### Abstract

We introduce an asymmetric classical Ginzburg-Landau model in a bounded interval, and study its dynamical behavior when perturbed by weak spatiotemporal noise. The Kramers escape rate from a locally stable state is computed as a function of the interval length. An asymptotically sharp second-order phase transition in activation behavior, with corresponding critical behavior of the rate prefactor, occurs at a critical length ℓ_{c}, similar to what is observed in symmetric models. The weak-noise exit time asymptotics, to both leading and subdominant orders, are analyzed at all interval lengthscales. The divergence of the prefactor as the critical length is approached is discussed in terms of a crossover from non-Arrhenius to Arrhenius behavior as noise intensity decreases. More general models without symmetry are observed to display similar behavior, suggesting that the presence of a "phase transition" in escape behavior is a robust and widespread phenomenon.

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

Pages (from-to) | 1537-1556 |

Number of pages | 20 |

Journal | Journal of Statistical Physics |

Volume | 114 |

Issue number | 5-6 |

State | Published - Mar 2004 |

### Fingerprint

### Keywords

- Droplet nucleation
- False vacuum
- Fluctuation determinant
- Fokker-Planck equation
- Instanton
- Non-Arrhenius behavior
- Spatiotemporal noise
- Stochastic escape problem
- Stochastic exit problem
- Stochastically perturbed dynamical systems

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*114*(5-6), 1537-1556.

**Critical behavior of the Kramers escape rate in asymmetric classical field theories.** / Stein, D. L.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 114, no. 5-6, pp. 1537-1556.

}

TY - JOUR

T1 - Critical behavior of the Kramers escape rate in asymmetric classical field theories

AU - Stein, D. L.

PY - 2004/3

Y1 - 2004/3

N2 - We introduce an asymmetric classical Ginzburg-Landau model in a bounded interval, and study its dynamical behavior when perturbed by weak spatiotemporal noise. The Kramers escape rate from a locally stable state is computed as a function of the interval length. An asymptotically sharp second-order phase transition in activation behavior, with corresponding critical behavior of the rate prefactor, occurs at a critical length ℓc, similar to what is observed in symmetric models. The weak-noise exit time asymptotics, to both leading and subdominant orders, are analyzed at all interval lengthscales. The divergence of the prefactor as the critical length is approached is discussed in terms of a crossover from non-Arrhenius to Arrhenius behavior as noise intensity decreases. More general models without symmetry are observed to display similar behavior, suggesting that the presence of a "phase transition" in escape behavior is a robust and widespread phenomenon.

AB - We introduce an asymmetric classical Ginzburg-Landau model in a bounded interval, and study its dynamical behavior when perturbed by weak spatiotemporal noise. The Kramers escape rate from a locally stable state is computed as a function of the interval length. An asymptotically sharp second-order phase transition in activation behavior, with corresponding critical behavior of the rate prefactor, occurs at a critical length ℓc, similar to what is observed in symmetric models. The weak-noise exit time asymptotics, to both leading and subdominant orders, are analyzed at all interval lengthscales. The divergence of the prefactor as the critical length is approached is discussed in terms of a crossover from non-Arrhenius to Arrhenius behavior as noise intensity decreases. More general models without symmetry are observed to display similar behavior, suggesting that the presence of a "phase transition" in escape behavior is a robust and widespread phenomenon.

KW - Droplet nucleation

KW - False vacuum

KW - Fluctuation determinant

KW - Fokker-Planck equation

KW - Instanton

KW - Non-Arrhenius behavior

KW - Spatiotemporal noise

KW - Stochastic escape problem

KW - Stochastic exit problem

KW - Stochastically perturbed dynamical systems

UR - http://www.scopus.com/inward/record.url?scp=3543028092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3543028092&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3543028092

VL - 114

SP - 1537

EP - 1556

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -