TY - JOUR

T1 - LIMITING BEHAVIORS OF HIGH DIMENSIONAL STOCHASTIC SPIN ENSEMBLES*

AU - Gao, Yuan

AU - Kirkpatrick, Kay

AU - Marzuola, Jeremy

AU - Mattingly, Jonathan

AU - Newhall, Katherine A.

N1 - Publisher Copyright:
© 2021 International Press. All Rights Reserved.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Langevin equation

KW - Metropolis-Hastings algorithm

KW - Stratonovich SDE

KW - harmonic map heat flow

UR - http://www.scopus.com/inward/record.url?scp=85105215588&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85105215588&partnerID=8YFLogxK

U2 - 10.4310/CMS.2021.v19.n2.a7

DO - 10.4310/CMS.2021.v19.n2.a7

M3 - Article

AN - SCOPUS:85105215588

SN - 1539-6746

VL - 19

SP - 453

EP - 494

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 2

ER -