Pitfalls driven by the sole use of local updates in dynamical systems

¹ Centre de Recherche en Épistémologie Appliquée (CREA), École Polytechnique and CNRS - Bld Victor, 32, 75015 Paris, France, EU
² GRIFE - EACH - Universidade de São Paulo - R. Arlindo Bétio, 1000, 03828-000, São Paulo, Brazil

^a serge.galam@polytechnique.edu

Received: 22 March 2011
Accepted: 5 July 2011

Abstract

The recent claim that the exit probability (EP) of a slightly modified version of the Sznadj model is a continuous function of the initial magnetization is questioned. This result has been obtained analytically and confirmed by Monte Carlo simulations, simultaneously and independently by two different groups (EPL, 82 (2008) 18006; 18007). It stands at odds with an earlier result which yielded a step function for the EP (Europhys. Lett., 70 (2005) 705). The dispute is investigated by proving that the continuous shape of the EP is a direct outcome of a mean-field treatment for the analytical result. As such, it is most likely to be caused by finite-size effects in the simulations. The improbable alternative would be a signature of the irrelevance of fluctuations in this system. Indeed, evidence is provided in support of the stepwise shape as going beyond the mean-field level. These findings yield new insight in the physics of one-dimensional systems with respect to the validity of a true equilibrium state when using solely local update rules. The suitability and the significance to perform numerical simulations in those cases is discussed. To conclude, a great deal of caution is required when applying updates rules to describe any system especially social systems.