Abstract
The short time behavior of nucleation probabilities is studied by representing nucleation as a diffusion process in a potential well with escape over a barrier. If initially all growing nuclei start at the bottom of the well, the first nucleation time on average is larger than the inverse nucleation frequency. Explicit expressions are obtained for the short time probability of first nucleation. For very short times these become independent of the shape of the potential well. They agree well with numerical results from an exact enumeration scheme. For a large number N of growing nuclei the average first nucleation time scales as 1/log N in contrast to the long-time nucleation frequency, which scales as 1/N. For linear potential wells closed form expressions are given for all times.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. N. Kolmogorov, On the statistical theory of the crystallization of metals, Bull. Acad. Sci. USSR, Math. Ser. 1:355(1937).
W. A. Johnson and P. A. Mehl, Reaction kinetics in processes of nucleation and growth, Trans. Am. Inst. Min. Engrs. 135:416(1939).
M. A. Avrami, Kinetics of phase change I-III, J. Chem. Phys. 7:1103(1939).
P. Dehghanpour and R. H. Schonmann, Metropolis dynamics relaxation via nucleation and growth, Commun. Math. Phys. 188:89(1997).
N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
N. G. van Kampen, Short first passage times, J. Statist. Phys. 70:15(1993).
G. Grimmett, Probability and Random Processes, (Clarendon, Oxford, 1981)
J. Koplik, S. Redner, and D. Wilkinson, Transport and dispersion in random networks with percolation disorder, Phys. Rev. A 37:2619(1998).
S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge 2001).
H. van Beijeren, Transport properties of stochastic Lorentz models, Rev. Mod. Phys. 54:195(1982).
The Laplace transform of this equation is identical to Eq. (2.2.30) of ref. 9, in the limit Pe<<-1.
H. B. Rosenstock, Random walks on lattices with traps, J. Math. Phys. 11:487(1970).
See, e.g., ref. 5, p. 222.
K. Brendel, G. Barkema, and H. van Beijeren, in preparation.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van Beijeren, H. The Uphill Turtle Race; On Short Time Nucleation Probabilities. Journal of Statistical Physics 110, 1397–1410 (2003). https://doi.org/10.1023/A:1022121616193
Issue Date:
DOI: https://doi.org/10.1023/A:1022121616193