Skip to main content
SpringerLink
Log in
Book cover

Classical Statistical Mechanics with Nested Sampling pp 61–96Cite as

Nested Sampling for Materials

  • Chapter
  • First Online:

  • 447 Accesses

Part of the book series: Springer Theses ((Springer Theses))

Abstract

This chapter develops nested sampling into a powerful tool for the calculation of pressure-temperature phase diagrams, and demonstrates how it may be applied to single species and binary systems, including the Lennard-Jones system, a binary Lennard-Jones alloy, and an EAM model for Aluminium. A comparison to parallel tempering is also presented.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Although \(V_u \) appears in Eq. (8.23) as an argument of \({ \Delta }_{\mathrm {NS}}\) but not of \(\widetilde{{ \Delta }}\), there is no inconsistency because (8.23) is only valid in the limit \(k_{\mathrm {B}}T/ PV_u \rightarrow 0\) where the value of \({ \Delta }_{\mathrm {NS}}\) is independent of \(V_u \).

  2. 2.

    Note that the behaviour of Lennard-Jonesium is different to that of a harmonic-crystal, where nearest neighbour atoms are connected by springs: at zero pressure the harmonic-crystal slightly favours the FCC lattice over the HCP lattice [21].

References

  1. J. Skilling, Nested sampling. AIP Conf. Proc. 735, 395 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  2. J. Skilling, Nested sampling for general Bayesian computation. Bayesian Anal. 1, 833 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. A.W. Jasper, N.E. Schultz, D.G. Truhlar, Analytic potential energy functions for simulating aluminum nanoparticles. J. Phys. Chem. B 109, 3915 (2005)

    Article  Google Scholar 

  4. D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, ser (Elsevier Science, Computational science series, 2001)

    MATH  Google Scholar 

  5. G.J. Martyna, D.J. Tobias, M.L. Klein, Constant pressure molecular dynamics algorithms. J. Chem. Phys. 101, 4177 (1994)

    Article  ADS  Google Scholar 

  6. M. Tuckerman, Statistical Mechanics and Molecular Simulations (Oxford University Press, 2008)

    Google Scholar 

  7. W. Wood, Monte Carlo calculations for hard disks in the isothermal-isobaric ensemble. J. Chem. Phys. 48, 415 (1968)

    Article  ADS  Google Scholar 

  8. L.B. Pártay, A.P. Bartók, G. Csányi, Nested sampling for materials: the case of hard spheres. Phys. Rev. E 89, 022302 (2014)

    Article  Google Scholar 

  9. R.M. Neal, MCMC using Hamiltonian dynamics, Handbook of Markov Chain Monte Carlo (CRC Press, New York, NY, 2011), p. 113

    Google Scholar 

  10. V.V. Brazhkin, A.G. Lyapin, V.N. Ryzhov, K. Trachenko, Y.D. Fomin, E.N. Tsiok, Where is the supercritical fluid on the phase diagram? Phys. Uspekhi 55, 1061 (2012)

    Article  ADS  Google Scholar 

  11. A. Bruce, N. Wilding, Scaling fields and universality of the liquid-gas critical point. Phys. Rev. Lett. 68, 193 (1992)

    Article  ADS  Google Scholar 

  12. D.A. Kofke, Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line. J. Chem. Phys. 98, 4149 (1993)

    Article  ADS  Google Scholar 

  13. E.A. Mastny, J.J. de Pablo, Melting line of the Lennard-Jones system, infinite size, and full potential. J. Chem. Phys. 127, 104504 (2007)

    Article  ADS  Google Scholar 

  14. G.C. McNeil-Watson, N.B. Wilding, Freezing line of the Lennard-Jones fluid: a phase switch Monte Carlo study. J. Chem. Phys. 124, 064504 (2006)

    Article  ADS  Google Scholar 

  15. A. Ahmed, R.J. Sadus, Effect of potential truncations and shifts on the solidliquid phase coexistence of Lennard-Jones fluids. J. Chem. Phys. 133, 124515 (2010)

    Article  ADS  Google Scholar 

  16. R. Agrawal, D.A. Kofke, Thermodynamic and structural properties of model systems at solid-fluid coexistence. Mol. Phys. 85, 43 (1995)

    Article  ADS  Google Scholar 

  17. M.A. Barroso, A.L. Ferreira, Solid-fluid coexistence of the Lennard-Jones system from absolute free energy calculations. J. Chem. Phys. 116, 7145 (2002)

    Article  ADS  Google Scholar 

  18. P.A. Apte, I. Kusaka, Direct calculation of solid-vapor coexistence points by thermodynamic integration: application to single component and binary systems. J. Chem. Phys. 124, 184106 (2006)

    Article  ADS  Google Scholar 

  19. B. Smit, Phase diagrams of Lennard-Jones fluids. J. Chem. Phys. 96, 8639 (1992)

    Article  ADS  Google Scholar 

  20. T. Kihara, S. Koba, Crystal structures and intermolecular forces of rare gases. J. Phys. Soc. Japan 7, 348 (1952)

    Article  ADS  Google Scholar 

  21. W.G. Hoover, Entropy for small classical crystals. J. Chem. Phys. 49, 1981 (1968)

    Article  ADS  Google Scholar 

  22. A.N. Jackson, A.D. Bruce, G.J. Ackland, Lattice-switch Monte Carlo method: application to soft potentials. Phys. Rev. E 65, 036710 (2002)

    Article  ADS  Google Scholar 

  23. F.H. Stillinger, Lattice sums and their phase diagram implications for the classical Lennard-Jones model. J. Chem. Phys. 115, 5208 (2001)

    Article  ADS  Google Scholar 

  24. P. Rein ten Wolde, M.J. Ruiz-Montero, D. Frenkel, Numerical calculation of the rate of crystal nucleation in a Lennard-Jones system at moderate undercooling. J. Chem. Phys. 104, 9932 (1996)

    Article  ADS  Google Scholar 

  25. S. Pronk, D. Frenkel, Can stacking faults in hard-sphere crystals anneal out spontaneously? J. Chem. Phys. 110, 4589 (1999)

    Article  ADS  Google Scholar 

  26. D. Frenkel, A.J.C. Ladd, New Monte Carlo method to compute the free energy of arbitrary solids, application to the fcc and hcp phases of hard spheres. J. Chem. Phys. 81, 3188 (1984)

    Article  ADS  Google Scholar 

  27. D. Bhatt, A.W. Jasper, N.E. Schultz, J.I. Siepmann, D.G. Truhlar, Critical properties of aluminum. J. Am. Chem. Soc. 128, 4224 (2006)

    Article  Google Scholar 

  28. Y. Akahama, M. Nishimura, K. Kinoshita, H. Kawamura, Y. Ohishi, Evidence of a fcc-hcp transition in aluminum at multimegabar pressure. Phys. Rev. Lett. 96, 045505 (2006)

    Article  ADS  Google Scholar 

  29. J.C. Boettger, S.B. Trickey, High-precision calculation of the equation of state and crystallographic phase stability for aluminum. Phys. Rev. B 53, 3007 (1996)

    Article  ADS  Google Scholar 

  30. G.V. Sin’ko, N.A. Smirnov, Ab initio calculations of elastic constants and thermodynamic properties of bcc, fcc, and hcp Al crystals under pressure. J. Phys. Condens. Matter 14, 6989 (2002)

    Article  ADS  Google Scholar 

  31. M.J. Tambe, N. Bonini, N. Marzari, Bulk aluminum at high pressure: a first-principles study. Phys. Rev. B 77, 172102 (2008)

    Article  ADS  Google Scholar 

  32. A.Z. Panagiotopoulos, Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble. Mol. Phys. 61, 813 (1987)

    Article  ADS  Google Scholar 

  33. A.A. Likalter, Critical points of metals of three main groups and selected transition metals. Phys. A 311, 137 (2002)

    Article  Google Scholar 

  34. V. Fortov, I. Iakubov, Non-Ideal Plasma (Plenum Press, New York, 2000)

    Book  Google Scholar 

  35. D. Errandonea, The melting curve of ten metals up to 12 GPa and 1600 K. J. Appl. Phys. 108, 033517 (2010)

    Article  ADS  Google Scholar 

  36. R. Boehler, M. Ross, Melting curve of aluminum in a diamond cell to 0.8 Mbar: implications for iron. Earth Planet. Sci. Lett. 153, 223 (1997)

    Article  ADS  Google Scholar 

  37. A. Hänström, P. Lazor, High pressure melting and equation of state of aluminium. J. Alloys Compd. 305, 209 (2000)

    Article  Google Scholar 

  38. J.W. Shaner, J.M. Brown, R.G. McQueen, Melting of metals above 100 GPa, High Pressure in Science and Technology (North-Holland, Amsterdam, 1984), p. 137

    Google Scholar 

  39. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087 (1953)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge, UK

    Robert John Nicholas Baldock

Authors
  1. Robert John Nicholas Baldock
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert John Nicholas Baldock .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Baldock, R.J.N. (2017). Nested Sampling for Materials. In: Classical Statistical Mechanics with Nested Sampling. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-66769-0_8

Download citation

Publish with us

Policies and ethics

Search

Navigation