论文标题
衍生成样品分布的空间相关的噪声驱动的自旋系统的非本地SPDE极限
Non-local SPDE limits of spatially-correlated-noise driven spin systems derived to sample a canonical distribution
论文作者
论文摘要
我们研究了由离散的马尔可夫跳跃过程驱动的随机旋转集合的宏观行为，该过程由大都市 - 危机算法动机，其中该建议是通过空间相关（彩色）噪声制成的，因此未能对称。但是，我们证明了一种情况，建议对称的失败是更高阶的效果。因此，从这些微观动力学中，我们将其作为限制得出，因为提案大小为零，并且旋转数量为无限，这是谐波映射热流的非本地随机版本（或过度抑制了landau-lipshitz方程）。该方程式在数学上既有良好，又是与动能相关的规范/吉布斯分布。由于自旋系统的限制几何形状与彩色噪声之间的相互作用引起的提案对称性的失败与不相关的白色噪声，驱动的系统形成鲜明对比。具体而言，噪声投影的选择保留旋转的大小对于维持适当的平衡分布至关重要。包括数值模拟以验证收敛属性并演示动力学。
We study the macroscopic behavior of a stochastic spin ensemble driven by a discrete Markov jump process motivated by the Metropolis-Hastings algorithm where the proposal is made with spatially correlated (colored) noise, and hence fails to be symmetric. However, we demonstrate a scenario where the failure of proposal symmetry is a higher order effect. Hence, from these microscopic dynamics we derive as a limit as the proposal size goes to zero and the number of spins to infinity, a non-local stochastic version of the harmonic map heat flow (or overdamped Landau-Lipshitz equation). The equation is both mathematically well-posed and samples the canonical/Gibbs distribution related to the kinetic energy. The failure of proposal symmetry due to interaction between the confining geometry of the spin system and the colored noise is in contrast to the uncorrelated, white-noise, driven system. Specifically, the choice of projection of the noise to conserve the magnitude of the spins is crucial to maintaining the proper equilibrium distribution. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.
