Far Eastern Mathematical Journal

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Solution of one-dimensional lattice gas models

Kharchenko Yu.N.

2019, issue 2, Ñ. 245–255

In this paper it is shown that the thermodynamic limit of the partition function of the statistical models under consideration on a one-dimensional lattice with an arbitrary finite number of interacting neighbors is expressed in terms of the principal eigenvalue of a matrix of finite size. The high sparseness of these matrices for any number of interactions makes it possible to perform an effective numerical analysis of the macro characteristics of these models.

Ising model, transfer matrix, statistical sum, free energy, singular curves, phase transitions

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[1] T.D. Lee, C.N. Yang, “Statistical theory of equations of state and phase transitions. I. Theory of condensation”, Phys. Rev., 87, (1952), 404–410.
[2] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York and London, 1982.
[3] E.H. Lieb, D.C. Mfttis, Mathematical Physics in One Dimension, Academic Press, New York and London, 1966.
[4] P. Cambardella, A. Dallmeyer, K. Maiti, M. Malagoli, W. Eberhard, K. Karn, and C. Carbone, “Ferromagnetism in one-dimensional monatomic metal chains”, Nature, 416:6878, (2002), 301–304.
[5] P.D. Andriushchenko and K.V. Nefedev, “Magnetic phase transition in the lattice Ising model”, Advanced Material Research, 718, (2013), 166–171.
[6] R. Huang, “Ising spins on randomly multy-branched Hustmi square lattice: Thermodynamics and phase transition in cross-dimensional range”, Physics Letters, A 380, (2016), 3333–3339.
[7] S. Shabnam, S. DasGupta, S.K. Roy, “Existence of a line of critical points in a twodimensional Lebwohl Lasher model”, Physics Letters, A 380, (2016), 667–671.
[8] F. Wang and D.P. Landau, “Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States”, Physics Letters, 86:2050, (2001).
[9] A.A. Dmitriev, V.V. Katrakhov, Iu.N. Kharchenko, Kornevye transfer-matritsy v modeliakh Izinga, Nauka, M., 2004.
[10] R. Khorn, Ch. Dzhonson, Matrichnyi analiz, Mir, M., 1989.
[11] V.V. Katrakhov, Yu.N. Kharchenko, “Two-dimensional four-line models of the Ising model type”, Theoretical and Mathematical Physics, 149:2, (2006), 1545–1558.
[12] U.B. Arnalds, J. Cyico, Y. Stopfel, V. Kapaklis, O. Barenbold, M.A. Verschuren, U. Volff, V. Neu, A. Bergman and B. Hjorvarsson, “A new look on the two-dimensional Ising model: thermal artificial spins”, New Journal of Physics, 18:2, (2016), 023008.
[13] Y. Fan, “One-dimensional Ising model with k-spin interactions”, European Journal of Physics, 32:6, (2011), 1643.
[14] Y. Shevchenko, A. Makarov, K. Nefedev, “Effect of long and shot-range interactions on the thermodynamics of dipolar spin ice”, Physics Letters, A 381, (2017), 428–434.

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