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Limiting behaviors of high dimensional stochastic spin ensembles
Communications in Mathematical Sciences ( IF 1 ) Pub Date : 2021-01-01 , DOI: 10.4310/cms.2021.v19.n2.a7
Yuan Gao 1 , Kay Kirkpatrick 2 , Jeremy Marzuola 1 , Jonathan Mattingly 3 , Katherine A. Newhall 1
Affiliation  

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.

中文翻译:

高维随机自旋集合的极限行为

统计物理学中的晶格自旋模型用于理解磁性。它们的哈密顿量是Dirichlet能量形式的离散形式,表示与谐波映射热流方程的关系。用哈密顿量定义的吉布斯分布用于大都会-哈丁斯(M-H)算法中,以生成趋于平衡状态的动力学。在逆温度较大的极限情况下,我们建立了离散的M‑H动力学与与哈密顿量相关的连续谐波图热流之间的关系。我们展示了M-H力度调和映射热流方程的收敛两个步骤:首先,固定的格子大小和一个M-H步建议大小的正确选择,M-H动力学起着梯度下降的作用,将被证明收敛于Langevin随机微分方程(SDE)系统。其次,在吉布斯分布中逆温度的适当缩放并将晶格大小设为无穷大后,将证明该SDE系统收敛于确定性谐波映射热流方程。我们的结果并非出乎意料,但显示了M‑H步骤与SDE Stratonovich公式之间的显着联系,并且揭示了轨迹运动失衡动力学与具有几何约束的规范PDE系统有关。结果表明,该SDE系统收敛于确定性谐波映射热流方程。我们的结果并非出乎意料,但显示了M‑H步骤与SDE Stratonovich公式之间的显着联系,并且揭示了轨迹运动失衡动力学与具有几何约束的规范PDE系统有关。结果表明,该SDE系统收敛于确定性谐波映射热流方程。我们的结果并非出乎意料,但显示了M‑H步骤与SDE Stratonovich公式之间的显着联系,并且揭示了轨迹运动失衡动力学与具有几何约束的规范PDE系统有关。
更新日期:2021-01-01
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