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Zero temperature dynamics of Ising model on a densely connected small world network

  • Statistical and Nonlinear Physics
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Abstract.

The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected small world network is studied where long range bonds are added randomly with a finite probability p. We find that in contrast to the sparsely connected networks and random graph, there is no freezing and an initial random configuration of the spins reaches the equilibrium configuration within a very few Monte Carlo time steps in the thermodynamic limit for any p ≠0. The residual energy and the number of spins flipped at any time shows an exponential relaxation to equilibrium. The persistence probability is also studied and it shows a saturation within a few time steps, the saturation value being 0.5 in the thermodynamic limit. These results are explained in the light of the topological properties of the network which is highly clustered and has a novel small world behaviour.

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References

  • D.J. Watts, S.H. Strogatz, Nature, 393, 440 (1998)

  • M.E.J. Newman, D.J. Watts, Phys. Rev. E 60, 7332 (1999)

    Article  Google Scholar 

  • R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

    Article  Google Scholar 

  • A. Barrat, M. Weigt, Eurphys. J. B 13, 547 (2000)

    Google Scholar 

  • M. Gitterman, J. Phys.A 33, 8373 (2000)

    Article  Google Scholar 

  • B.J. Kim et al., Phys. Rev. E 64, 056135 (2001)

    Article  Google Scholar 

  • C.P. Herrero, Phys. Rev. E 65, 066110 (2002)

    Article  Google Scholar 

  • H. Hong, B.J. Kim, M.Y Choi, Phys. Rev. E 66, 011107 (2002)

    Article  Google Scholar 

  • J. Viana Lopes, Yu. G. Pogorelov, J.M.B. Lopes dos Santos, R. Toral, Phys. Rev. E 70 026112 (2004)

    Google Scholar 

  • S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 66, 016104 (2002)

    Article  Google Scholar 

  • M. Leone, A. Vazquez, A. Vespignani, R. Zecchina, Eur. Phys. J. B 28, 191 (2002)

    Article  Google Scholar 

  • A. Aleksiejuk, J.A. Holyst, D. Stauffer, Physica A 310, 260 (2002); G. Bianconi, Phys. Lett. A 303, 166 (2002); J. Viana Lopes, Yu. G. Pogorelov, J.M.B. Lopes dos Santos, R. Toral, Phys. Rev. E 70, 026112 (2004)

    Article  Google Scholar 

  • P. Svenson, Phys. Rev. E 64, 036122 (2001)

    Article  Google Scholar 

  • O. Haggstrom, Physica A 310, 275 (2002)

    Article  Google Scholar 

  • D. Boyer, O. Miramontes, Phys. Rev. E 67, R035102 (2003)

  • P. Svenson, D.A. Johnson, Phys. Rev. E 65, 036105 (2002)

    Article  Google Scholar 

  • J.Y. Zhu, H. Zhu, Phys. Rev. E 67, 026125 (2003)

    Article  Google Scholar 

  • D. Jeong, M.Y. Choi, H. Park, Phys. Rev. E 71, 036103 (2005)

    Article  Google Scholar 

  • C. Castellano, V. Loreto, A. Barrat, F. Cecconi, D. Parisi, Phys. Rev. E 71 066107 (2005)

    Google Scholar 

  • P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977)

    Article  Google Scholar 

  • J.D. Gunton, M. San Miguel, P.S. Sahni, Phase Transitions and critical phenomena, Vol. 8, edited by C. Domb, J.L. Lebowitz (Academic, NY, 1983)

  • A.J. Bray, Adv. Phys. 43, 357 (1994) and the references therein

    Google Scholar 

  • For a review, see S.N. Majumdar, Curr. Sci. 77, 370 (1999)

    Google Scholar 

  • B. Derrida, A.J. Bray, C. Godreche, J. Phys. A 27, L357 (1994)

  • D. Stauffer, J. Phys. A 27, 5029 (1994)

    Article  Google Scholar 

  • P.L. Krapivsky, E. Ben-Naim, S. Redner, Phys. Rev. E 50, 2474 (1994)

    Article  Google Scholar 

  • V. Spirin, P.L. Krapivsky, S. Redner, Phys. Rev. E 63, 036118 (2001)

    Article  Google Scholar 

  • B. Derrida, Phys. Rev. E 55, 3705 (1997)

    Article  Google Scholar 

  • P.L. Krapivsky, E. Ben-Naim, Phys. Rev. E 56, 3788 (1997)

    Article  Google Scholar 

  • N.R. da Silva, J.M. Silva, Phys. Lett. A 135, 373 (1989)

    Article  Google Scholar 

  • Tao Zhou, Bing-Hong Wang, P.M. Hui, K.P. Chan, e-print: arXiv: cond-mat/0405258; in the first version of this paper the authors used the term “super small world"

  • P. Sen, K. Banerjee, T. Biswas, Phys. Rev. E 66, 037102 (2002)

    Article  Google Scholar 

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Correspondence to Parongama Sen.

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Das, P., Sen, P. Zero temperature dynamics of Ising model on a densely connected small world network. Eur. Phys. J. B 47, 391–396 (2005). https://doi.org/10.1140/epjb/e2005-00337-6

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  • DOI: https://doi.org/10.1140/epjb/e2005-00337-6

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