Nonlocal stochastic-partial-differential-equation limits of spatially correlated noise-driven spin systems derived to sample a canonical distribution
Nonlocal stochastic-partial-differential-equation limits of spatially correlated noise-driven spin systems derived to sample a canonical distribution
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为采样正则分布而导出的空间相关噪声驱动自旋系统的非局部随机偏微分方程极限
DOI：
10.1103/physreve.102.052112
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发表时间：
2020
影响因子：
2.4
通讯作者：
Newhall, Katherine A.
中科院分区：
文献类型：
--
作者：
Gao, Yuan;Marzuola, Jeremy L.;Mattingly, Jonathan C.;Newhall, Katherine A.
对于一个嘈杂的自旋系统，我们推导出一个非本地的随机版本的过阻尼朗道-Lipshitz方程设计尊重底层的哈密顿结构和样本的正则或吉布斯分布，而被驱动的空间相关的（有色）噪声，正则化的动态，使这个随机偏微分方程数学适定性。我们开始从一个微观离散时间模型的Metropolis-Hastings算法的有限数量的自旋周期性边界条件，其值分布在单位球。因此，我们提出了一个未来的状态，系统通过添加到每个自旋有色噪声投影到球，然后接受这个建议的状态与建议和当前状态的正则分布的比率给出的概率。对于不相关（白色）噪声，该过程保证对正则分布进行采样。我们证明，对于有色噪声，用于将噪声投影到球体上并保持自旋大小的方法会影响系统的平衡分布，因为着色投影噪声不等同于投影有色噪声。在一个特定的情况下，我们表明这种对称性的破坏消失与消失的建议大小;由此产生的连续时间系统的随机微分方程样本的正则分布，并保持自旋的幅度，而被驱动的有色噪声。取无限多个自旋的连续极限，我们就得到了前面提到的过阻尼朗道-利普希茨方程。包括数值模拟，以验证收敛性能和演示的动态。
For a noisy spin system, we derive a nonlocal stochastic version of the overdamped Landau-Lipshitz equation designed to respect the underlying Hamiltonian structure and sample the canonical or Gibbs distribution while being driven by spatially correlated (colored) noise that regularizes the dynamics, making this Stochastic partial differential equation mathematically well-posed. We begin from a microscopic discrete-time model motivated by the Metropolis-Hastings algorithm for a finite number of spins with periodic boundary conditions whose values are distributed on the unit sphere. We thus propose a future state of the system by adding to each spin colored noise projected onto the sphere, and then accept this proposed state with probability given by the ratio of the canonical distribution at the proposed and current states. For uncorrelated (white) noise this process is guaranteed to sample the canonical distribution. We demonstrate that for colored noise, the method used to project the noise onto the sphere and conserve the magnitude of the spins impacts the equilibrium distribution of the system, as coloring projected noise is not equivalent to projecting colored noise. In a specific scenario we show this break in symmetry vanishes with vanishing proposal size; the resulting continuous-time system of Stochastic differential equations samples the canonical distribution and preserves the magnitude of the spins while being driven by colored noise. Taking the continuum limit of infinitely many spins we arrive at the aforementioned version of the overdamped Landau-Lipshitz equation. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.