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Journal of Statistical Physics

, Volume 78, Issue 1–2, pp 267–283 | Cite as

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

  • O. Penrose
Articles

Abstract

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series n Q n z n , whereQ n , which depends on the temperatureT, is the “partition function” of a cluster of sizen. At low temperatures this series has a finite radius of convergencez s . Some theorems are proved showing that ifQ n , considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromz s to +∞ and that (a) the imaginary part ofP(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 ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.

Key Words

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

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References

  1. 1.
    T. J. I'A. Bromwich,Infinite Series (Macmillan, 1908), Section 112, p. 293.Google Scholar
  2. 2.
    R. Becker and W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen,Ann. Phys. 24:719–752 (1935).Google Scholar
  3. 3.
    C. Borgs, Private communication (1994).Google Scholar
  4. 4.
    J. Bricmont, K. Gawedski, O. Gabber, and A. Kupiainen, Private communication (1994).Google Scholar
  5. 5.
    H. S. Carslaw and J. C. Jaeger,Operational Methods in Applied Mathematics, 2nd ed. (Oxford University Press, Oxford, 1948), p. 354.Google Scholar
  6. 6.
    M. E. Fisher, The theory of condensation and the critical point,Physics 3:255–283 (1967).Google Scholar
  7. 7.
    B. Gaveau and L. S. Schulman, Metastable decay rates and analytic continuation,Lett. Math. Phys. 18:201–208 (1989).Google Scholar
  8. 8.
    R. B. Griffiths, Microcanonical ensemble in quantum statistical mechanics,J. Math. Phys. 6:1447–1461 (1965).Google Scholar
  9. 9.
    C. C. A. Günther, P. A. Rikvold, and M. A. Novotny, Transfer-matrix study of metastability in thed=2 Ising model,Phys. Rev. Lett. 71:3898–3901 (1993).Google Scholar
  10. 10.
    T. L. Hill,Statistical Mechanics: Principles and Selected Applications (McGraw-Hill, New York, 1956), Section 26.Google Scholar
  11. 11.
    S. N. Isakov, Nonanalytic features of the first-order phase transition in the Ising model,Commun. Math. Phys. 95:427–443 (1984).Google Scholar
  12. 12.
    J. S. Langer, Statistical theory of the decay of metastable states,Ann. Phys. 54:258–275 (1969).Google Scholar
  13. 13.
    T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions, II. Lattice gas and Ising model,Phys. Rev. 87, 410–419 (1952).Google Scholar
  14. 14.
    S. Mandelbrojt,C. R. Acad. Sci. Paris 209:977 (1939).Google Scholar
  15. 15.
    J. C. Maxwell, On the dynamical evidence of the molecular constitution of bodies, inThe Scientific Papers of James Clerk Maxwell, W. A. Niven ed. (1890) (reprinted, Dover, New York, 1965), Vol. 2, pp. 418–438.Google Scholar
  16. 16.
    R. J. McCraw and L. S. Schulman, Metastability in the two-dimensional Ising model,J. Stat. Phys. 18:293–308 (1978).Google Scholar
  17. 17.
    A. Messiah,Quantum Mechanics (North-Holland, Amsterdam, 1965), Section X15–16, pp. 399–402.Google Scholar
  18. 18.
    C. M. Newman and L. S. Schulman, Complex free energies and metastable lifetimes,J. Stat. Phys. 23:131–148 (1980).Google Scholar
  19. 19.
    O. Penrose, Metastable states for the Becker-Döring cluster equations,Commun. Math. Phys. 121:527–540 (1989).Google Scholar
  20. 20.
    O. Penrose and J. L. Lebowitz, Rigorous treatment of metastable states in the van der Waals-Maxwell theory,J. Stat. Phys. 3:211–241 (1971).Google Scholar
  21. 21.
    A. C. Pipkin,A Course on Integral Equations (Springer, 1991), p. 174.Google Scholar
  22. 22.
    G. Roepstorff and L. S. Schulman, Metastability and analyticity in a dropletlike model,J. Stat. Phys. 34:35–56 (1984).Google Scholar
  23. 23.
    L. S. Schulman, A stochastic process with metastability and complex free energy,Phys. Rep. 77:359–362 (1981).Google Scholar
  24. 24.
    J. G. Taylor and J. Gunson, Unstable particles in a general field theory,Phys. Rev. 119:1121–1125 (1960); single-particle singularities in scattering and production amplitudes,Nuovo Cimento 15:806 (1960).Google Scholar
  25. 25.
    U. Weiss,Quantum Dissipative Systems (World Scientific, Singapore, 1993), esp. Section 8.4.Google Scholar
  26. 26.
    W. Zwerger, Dynamical interpretation of a classical complex free energy,J. Phys. A: Math. Gen. 18:2079–2085 (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • O. Penrose
    • 1
  1. 1.Department of MathematicsHeriot-Watt University, RiccartonEdinburghScotland, UK

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