Abstract
We consider a dynamic mean-field ferromagnetic model in the low-temperature regime in the neighborhood of the zero magnetization state. We study the random time it takes for the system to make a decision, i.e., to exit the neighborhood of the unstable equilibrium and approach one of the two stable equilibrium points. We prove a limit theorem for the distribution of this random time in the thermodynamic limit.
Article PDF
Similar content being viewed by others
References
Arnold, V.I.: Ordinary differential equations (Translated from the Russian by Roger Cooke, Second printing of the 1992 edition). Universitext. Springer, Berlin (2006)
Bakhtin Y.: Exit asymptotics for small diffusion about an unstable equilibrium. Stoch. Process. Appl. 118(5), 839–851 (2008)
Bakhtin, Y.: Noisy heteroclinic networks. Probab. Theory Relat. Fields, in print (2010)
Bakhtin, Y.: Small noise limit for diffusions near heteroclinic networks. Dyn. Syst., in print (2010)
Buterakos, L.A.: The exit time distribution for small random perturbations of dynamical systems with a repulsive type stationary point. PhD thesis, Virginia Polytechnic Institute and State University (2003)
Caderoni P., Pellegrinotti A., Presutti E., Vares M.E.: Transient bimodality in interacting particle systems. J. Stat. Phys. 55(3–4), 523–577 (1989)
Davydov Y., Rotar V.: On asymptotic proximity of distributions. J. Theor. Probab. 22(1), 82–98 (2009)
Day M.V.: On the exit law from saddle points. Stoch. Process. Appl. 60(2), 287–311 (1995)
De Masi A., Orlandi E., Presutti E., Triolo L.: Glauber evolution with Kac potentials. III. Spinodal decomposition. Nonlinearity 9(1), 53–114 (1996)
De Masi A., Pellegrinotti A., Presutti E., Vares M.E.: Spatial patterns when phases separate in an interacting particle system. Ann. Probab. 22(1), 334–371 (1994)
Eizenberg A.: The exit distributions for small random perturbations of dynamical systems with a repulsive type stationary point. Stochastics 12(3–4), 251–275 (1984)
Ellis, R.S.: Entropy, large deviations, and statistical mechanics. In: Classics in Mathematics. Springer, Berlin (Reprint of the 1985 original, 2006)
Ethier, S.N., Kurtz, Thomas G.: Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)
Kifer Y.: The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Israel J. Math. 40(1), 74–96 (1981)
Liggett, T.M.: Interacting particle systems. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276. Springer, New York (1985)
Presutti, E.: Scaling limits in statistical mechanics and microstructures in continuum mechanics. In: Theoretical and Mathematical Physics. Springer, Berlin (2009)
Rabinovich, M.I., Huerta, R., Varona, P., Afraimovich, V.S.: Transient cognitive dynamics, metastability, and decision making. PLoS Comput. Biol. 4(5), e1000072, 9 (2008)
Ratcliff R., McKoon G.: The diffusion decision model: theory and data for two-choice decision tasks (PMID: 18085991). Neural Comput. 20(4), 873–922 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jean Bellissard.
Rights and permissions
About this article
Cite this article
Bakhtin, Y. Decision Making Times in Mean-Field Dynamic Ising Model. Ann. Henri Poincaré 13, 1291–1303 (2012). https://doi.org/10.1007/s00023-011-0148-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-011-0148-6