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

DOI: 10.1007/s10955-011-0143-9

Cite this article as:
Pigeard de Almeida Prado, F. & Schütz, G.M. J Stat Phys (2011) 142: 984. doi:10.1007/s10955-011-0143-9


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).


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

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

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