Abstract
Contrary to the actual nonlinear Glauber model, the linear Glauber model (LGM) is exactly solvable, although the detailed balance condition is not generally satisfied. This motivates us to address the issue of writing the transition rate (\(w_j\)) in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least. The advantage of this work is that, by studying the LGM analytically, we will be able to anticipate how the kinetic properties of an arbitrary Ising system depend on the temperature and the coupling constants. The analytical expressions for the optimal values of the parameters involved in the linear \(w_j\) are obtained using a simple Moore–Penrose pseudoinverse matrix. This approach is quite general, in principle applicable to any system and can reproduce the exact results for one dimensional Ising system. In the continuum limit, we get a linear time-dependent Ginzburg–Landau equation from the Glauber’s microscopic model of non-conservative dynamics. We analyze the critical and dynamic properties of the model, and show that most of the important results obtained in different studies can be reproduced by our new mathematical approach. We will also show in this paper that the effect of magnetic field can easily be studied within our approach; in particular, we show that the inverse of relaxation time changes quadratically with (weak) magnetic field and that the fluctuation-dissipation theorem is valid for our model.
Similar content being viewed by others
References
Glauber, R.J.: Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294 (1963)
Michael, T., Trimper, S., Schulz, M.: Glauber model in a quantum representation. Phys. Rev. E 73, 062101 (2006)
Grynberg, M.D., Stinchcombe, R.B.: Nonuniversal disordered Glauber dynamics. Phys. Rev. E 87, 062102 (2013)
Uchida, M., Shirayama, S.: Effect of initial conditions on Glauber dynamics in complex networks. Phys. Rev. E 75, 046105 (2007)
Kong, X.-M., Yang, Z.R.: Critical dynamics of the kinetic Glauber–Ising model on hierarchical lattices. Phys. Rev. E 69, 016101 (2004)
Godreche, C., Luck, J.M.: Response of non-equilibrium systems at criticality: exact results for the Glauber–Ising chain. J. Phys. A Math. Gen. 33, 1151 (2000)
Pini, M.G., Rettori, A.: Effect of antiferromagnetic exchange interactions on the Glauber dynamics of one-dimensional Ising models. Phys. Rev. B 76, 064407 (2007)
Gleeson, J.P.: High-accuracy approximation of binary-state dynamics on networks. Phys. Rev. Lett. 107, 068701 (2011)
Goncalves, L.L., de Haro, M.L., Tagoena-Mart-nez, J., Stinchcombe, R.B.: Nagel scaling, elaxation, and universality in the kinetic Ising model on an alternating isotopic chain. Phys. Rev. Lett. 84, 1507 (2000)
Stanley, H.E., Stauffer, D., Kertesz, J., Herrmann, H.J.: Dynamics of spreading phenomena in two-dimensional Ising models. Phys. Rev. Lett. 59, 2326 (1987)
Fisher, D.S., Le Doussal, P., Monthus, C.: Random walks, reaction-diffusion, and nonequilibrium dynamics of spin chains in one-dimensional random environments. Phys. Rev. Lett. 80, 3539 (1998)
K-t, Leung, Neda, Z.: Response in kinetic Ising model to oscillating magnetic fields. Phys. Lett. A 246, 505 (1998)
Vojta, T.: Chaotic behavior and damage spreading in the Glauber Ising model: a master equation approach. Phys. Rev. E 55, 5157 (1997)
Puri, S.: In: Puri, S., Wadhawan, V. (ed) Kinetics of Phase Transitions. CRC Press, Boca Raton (2009)
Krapivsky, P.L., Redner, S., Ben-Naim, E.: A Kinetic View of Statistical Physics, chap. 2, 8 and 9. Cambridge University Press, Cambridge (2010)
Schadschneider, A., Chowdhury, D., Nishinari, K.: Stochastic Transport in Complex Systems: From Molecules to Vehicles, chap. 2. Elsevier, Amsterdam (2011)
Scheucher, M., Spohn, H.: A soluble kinetic model for spinodal decomposition. J. Stat. Phys. 53, 279 (1988)
de Oliveira, M.J.: Linear Glauber model. Phys. Rev. E 67, 066101 (2003)
Hase, M.O., Salinas, S.R., Tome, T., de Oliveira, M.J.: Fluctuation-dissipation theorem and the linear Glauber model. Phys. Rev. E 73, 056117 (2006)
Campbell, S.L., Meyer, C.D.: Generalized Inverse of Linear Transformations. SIAM, Philadelphia (2008)
Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferromagnet part I. Phys. Rev. 60, 252 (1941)
Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117 (1944)
Salman, Z., Adler, J.: High and low temperature series estimates for the critical temperature of the 3D Ising model. Int. J. Mod. Phys. C 09, 195 (1998)
Livet, F.: The cluster updating Monte Carlo algorithm applied to the 3D Ising problem. Europhys. Lett. 16, 139 (1991)
Talapov, A.L., Blote, H.W.J.: The magnetization of the 3D Ising model. J. Phys. A Math. Gen. 29, 5727 (1996)
Bray, A.J.: Theory of phase ordering kinetics. Adv. Phys. 43, 357 (1994)
Sahoo, S., Chatterjee, S.: Optimal linear Kawasaki model. arXiv:1404.6027 (2014)
Acknowledgments
SS thanks Prof. S. Ramasesha for his financial support through his various projects from IFCPAR and DST, India, and SKG thanks CSIR, India for financial support.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sahoo, S., Ganguly, S.K. Optimal Linear Glauber Model. J Stat Phys 159, 336–357 (2015). https://doi.org/10.1007/s10955-015-1188-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-015-1188-y