Journal of Statistical Physics

, Volume 142, Issue 5, pp 984–999

Loss of Ergodicity in the Transition from Annealed to Quenched Disorder in a Finite Kinetic Ising Model

  • Fernando Pigeard de Almeida Prado
  • Gunter M. Schütz
Article

Abstract

We consider a kinetic Ising model which represents a generic agent-based model for various types of socio-economic systems. We study the case of a finite (and not necessarily large) number of agents N as well as the asymptotic case when the number of agents tends to infinity. The main ingredient are individual decision thresholds which are either fixed over time (corresponding to quenched disorder in the Ising model, leading to nonlinear deterministic dynamics which are generically non-ergodic) or which may change randomly over time (corresponding to annealed disorder, leading to ergodic dynamics). We address the question how increasing the strength of annealed disorder relative to quenched disorder drives the system from non-ergodic behavior to ergodicity. Mathematically rigorous analysis provides an explicit and detailed picture for arbitrary realizations of the quenched initial thresholds, revealing an intriguing “jumpy” transition from non-ergodicity with many absorbing sets to ergodicity. For large N we find a critical strength of annealed randomness, above which the system becomes asymptotically ergodic. Our theoretical results suggests how to drive a system from an undesired socio-economic equilibrium (e.g. high level of corruption) to a desirable one (low level of corruption).

Keywords

Ising model Agent-based model Ergodicity Annealed and quenched dynamics Phase transition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Durlauf, S.: Nonergodic economic growth. Rev. Econ. Stud. 60, 349–366 (1993) MATHCrossRefGoogle Scholar
  2. 2.
    Brock, W.A., Durlauf, S.: Discrete choice with social interactions. Rev. Econ. Stud. 68, 235–260 (2001) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brock, W.A., Durlauf, S.: Interactions based models. In: Heckman, J., Learmer, E. (eds.) Handbook of Econometrics: vol. V. North Holland, Amsterdam (2001) Google Scholar
  4. 4.
    Föllmer, H.: Random economies with many interacting agents. J. Math. Econ. 1, 51–62 (1974) MATHCrossRefGoogle Scholar
  5. 5.
    Nadal, J.-P., Phan, D., Gordon, M.B.: Multiple equilibria in a monopoly market with heterogeneous agents and externalities. Quant. Finance 5, 557–568 (2005) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Kaizoji, T.: Speculative bubbles and crashes in stock markets: An interacting-agent model of speculative activity. Physica A 287, 493–506 (2000) ADSCrossRefGoogle Scholar
  7. 7.
    Levy, M.: Social phase transitions. J. Econ. Behav. Organ. 57, 71–87 (2005) CrossRefGoogle Scholar
  8. 8.
    Lux, T.: Herd behavior, bubbles and crashes. Econ. J. 105, 881–896 (1995) CrossRefGoogle Scholar
  9. 9.
    Glaeser, E., Scheinkman, J.A.: Measuring social interactions. In: Durlauf, S., Young, P. (eds.) Social Dynamics. MIT Press, Cambridge (2001) Google Scholar
  10. 10.
    Glaeser, E., Sacerdote, B., Scheinkman, J.A.: Crime and social interactions. Q. J. Econ. CXI, 507–548 (1996) CrossRefGoogle Scholar
  11. 11.
    Schneider, T., Pytte, E.: Random-field instability of the ferromagnetic state. Phys. Rev. B 15, 1519–1522 (1977) ADSCrossRefGoogle Scholar
  12. 12.
    Weisbuch, G., Salomon, S., Stauffer, D.: Social percolators and self organized criticality. In: Kirman, A., Zimmermann, J.-B. (eds.) Economics with Heterogeneous Agents. Springer, Berlin (2001) Google Scholar
  13. 13.
    Gordon, M.B., Nadal, J.-P., Phan, D., Vannimenus, J.: Seller’s dilemma due to social interactions between customers. Phys. A Stat. Mech. Appl. 356, 628–640 (2005) CrossRefGoogle Scholar
  14. 14.
    Yin, C.-C.: Equilibria of collective action in different distributions of protest thresholds. Public Choice 97, 535–567 (1998) CrossRefGoogle Scholar
  15. 15.
    Galam, S.: Rational group decision making. Physica (Amsterdam) 238A, 66 (1997) ADSGoogle Scholar
  16. 16.
    Michard, Q., Bouchaud, J.-P.: Theory of collective opinion shifts: from smooth trends to abrupt swings. Eur. Phys. J. B 47, 151 (2005) ADSCrossRefGoogle Scholar
  17. 17.
    Krzakala, F., Ricci-Tersenghi, F., Zdeborová, L.: Elusive spin-glass phase in the random field Ising model. Phys. Rev. Lett. 104, 207208 (2010) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Fernando Pigeard de Almeida Prado
    • 1
  • Gunter M. Schütz
    • 2
    • 3
  1. 1.Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeiro PretoUniversidade de São PauloSão PauloBrazil
  2. 2.Institut für FestkörperforschungForschungszentrum JülichJülichGermany
  3. 3.Interdisziplinäres Zentrum für Komplexe SystemeUniversität BonnBonnGermany

Personalised recommendations