[论文解读] WIGNER FUNCTIONS AND STOCHASTICALLY PERTURBED LATTICE DYNAMICS

Giada Basile, Stefano Olla+1 more|arXiv (Cornell University)|May 20, 2008

Quantum chaos and dynamical systems参考文献 13被引用 8

一句话总结

本文将Wigner函数新颖地应用于具有小随机扰动的晶格动力学，表明在半经典极限下，Wigner函数遵循带有非弹性碰撞的线性输运方程演化。关键结果是，在平衡态下预测能量流相关性的衰减速率为1/√t，与先前采用不同方法的研究结果一致。

ABSTRACT

We consider lattice dynamics with a small stochastic perturbation of order and prove that for a space-time scale of order 1 the Wigner function evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain the transport equation predicts a slow decay, as 1/ √ t, for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method. Wigner functions are a very convenient and versatile tool in the analysis of wave equations. For multi-component linear wave equations the semiclassical part of the solution is covered by the time evolved Wigner function, see (9) with refinements in (5). If the coefficients of the wave equation areweakly random, then in the semiclassical limit the Wigner function is governed by a transport equation, which accounts for the finite life-time of the modes. We refer to the very informative survey (11) and to (1), (8) for two completely worked out benchmarks. A similar, but more involved scheme works for weakly nonlinear wave equations, see (13, 12). In our contribution we develop a novel application for Wigner functions. Rather than stochastically perturbing the coefficients of the wave e we add sto- chastic terms to the equation. They can be written down most easily for a discrete wave equation (lattice dynamics) which is the only case considered here. As a Hamiltonian system the lattice dynamics conserves energy and, depending on the couplings, also momentum. The basic idea is to have the added stochastic terms respect locally such conservation laws. In the context of interacting mechanical particles related models have been studied, e.g., in (10). But in the context of wave equations such an approach is very recent (4, 2, 3). To have a closed equation for the evolution of the Wigner function the stochas- tic part of the generator has to be of order e with e the semiclassical parameter, 0 < e ≪ 1. We will prove that in the limit e → 0 the Wigner function is gov- erned by a linear transport equation. In the cases mentioned above the collision operator of the transport equation describes elastic collisions, while in our case the collisions are inelastic with energy conserved only on average. The Wigner function evolution is very efficient for the understanding of the long-time properties of the stochastic wave dynamics. We will provide only one

研究动机与目标

• 将Wigner函数的应用从随机系数扩展到波方程中的随机扰动。
• 分析在满足局部守恒定律的小随机噪声作用下，晶格动力学的长时间行为。
• 在半经典极限（ε → 0）下，推导出Wigner函数的闭合输运方程。
• 表征所得输运方程中碰撞的性质为非弹性，仅在平均意义下保持能量守恒。
• 为理解随机波系统中能量输运与相关性衰减，提供一种新的分析框架。

提出的方法

• 将晶格动力学建模为在波方程中直接加入小随机扰动的哈密顿系统。
• 确保随机项尊重能量与动量的局部守恒，以模拟波系统中的物理一致性。
• 使用半经典参数 ε ≪ 1 来缩放生成元中的随机部分，从而实现严格的极限。
• 在 ε → 0 极限下推导Wigner函数的演化方程，证明其收敛于线性输运方程。
• 分析输运方程中的碰撞算符，确认其描述的是仅在平均意义下能量守恒的非弹性碰撞过程。
• 应用Wigner函数形式体系，提取长时间相关性特性，特别是能量流相关性。

实验结果

研究问题

• RQ1 在晶格动力学中加入小随机噪声，如何影响波模式的长时间演化？
• RQ2 Wigner函数形式体系能否扩展到具有守恒定律的随机波方程，而不仅仅是随机系数？
• RQ3 在随机扰动下，Wigner函数输运方程中出现的碰撞算符类型为何？
• RQ4 所得输运方程是否能预测平衡态下能量流相关性的特定衰减速率？
• RQ5 在半经典极限下，随机扰动如何影响波模式的有效寿命与散射行为？

主要发现

• 在半经典极限（ε → 0）下，Wigner函数遵循带有非弹性碰撞的线性输运方程演化。
• 输运方程中的碰撞算符描述的是仅在平均意义下能量守恒的非弹性过程。
• 对于能量与动量守恒的链式系统，该模型预测在平衡态下能量流相关函数的衰减速率为1/√t。
• 该1/√t衰减速率与先前通过不同分析方法获得的结果一致，验证了Wigner函数方法的有效性。
• Wigner函数为分析随机扰动波系统长时间特性提供了一种高效且通用的工具。
• 该方法成功捕捉了随机噪声下波模式的有效动力学，包括其有限寿命与散射行为。