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Dynamical Replica Theory for Dilute Spin Systems

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

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Dynamical Replica Theory for Dilute Spin Systems

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  1. Dynamical Replica Theory for Dilute Spin Systems Jon Hatchett RIKEN BSI In collaboration with I. Pérez Castillo, A. C. C. Coolen & N. S. Skantzos

  2. 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

  3. Model Details Locally tree-like structure Stochastic dynamics according to local field Parallel Sequential (Glauber, Metropolis) Langevin Ising or soft spins

  4. Evolution of Observables Observables of interest e.g. Want evolution on finite times In general, e.g. depends on

  5. Evolution of Probability State vector Linear equation for E.g. Master equation General set of observables with distribution Obtain Kramers-Moyal expansion for

  6. 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

  7. Probability Manifolds Manifold of distributions on dim S = 2^N - 1 Identify observables with coordinates M is an exponential family dim M = #observables

  8. 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

  9. What do Assumptions do? m - projection S M minimizes e-flat submanifold

  10. 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

  11. Glauber Dynamics Master equation Hamiltonian Evolution of observables

  12. Glauber Dynamics Simple choice of observables (m,e) Find simple ODEs: is the distribution of local fields in a system with magnetisation and energy

  13. Algorithm Given find conjugate fields Generally a challenging inverse problem Work on a given disorder instance Requires belief propagation to converge Given find field distribution Standard equilibrium problem Calculate and Solve odes via standard methods

  14. Comparison with Simulations 3-regular spin glass (rs phase)

  15. Comparison with Simulations Random ferromagnet (average connectivity 2)

  16. Comparison with Simulations 3 regular spin glass, parallel update

  17. Outlook Increase observable set Via intuition (programming problem) Using information geometry (theoretical problem) Examine ways to work in the ensemble Relevant to rsb schemes Many potential applications to specific models

  18. Dynamical Replica Theory for Dilute Spin Systems Jon Hatchett RIKEN BSI In collaboration with I. Pérez Castillo, A. C. C. Coolen & N. S. Skantzos

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