TY - JOUR

T1 - Long-time predictability in disordered spin systems following a deep quench

AU - Ye, J.

AU - Gheissari, R.

AU - Machta, J.

AU - Newman, C. M.

AU - Stein, D. L.

N1 - Funding Information:
J.M. was supported in part by NSF Grants No. DMR-1208046 and No. DMR-1507506. R.G., C.M.N., and D.L.S. were supported in part by NSF Grant No. DMS-1207678. D.L.S. thanks the Guggenheim Foundation for a fellowship that partially supported this research. Simulations were performed on the Courant Institute of Mathematical Sciences computer cluster. J.M. and D.L.S. thank the Aspen Center for Physics (under NSF Grant 1066293), where some of this work was done.

PY - 2017/4/4

Y1 - 2017/4/4

N2 - We study the problem of predictability, or "nature vs nurture," in several disordered Ising spin systems evolving at zero temperature from a random initial state: How much does the final state depend on the information contained in the initial state, and how much depends on the detailed history of the system? Our numerical studies of the "dynamical order parameter" in Edwards-Anderson Ising spin glasses and random ferromagnets indicate that the influence of the initial state decays as dimension increases. Similarly, this same order parameter for the Sherrington-Kirkpatrick infinite-range spin glass indicates that this information decays as the number of spins increases. Based on these results, we conjecture that the influence of the initial state on the final state decays to zero in finite-dimensional random-bond spin systems as dimension goes to infinity, regardless of the presence of frustration. We also study the rate at which spins "freeze out" to a final state as a function of dimensionality and number of spins; here the results indicate that the number of "active" spins at long times increases with dimension (for short-range systems) or number of spins (for infinite-range systems). We provide theoretical arguments to support these conjectures, and also study analytically several mean-field models: the random energy model, the uniform Curie-Weiss ferromagnet, and the disordered Curie-Weiss ferromagnet. We find that for these models, the information contained in the initial state does not decay in the thermodynamic limit - in fact, it fully determines the final state. Unlike in short-range models, the presence of frustration in mean-field models dramatically alters the dynamical behavior with respect to the issue of predictability.

AB - We study the problem of predictability, or "nature vs nurture," in several disordered Ising spin systems evolving at zero temperature from a random initial state: How much does the final state depend on the information contained in the initial state, and how much depends on the detailed history of the system? Our numerical studies of the "dynamical order parameter" in Edwards-Anderson Ising spin glasses and random ferromagnets indicate that the influence of the initial state decays as dimension increases. Similarly, this same order parameter for the Sherrington-Kirkpatrick infinite-range spin glass indicates that this information decays as the number of spins increases. Based on these results, we conjecture that the influence of the initial state on the final state decays to zero in finite-dimensional random-bond spin systems as dimension goes to infinity, regardless of the presence of frustration. We also study the rate at which spins "freeze out" to a final state as a function of dimensionality and number of spins; here the results indicate that the number of "active" spins at long times increases with dimension (for short-range systems) or number of spins (for infinite-range systems). We provide theoretical arguments to support these conjectures, and also study analytically several mean-field models: the random energy model, the uniform Curie-Weiss ferromagnet, and the disordered Curie-Weiss ferromagnet. We find that for these models, the information contained in the initial state does not decay in the thermodynamic limit - in fact, it fully determines the final state. Unlike in short-range models, the presence of frustration in mean-field models dramatically alters the dynamical behavior with respect to the issue of predictability.

UR - http://www.scopus.com/inward/record.url?scp=85017187583&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017187583&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.95.042101

DO - 10.1103/PhysRevE.95.042101

M3 - Article

C2 - 28505767

AN - SCOPUS:85017187583

VL - 95

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 4

M1 - 042101

ER -