Dynamical Replica Theory for Dilute Spin Systems

Jon HatchettRIKEN 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

m - projection

e-flat submanifold

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 Dynamics•Simple choice of observables (m,e)

•Find simple ODEs:

is the distribution of local fields

in a system with magnetisation and energy

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

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

Comparison with Simulations•Random ferromagnet (average connectivity 2)

Comparison with Simulations•3 regular spin glass, parallel update

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

Dynamical Replica Theory for Dilute Spin Systems

Jon Hatchett

RIKEN BSIIn collaboration with I. Pérez Castillo,

A. C. C. Coolen & N. S. Skantzos