超扩散能量传输中的边界热化
Boundary Thermalization in Superdiffusive Energy Transport
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中文总结 AI 辅助
研究有限一维无钉调和链在特定时间尺度下的能量传输，通过随机动量交换和端点热浴实现超扩散。证明平均微观能量分布收敛到求解分数热方程温度场，给出宏观边界条件严格推导及新边界条件并受物理模型启发。
中文摘要 AI 辅助
我们研究了一个有限的一维无钉调和链中的能量传输，该链具有随机最近邻动量交换，并在其端点处有朗之万热浴。已知此类系统表现出由长波长声学模式驱动的超扩散传输，导致分数宏观行为。虽然分数热方程已针对无限链严格推导出来，但由于分数拉普拉斯算子的非局部性，与热浴接触的有限系统的相应边界条件仍不清楚。在超扩散时间尺度$t\sim n^{3/2}$（其中$n$是系统大小）下，我们证明，当$n\to+\infty$时，平均微观能量分布收敛到一个在$[0,1]$上求解分数热方程的温度场，其生成器由诺伊曼分数拉普拉斯算子和热浴引起的附加非局部边界项给出。我们的结果为开放链中超扩散热传输的宏观边界条件提供了严格推导，并引入了受物理模型启发的分数拉普拉斯算子的新边界条件。
英文摘要
We study energy transport in a finite one-dimensional unpinned harmonic chain with stochastic nearest-neighbor momentum exchanges and Langevin heat baths at its endpoints. Such systems are known to exhibit superdiffusive transport driven by long-wavelength acoustic modes, leading to fractional macroscopic behavior. While fractional heat equations have been rigorously derived for infinite chains, the corresponding boundary conditions for finite systems in contact with heat baths remain unclear due to the nonlocality of the fractional Laplacian. Under the superdiffusive time scaling $t\sim n^{3/2}$, where $n$ is the system size, we prove that the averaged microscopic energy profile converges, as $n\to+\infty$, to a temperature field solving a fractional heat equation on $[0,1]$, with the generator given by a Neumann fractional Laplacian and additional nonlocal boundary terms induced by the heat baths. Our results provide a rigorous derivation of macroscopic boundary conditions for superdiffusive heat transport in open chains and introduce new boundary conditions for fractional Laplacians, that are motivated by a physical model.
