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LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN ENSEMBLES*

Publication ,  Journal Article
Gao, Y; Kirkpatrick, K; Marzuola, J; Mattingly, J; Newhall, KA
Published in: Communications in Mathematical Sciences
January 1, 2021

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.

Duke Scholars

Published In

Communications in Mathematical Sciences

DOI

EISSN

1945-0796

ISSN

1539-6746

Publication Date

January 1, 2021

Volume

19

Issue

2

Start / End Page

453 / 494

Related Subject Headings

  • Applied Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 1502 Banking, Finance and Investment
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

Citation

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Gao, Y., Kirkpatrick, K., Marzuola, J., Mattingly, J., & Newhall, K. A. (2021). LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN ENSEMBLES*. Communications in Mathematical Sciences, 19(2), 453–494. https://doi.org/10.4310/CMS.2021.v19.n2.a7
Gao, Y., K. Kirkpatrick, J. Marzuola, J. Mattingly, and K. A. Newhall. “LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN ENSEMBLES*.” Communications in Mathematical Sciences 19, no. 2 (January 1, 2021): 453–94. https://doi.org/10.4310/CMS.2021.v19.n2.a7.
Gao Y, Kirkpatrick K, Marzuola J, Mattingly J, Newhall KA. LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN ENSEMBLES*. Communications in Mathematical Sciences. 2021 Jan 1;19(2):453–94.
Gao, Y., et al. “LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN ENSEMBLES*.” Communications in Mathematical Sciences, vol. 19, no. 2, Jan. 2021, pp. 453–94. Scopus, doi:10.4310/CMS.2021.v19.n2.a7.
Gao Y, Kirkpatrick K, Marzuola J, Mattingly J, Newhall KA. LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN ENSEMBLES*. Communications in Mathematical Sciences. 2021 Jan 1;19(2):453–494.

Published In

Communications in Mathematical Sciences

DOI

EISSN

1945-0796

ISSN

1539-6746

Publication Date

January 1, 2021

Volume

19

Issue

2

Start / End Page

453 / 494

Related Subject Headings

  • Applied Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 1502 Banking, Finance and Investment
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics