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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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Volume 19, Issue 3-4


Classical Magnetism and an Integral Formula Involving Modified Bessel Functions

Orion Ciftja
Published Online: 2018-03-31 | DOI: https://doi.org/10.1515/ijnsns-2017-0193


We study an integral expression that is encountered in some classical spin models of magnetism. The idea is to calculate the key integral that represents the building block for the expression of the partition function of these models. The general calculation allows one to have a better look at the internal structure of the quantity of interest which, in turn, may lead to potentially new useful insights. We find out that application of two different approaches to solve the problem in a general-case scenario leads to an interesting integral formula involving modified Bessel functions of the first kind which appears to be new. We performed Monte Carlo simulations to verify the correctness of the integral formula obtained. Additional numerical integration tests lead to the same result as well. The approach under consideration, when generalized, leads to a linear integral equation that might be of interest to numerical studies of classical spin models of magnetism that rely on the well-established transfer-matrix formalism.

Keywords: General Physics; Theory and Modeling; Function Theory and Analysis

PACS: numbers; 01.55.+b; 73.43.Cd; 02.30.-f


  • [1]

    H. E. Stanley, Phys. Rev. 158 (1967), 537.Google Scholar

  • [2]

    Nanomagnetism, (NATO Asi Series E: Applied Sciences, Vol. 247, A. Hernando, Kluwer Academic, Dordrecht, 1993.Google Scholar

  • [3]

    O. Kahn, Magnetism Molecular, Wiley-VCH, York New, 1993.Google Scholar

  • [4]

    K. L. Taft, C. D. Delfs, G. C. Papaefthymiou, S. Foner, D. Gatteschi and S. J. Lippard , J. Am Chem. Soc. 116 (1994), 823.Google Scholar

  • [5]

    D. Gatteschi, A. Caneschi, L. Pardi and R. Sessoli, Science 265 (1994), 1054.

  • [6]

    M. E. Fisher, Am. J. Phys. 32 (1964), 343.

  • [7]

    G. B. Arfken and H. J. Weber, Mathematical Methods For Physicists, Fifth Edition, Harcourt/Academic Press, San Diego, 2001.Google Scholar

  • [8]

    A. L. Barra, A. Caneschi, A. Cornia, F. Fabrizi de Biani, D. Gatteschi, C. Sangregorio, R. Sessoli and R. Sorace, J. Am. Chem. Soc. 121 (1999), 5302.

  • [9]

    A. Bino, D. C. Johnston, D. P. Goshorn, T. R. Halbert and E. I. Stiefel, Science 241, 1479 (1988).

  • [10]

    O. Ciftja, Physica A 286 (2000), 541.

  • [11]

    M. A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th Edition, Dover Publications Inc, New York, 1972.Google Scholar

  • [12]

    L. C. Andrews, Special Functions of Mathematics for Engineers, Second Edition, Oxford University Press, Oxford, 1998.

  • [13]

    R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume 1, Chapter 3, Wiley-Interscience, New York, 1953.

  • [14]

    G. S. Joyce, Phys. Rev. 155 (1967), 478.

  • [15]

    M. Blume, P. Heller and N. A. Lurie, Phys. Rev. B 11 (1975), 4483.Google Scholar

  • [16]

    O. Ciftja, M. Luban, M. Auslender and J. H. Luscombe, Phys. Rev. B 60 (1999), 10122.Google Scholar

  • [17]

    P. J. Cregg, J. L. Garcia-Palacios and P. Svedlindh, Phys J. Math A:. Theor. 41 (2008), 435202.Google Scholar

  • [18]

    P. A. Martin, J. Phys Math A:. Theor. 41 (2008), 015207.Google Scholar

  • [19]

    O. Ciftja, Physica B 407 (2012), 2803.

  • [20]

    A. Al-Sharif, M. Hajja and P. T. Krasopoulos, Math Results. 55 (2009), 231.

  • [21]

    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Fourth Edition, Academic Press, New York,1965.Google Scholar

  • [22]

    P. Prudnikov, A. Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Volume 2, Special Functions, Third Printing, Gordon & Breach Science Publishers, New York, 1992.Google Scholar

  • [23]

    See http://www.wolfram.com/mathematica/

  • [24]

    M. L. Glasser and E. Montaldi, J. Math Anal. and Appl. 183 (1994), 577.Google Scholar

  • [25]

    F. W. J. Olver and L. C. Maximon, Chapter 10: Bessel Functions in Handbook NIST of Functions Mathematical, pages 215-286, Cambridge University Press, New York, 2010. http://dlmf.nist.gov/10.

About the article

Received: 2017-08-31

Accepted: 2015-03-15

Published Online: 2018-03-31

Published in Print: 2018-06-26

This research was supported in part by National Science Foundation (NSF) Grants No. DMR-1410350 and DMR-1705084.

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 409–414, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0193.

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