Dynamical Replica Theory for Dilute Spin Systems：稀自旋系统的动力学复制理论.ppt

Dynamical Replica Theory for Dilute Spin Systems Jon Hatchett RIKEN BSI In collaboration with I. Pérez Castillo, A. C. C. Coolen N. S. Skantzos Motivation Order parameter flow in physical systems Non-equilibrium phenomena Temperature cycling Slow relaxations Non-detailed balance systems Biological models Complex networks Analysis of algorithms Model Details Locally tree-like structure Stochastic dynamics according to local field Parallel Sequential (Glauber, Metropolis) Langevin Ising or soft spins Evolution of Observables Observables of interest e.g. Want evolution on finite times In general, e.g. depends on Evolution of Probability State vector Linear equation for E.g. Master equation General set of observables with distribution Obtain Kramers-Moyal expansion for Assumptions of DRT Retain Liouville term in KM expansion Exact if observables are well-chosen Deterministic evolution of order parameters Self-averaging observables at all times Expect this to be exact (for certain observables) Equipartitioning in the subshell average In general, this is an approximation Probability Manifolds Manifold of distributions on dim S = 2^N - 1 Identify observables with coordinates M is an exponential family dim M = #observables Geometry of S and M Two important coordinate systems for M Exponential family coordinates Mixture family coordinates Connected via Legendre transformation Geodesic for exponential family Geodesic for mixture family What do Assumptions do? S M minimizes What do Assumptions do? If then theory is exact E.g. at equilibrium For some sets of initial conditions For some simpler systems, observable sets This condition is not necessary for exactness Approximation is “best possible” Since is unknown, the KL divergence is not accessible Glauber Dynamics Master equation Hamiltonian Evolution of observables Glauber D