Skip to main content
Duke University Libraries
DukeSpace Scholarship by Duke Authors
  • Login
  • Ask
  • Menu
  • Login
  • Ask a Librarian
  • Search & Find
  • Using the Library
  • Research Support
  • Course Support
  • Libraries
  • About
View Item 
  •   DukeSpace
  • Duke Scholarly Works
  • Scholarly Articles
  • View Item
  •   DukeSpace
  • Duke Scholarly Works
  • Scholarly Articles
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

Limiting Behaviors of High Dimensional Stochastic Spin Ensemble

Thumbnail
View / Download
513.6 Kb
Authors
Gao, Y
Kirkpatrick, K
Marzuola, J
Mattingly, J
Newhall, KA
Repository Usage Stats
117
views
27
downloads
Abstract
Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. The Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the convergence of the M-H dynamics to the Harmonic map heat flow equation in two steps: First, with fixed lattice size and proper choice of proposal size in one M-H step, the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations (SDE). Second, with proper scaling of the inverse temperature in the Gibbs distribution and taking the lattice size to infinity, it will be shown that this SDE system converges to the deterministic Harmonic map heat flow equation. Our results are not unexpected, but show remarkable connections between the M-H steps and the SDE Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.
Type
Journal article
Subject
math.PR
math.PR
math-ph
math.MP
65C05, 58J65, 82C05, 60J10, 60H10, 60J60
Permalink
https://hdl.handle.net/10161/17194
Collections
  • Scholarly Articles
More Info
Show full item record

Scholars@Duke

Mattingly

Jonathan Christopher Mattingly

Kimberly J. Jenkins Distinguished University Professor of New Technologies
Jonathan Christopher  Mattingly grew up in Charlotte, NC where he attended Irwin Ave elementary and Charlotte Country Day.  He graduated from the NC School of Science and Mathematics and received a BS is Applied Mathematics with a concentration in physics from Yale University. After two years abroad with a year spent at ENS Lyon studying nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to attend Princeton University where he obtained a PhD in Applied and
Open Access

Articles written by Duke faculty are made available through the campus open access policy. For more information see: Duke Open Access Policy

Rights for Collection: Scholarly Articles


Works are deposited here by their authors, and represent their research and opinions, not that of Duke University. Some materials and descriptions may include offensive content. More info

Make Your Work Available Here

How to Deposit

Browse

All of DukeSpaceCommunities & CollectionsAuthorsTitlesTypesBy Issue DateDepartmentsAffiliations of Duke Author(s)SubjectsBy Submit DateThis CollectionAuthorsTitlesTypesBy Issue DateDepartmentsAffiliations of Duke Author(s)SubjectsBy Submit Date

My Account

LoginRegister

Statistics

View Usage Statistics
Duke University Libraries

Contact Us

411 Chapel Drive
Durham, NC 27708
(919) 660-5870
Perkins Library Service Desk

Digital Repositories at Duke

  • Report a problem with the repositories
  • About digital repositories at Duke
  • Accessibility Policy
  • Deaccession and DMCA Takedown Policy

TwitterFacebookYouTubeFlickrInstagramBlogs

Sign Up for Our Newsletter
  • Re-use & Attribution / Privacy
  • Harmful Language Statement
  • Support the Libraries
Duke University