### Abstract

The problem is considered of a Brownian particle confined in a potential well of forces, which escapes the potential barrier as the result of white noise forces acting on it. The problem is characterized by a diffusion process in a force field and is described by Langevin's stochastic differential equation. Potential wells with many transition states are considered and the expected exit time of the particle from the well as well as the probability distribution of the exit points are computed. The method relates these quantities to the solutions of certain singularly perturbed elliptic boundary value problems which are solved asymptotically. The results are then applied to the calculation of chemical reaction rates by considering the breaking of chemical bonds caused by random molecular collisions, and to the calculation of the diffusion matrix in crystals by considering random atomic migration in the periodic force field of the crystal lattice, caused by thermal vibrations of the lattice.

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

Pages (from-to) | 604-623 |

Number of pages | 20 |

Journal | SIAM J Appl Math |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1979 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*SIAM J Appl Math*,

*36*(3), 604-623. https://doi.org/10.1137/0136043

}

*SIAM J Appl Math*, vol. 36, no. 3, pp. 604-623. https://doi.org/10.1137/0136043

**EXIT PROBLEM : A NEW APPROACH TO DIFFUSION ACROSS POTENTIAL BARRIERS.** / Schuss, Zeev; Matkowsky, Bernard J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - EXIT PROBLEM

T2 - A NEW APPROACH TO DIFFUSION ACROSS POTENTIAL BARRIERS.

AU - Schuss, Zeev

AU - Matkowsky, Bernard J.

PY - 1979/1/1

Y1 - 1979/1/1

N2 - The problem is considered of a Brownian particle confined in a potential well of forces, which escapes the potential barrier as the result of white noise forces acting on it. The problem is characterized by a diffusion process in a force field and is described by Langevin's stochastic differential equation. Potential wells with many transition states are considered and the expected exit time of the particle from the well as well as the probability distribution of the exit points are computed. The method relates these quantities to the solutions of certain singularly perturbed elliptic boundary value problems which are solved asymptotically. The results are then applied to the calculation of chemical reaction rates by considering the breaking of chemical bonds caused by random molecular collisions, and to the calculation of the diffusion matrix in crystals by considering random atomic migration in the periodic force field of the crystal lattice, caused by thermal vibrations of the lattice.

AB - The problem is considered of a Brownian particle confined in a potential well of forces, which escapes the potential barrier as the result of white noise forces acting on it. The problem is characterized by a diffusion process in a force field and is described by Langevin's stochastic differential equation. Potential wells with many transition states are considered and the expected exit time of the particle from the well as well as the probability distribution of the exit points are computed. The method relates these quantities to the solutions of certain singularly perturbed elliptic boundary value problems which are solved asymptotically. The results are then applied to the calculation of chemical reaction rates by considering the breaking of chemical bonds caused by random molecular collisions, and to the calculation of the diffusion matrix in crystals by considering random atomic migration in the periodic force field of the crystal lattice, caused by thermal vibrations of the lattice.

UR - http://www.scopus.com/inward/record.url?scp=0018478697&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0018478697&partnerID=8YFLogxK

U2 - 10.1137/0136043

DO - 10.1137/0136043

M3 - Article

VL - 36

SP - 604

EP - 623

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -