Abstract
This chapter discuss several features and connections arising in a class of Ising-based models, namely the Glauber-Ising time dependent model, the Q2R cellular automata, the Schelling model for social segregation, the decision-choice model for social sciences and economics and finally the bootstrap percolation model for diseases dissemination. Although all these models share common elements, like discrete networks and boolean variables, and more important the existence of an Ising-like transition; there is also an important difference given by their particular evolution rules. As a result, the above implies the fact that macroscopic variables like energy and magnetization will show a dependence on the particular model chosen. To summarize, we will discuss and compare the time dynamics for these variables, exploring whether they are conserved, strictly decreasing (or increasing) or fluctuating around a macroscopic equilibrium regime.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Q by four, quatre, in french, 2 by two steps automata rule as explicitly written below, and R by reversible.
- 2.
The so called “perceive overall incentive agent function”, by Bouchaud [7].
- 3.
This two-step rule may be naturally re-written as a one-step rule with the aid of an auxiliary dynamical variable [9].
- 4.
The criteria (8) may be unified in a single criteria [17] (multiplying both sides of the two inequalities by \(S_k\)): \( \mathrm{an\, individual\,} S_k \, \mathrm{is\, unhappy\, at \, the\, node\,} k \mathrm{\, if\,, and \, only \, if }, \, \, S_k \sum _{i \in V_k}S_i \le |V_k| - 2 \theta _k ,\) which is a kind of energy density instead of the threshold criteria found in Glauber dynamics (4).
- 5.
In statistical physics, \(\beta \) is the inverse of the thermodynamical temperature, \(\beta \sim 1/T\).
- 6.
Q2R is a micro canonical description of the Ising transition, therefore we use the energy in absence of any temperature. In [10] it is shown the excellent agreement among the Q2R bifurcation diagram with the Ising thermodynamical transition.
- 7.
Notice that, as already said, the total magnetization is constant in the Schelling model . Therefore we cannot match the Schelling transitions observed here with the phase transition for the cases of the Glauber-Ising and the Q2R models .
References
W. Lenz, Physikalische Zeitschrift 21, 613–615 (1920)
E. Ising, Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 31, 253–258 (1925)
S.G. Brush, History of the Lenz-Ising Model Rev. Mod. Phys. 39, 883 (1967)
R.J. Glauber, Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294–307 (1963). doi:10.1063/1.1703954
W.A. Brock, S.N. Durlauf, A formal model of theory choice in science. Econ. Theory 14, 113–130 (1999)
W.A. Brock, S.N. Durlauf, Interactions-based models, ed. by J.J. Heckman, E. Leader. Handbook of Econometrics, 5, Chapter 54, pp. 3297–3380 (2001)
J.-P. Bouchaud, Crises and collective socio-economic phenomena: simple models and challenges. J. Stat. Phys. 151, 567–606 (2013)
G. Vichniac, Simulating physics with cellular automata. Physica D 10, 96–116 (1984)
Y. Pomeau, Invariant in cellular automata. J. Phys. A: Math. Gen. 17, L415–L418 (1984)
E. Goles, S. Rica, Irreversibility and spontaneous appearance of coherent behavior in reversible systems. Eur. Phys. J. D 62, 127–137 (2011)
L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)
C.N. Yang, The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85, 808–816 (1952)
F. Urbina, S. Rica, E. Tirapegui, Coarse-graining and master equation in a reversible and conservative system. Discontin. Nonlinearity Complex. 4(2), 199–208 (2015)
T.C. Schelling, Models of segregation. Am. Econ. Rev. 59, 488–493 (1969)
T.C. Schelling, Dynamic models of segregation. J. Math. Sociol. 1, 143–186 (1971)
T.C. Schelling, Micromotives and macrobehavior (W.W. Norton, New York, 2006)
N. Goles Domic, E. Goles, S. Rica, Dynamics and complexity of the Schelling segregation model. Phys. Rev. E 83, 056111 (2011)
J. Chalupa, P.L. Leath, G.R. Reich, Bootstrap percolation on a Bethe lattice. J. Phys. C Solid State Phys. 12, L31–L35 (1979)
D. Stauffer, S. Solomon, Ising, Schelling and self-organising segregation. Eur. Phys. J. B 57, 473 (2007)
A. Singh, D. Vainchtein, H. Weiss, Schelling’s segregation model: parameters, scaling, and aggregation 21, 341–366 (2009)
L. Gauvin, J. Vannimenus, J.-P. Nadal, Eur. Phys. J. B 70, 293 (2009)
V. Cortez, P. Medina, E. Goles, R. Zarama, S. Rica, Attractors, statistics and fluctuations of the dynamics of the Schelling’s model for social segregation. Eur. Phys. J. B 88, 25 (2015)
J. Balogh, B. Bollobás, R. Morris, Majority bootstrap percolation on the hypercube. Comb. Probab. Comput. 18, 17–51 (2009)
J. Balogh, B. Pittel, Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30, 257–286 (2007)
Acknowledgments
The authors acknowledge Eric Goles, Pablo Medina, Iván Rapaport, and Enrique Tirapegui for their partial contribution to some aspects of the present paper, as well as Aldo Marcareño, Andrea Repetto, Gonzalo Ruz and Romualdo Tabensky for fruitful discussions and valuable comments on different topics of this work. This work is supported by Núcleo Milenio Modelos de Crisis NS130017 CONICYT (Chile). FU. also thanks CONICYT-Chile under the Doctoral scholarship 21140319 and Fondequip AIC-34.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Mora, F., Urbina, F., Cortez, V., Rica, S. (2016). Around the Ising Model. In: Tlidi, M., Clerc, M. (eds) Nonlinear Dynamics: Materials, Theory and Experiments. Springer Proceedings in Physics, vol 173. Springer, Cham. https://doi.org/10.1007/978-3-319-24871-4_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-24871-4_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24869-1
Online ISBN: 978-3-319-24871-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)