 |

Externally published | Yes |

### Bibliographical note

Funding Information:We thank Alexandre Chorin, Matthias Heymann, Jose Koiller and Paul Wright for useful discussions and suggestions. We are especially grateful to Ray Kapral for sharing his insight on the Kac-Zwanzig model and to Zeev Schuss for introducing us to the work of Pollak et al and sharing some unpublished results. Partial support from NSF through Grants DMS01-01439, DMS02-09959 and DMS02-39625, and from ONR through Grant N-00014-04-1-0565 is gratefully acknowledged.

### Funding

We thank Alexandre Chorin, Matthias Heymann, Jose Koiller and Paul Wright for useful discussions and suggestions. We are especially grateful to Ray Kapral for sharing his insight on the Kac-Zwanzig model and to Zeev Schuss for introducing us to the work of Pollak et al and sharing some unpublished results. Partial support from NSF through Grants DMS01-01439, DMS02-09959 and DMS02-39625, and from ONR through Grant N-00014-04-1-0565 is gratefully acknowledged.

Funders | Funder number |
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National Science Foundation | DMS01-01439, DMS02-39625, DMS02-09959 |

Office of Naval Research | N-00014-04-1-0565 |

## Keywords

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