### Abstract

The grand potential P(z)/kT of the cluster model at fugacity z, neglecting interactions between clusters, is defined by a power series ?_{n}Q_{n}z^{n}, where Q_{n}, which depends on the temperature T, is the "partition function" of a cluster of size n. At low temperatures this series has a finite radius of convergence z_{s}. Some theorems are proved showing that if Q_{n}, considered as a function of n, is the Laplace transform of a function with suitable properties, then P(z) can be analytically continued into the complex z plane cut along the real axis from z_{s} to +8 and that (a) the imaginary part of P(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part of P(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term. © 1995 Plenum Publishing Corporation.

Original language | English |
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Pages (from-to) | 267-283 |

Number of pages | 17 |

Journal | Journal of Statistical Physics |

Volume | 78 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jan 1995 |

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### Keywords

- analytic continuation
- asymptotic expansions
- cluster model
- complex fugacity plane
- lattice gases
- Metastability

### Cite this

*Journal of Statistical Physics*,

*78*(1-2), 267-283. https://doi.org/10.1007/BF02183348

}

*Journal of Statistical Physics*, vol. 78, no. 1-2, pp. 267-283. https://doi.org/10.1007/BF02183348

**Metastable decay rates, asymptotic expansions, and analytic continuation of thermodynamic functions.** / Penrose, O.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Metastable decay rates, asymptotic expansions, and analytic continuation of thermodynamic functions

AU - Penrose, O.

PY - 1995/1

Y1 - 1995/1

N2 - The grand potential P(z)/kT of the cluster model at fugacity z, neglecting interactions between clusters, is defined by a power series ?nQnzn, where Qn, which depends on the temperature T, is the "partition function" of a cluster of size n. At low temperatures this series has a finite radius of convergence zs. Some theorems are proved showing that if Qn, considered as a function of n, is the Laplace transform of a function with suitable properties, then P(z) can be analytically continued into the complex z plane cut along the real axis from zs to +8 and that (a) the imaginary part of P(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part of P(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term. © 1995 Plenum Publishing Corporation.

AB - The grand potential P(z)/kT of the cluster model at fugacity z, neglecting interactions between clusters, is defined by a power series ?nQnzn, where Qn, which depends on the temperature T, is the "partition function" of a cluster of size n. At low temperatures this series has a finite radius of convergence zs. Some theorems are proved showing that if Qn, considered as a function of n, is the Laplace transform of a function with suitable properties, then P(z) can be analytically continued into the complex z plane cut along the real axis from zs to +8 and that (a) the imaginary part of P(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part of P(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term. © 1995 Plenum Publishing Corporation.

KW - analytic continuation

KW - asymptotic expansions

KW - cluster model

KW - complex fugacity plane

KW - lattice gases

KW - Metastability

UR - http://www.scopus.com/inward/record.url?scp=21844523337&partnerID=8YFLogxK

U2 - 10.1007/BF02183348

DO - 10.1007/BF02183348

M3 - Article

VL - 78

SP - 267

EP - 283

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -