Skip to main content
Log in

Packing Densities and Simulated Tempering for Hard Core Gibbs Point Processes

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Monotonicity and convergence properties of the intensity of hard core Gibbs point processes are investigated and compared to the closest packing density. For such processes simulated tempering is shown to be an efficient alternative to commonly used Markov chain Monte Carlo algorithms. Various spatial characteristics of the pure hard core process are studied based on samples obtained with the simulated tempering algorithm.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Alder, B. J. and Wainwright, T. E. (1957). Phase transition of a hard sphere system, Journal of Chemical Physics, 27, 1208-1209.

    Google Scholar 

  • Alder, B. J. and Wainwright, T. E. (1962). Phase transition in elastic disks, Phys. Rev., 127, 359-361.

    Google Scholar 

  • Allen, M. P. and Tildesley, D. J. (1987). Computer Simulation of Liquids, Oxford University Press, Oxford.

    Google Scholar 

  • Bourbaki, N. (1958). Élements de Mathématique, Fonctions d'une variable réelle (Théorie Élémentaire), Hermann, Paris.

    Google Scholar 

  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, Springer, New York.

    Google Scholar 

  • Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns, Academic Press, London.

    Google Scholar 

  • Ferguson, S. P. and Hales, T. C. (1998). A formulation of the Kepler conjecture (manuscript).

  • Fernández, J. F., Alonso, J. J. and Stankiewicz, E. (1995). One-stage continuous melting transition in two dimensions, Phys. Rev. Lett., 75, 3477-3480.

    Google Scholar 

  • Georgii, H.-O. (1994). Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction, Probab. Theory Related Fields, 99, 171-195.

    Google Scholar 

  • Georgii, H.-O. (1995). The equivalence of ensembles for classical systems of particles, J. Statist. Phys., 80, 1341-1378.

    Google Scholar 

  • Georgii, H.-O. (2000). Phase transition and percolation in Gibbsian particle models, Statistical Phyisics and Spatial Statistics (eds. Klaus R. Mecke and Dietrich Stoyan), Springer Lecure Notes in Physics, 554, 267-294, Springer, Berlin.

    Google Scholar 

  • Georgii, H.-O. and Häggström, O. (1996). Phase transition in continuum Potts models, Comm. Math. Phys., 181, 507-528.

    Google Scholar 

  • Geyer, C. J. (1992). Practical Markov chain Monte Carlo (with discussion), Statist. Sci., 7, 473-511.

    Google Scholar 

  • Geyer, C. J. (1999). Likelihood inference for spatial point processes, Stochastic Geometry: Likelihood and Computations (eds. O. E. Barndorff-Nielsen, W. S. Kendall and M. N. M. van Lieshout), 79-140, Chapman and Hall/CRC, London.

    Google Scholar 

  • Geyer, C. J. and Møller, J. (1994). Simulation procedures and likelihood inference for spatial point processes, Scand. J. Statist., 21, 359-373.

    Google Scholar 

  • Geyer, C. J. and Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to pedigree analysis, J. Amer. Statist. Assoc., 90, 909-920.

    Google Scholar 

  • Hales, T. C. (1997a). Sphere packings I, Discrete Comput. Geom., 17, 1-51.

    Google Scholar 

  • Hales, T. C. (1997b). Sphere packings II, Discrete Comput. Geom., 18, 135-149.

    Google Scholar 

  • Hales, T. C. (1998a). Sphere packings III (manuscript).

  • Hales, T. C. (1998b). Sphere packings IV (manuscript).

  • Hales, T. C. (1998c). The Kepler conjecture (manuscript).

  • Hales, T. C. (1998d). An overview of the Kepler conjecture (manuscript).

  • Hansen, J.-P. and McDonald, I. R. (1986). Theory of Simple Liquids, Academic Press, London.

    Google Scholar 

  • Hoover, W. G. and Ree, F. H. (1969). Melting transition and communal entropy for hard spheres, Journal of Chemical Physics, 49, 3609-3617.

    Google Scholar 

  • Marinari, E. and Parisi, G. (1992). Simulated tempering: A new Monte Carlo scheme, Europhys. Lett., 19, 451-458.

    Google Scholar 

  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines, Journal of Chemical Physics, 21, 1087-1092.

    Google Scholar 

  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability, Springer, London.

    Google Scholar 

  • Mitus, A. C., Weber, H. and Marx, D. (1997). Local structure analysis of the hard-disk fluid near melting, Phys. Rev. E, 55, 6855-6859.

    Google Scholar 

  • Møller, J. (1999). Markov chain Monte Carlo and spatial point processes, Stochastic Geometry: Likelihood and Computations (eds. O. E. Barndorff-Nielsen, W. S. Kendall and M. N. M. van Lieshout), 141-172, Chapman and Hall/CRC, London.

    Google Scholar 

  • Ripley, B. D. (1976). The second-order analysis of stationary point processes, J. Appl. Probab., 13, 255-266.

    Google Scholar 

  • Ruelle, D. (1988). Statistical Mechanics: Rigorous Results, Addison-Wesley, Redwood City, California.

    Google Scholar 

  • Salsburg, Z. W., Rudd, W. G. and Stillinger, F. H. (1967). Rigid disks at high density, II, Journal of Chemical Physics, 47, 4534-4539.

    Google Scholar 

  • Sokal, A. D. (1997). Monte Carlo methods in statistical mechanics: Foundations and new algorithms, Functional integration (Cargèse, 1996), NATO Adv. Sci. Inst. Ser. B. Phys., 361, 131-192.

    Google Scholar 

  • Stillinger, F. H., Salsburg, Z. W. and Kornegay, R. L. (1965). Rigid disks at high density, Journal of Chemical Physics, 43, 932-943.

    Google Scholar 

  • Stoyan, D. and Schlather, M. (2000). Random sequential adsorption: Relation-ship to dead leaves and characterization of variability, J. Statist. Phys., 100, 969-979.

    Google Scholar 

  • Stoyan, D. and Stoyan, H. (1994). Fractals, Random Shapes and Point Fields, Wiley, Chichester.

    Google Scholar 

  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd ed., Wiley, New York.

    Google Scholar 

  • Strandburg, K. J. (1988). Two-dimensional melting, Rev. Modern. Phys., 60, 161-207.

    Google Scholar 

  • Tóth, L. F. (1972). Lagerungen in der Ebene, auf der Kugel und im Raum, Springer, Heidelberg.

    Google Scholar 

  • Truskett, T. M., Torquato, S., Sastry, S., Debenetti, P. G. and Stillinger, F. H. (1998). A structural precursor to freezing in the hard-disk and hard-sphere systems, Phys. Rev. E, 58, 3083-3088.

    Google Scholar 

  • Weber, H., Marx, D. and Binder, K. (1995). Melting transition in two dimensions: A finite-size analysis of bond-orientational order in hard disks, Phys. Rev. B, 51, 14636-14651.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

About this article

Cite this article

Mase, S., Møller, J., Stoyan, D. et al. Packing Densities and Simulated Tempering for Hard Core Gibbs Point Processes. Annals of the Institute of Statistical Mathematics 53, 661–680 (2001). https://doi.org/10.1023/A:1014662415827

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014662415827

Navigation