IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v390y2011i13p2443-2451.html
   My bibliography  Save this article

Exact and approximate solutions for the dilute Ising model

Author

Listed:
  • Serva, Maurizio

Abstract

The ground state energy and entropy of the dilute mean field Ising model is computed exactly by a single order parameter as a function of the dilution coefficient. An analogous exact solution is obtained in the presence of a magnetic field with random locations. Results lead to a complete understanding of the geography of the associated random graph. In particular, we give the size of the giant component (continent) and the number of isolated clusters of connected spins of all given size (islands). We also compute the average number of bonds per spin in the continent and in the islands. Then, we tackle the problem of solving the dilute Ising model at strictly positive temperature. In order to obtain the free energy as a function of the dilution coefficient and the temperature, it is necessary to introduce a second order parameter. We are able to find out the exact solution in the paramagnetic region and exactly determine the phase transition line. In the ferromagnetic region we provide a solution in terms of an expansion with respect to the second parameter which can be made as accurate as necessary. All results are reached in the replica frame by a strategy which is not based on multi-overlaps.

Suggested Citation

  • Serva, Maurizio, 2011. "Exact and approximate solutions for the dilute Ising model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(13), pages 2443-2451.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:13:p:2443-2451
    DOI: 10.1016/j.physa.2011.02.050
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437111002007
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2011.02.050?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Barré, Julien & Ciani, Antonia & Fanelli, Duccio & Bagnoli, Franco & Ruffo, Stefano, 2009. "Finite size effects for the Ising model on random graphs with varying dilution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(17), pages 3413-3425.
    2. M. O. Hase & J. R.L. de Almeida & S. R. Salinas, 2005. "Replica-symmetric solutions of a dilute Ising ferromagnet in a random field," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 47(2), pages 245-249, September.
    3. Serva, Maurizio, 2010. "Magnetization densities as replica parameters: The dilute ferromagnet," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2700-2707.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. de Almeida, M.L. & Albuquerque, E.L. & Fulco, U.L. & Serva, M., 2014. "A percolation system with extremely long range connections and node dilution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 416(C), pages 273-278.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Serva, Maurizio, 2010. "Magnetization densities as replica parameters: The dilute ferromagnet," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2700-2707.
    2. de Almeida, M.L. & Albuquerque, E.L. & Fulco, U.L. & Serva, M., 2014. "A percolation system with extremely long range connections and node dilution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 416(C), pages 273-278.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:390:y:2011:i:13:p:2443-2451. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.