Yuan Gao, Kay Kirkpatrick, Jeremy Marzuola, Jonathan Mattingly, Katherine A. Newhall

Research output: Contribution to journalArticlepeer-review


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 sto chastic 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 tra jectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.

Original languageEnglish (US)
Pages (from-to)453-494
Number of pages42
JournalCommunications in Mathematical Sciences
Issue number2
StatePublished - 2021


  • Langevin equation
  • Metropolis-Hastings algorithm
  • Stratonovich SDE
  • harmonic map heat flow

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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