Chalker JT, Wang ZJ
Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United Kingdom.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jan;61(1):196-203. doi: 10.1103/physreve.61.196.
We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. 79, 1797 (1997)]. We calculate analytically the ensemble-averaged one-particle Green function and the eigenvalue density for this Fokker-Planck operator, using a diagrammatic expansion developed for resolvents of non-Hermitian random operators, together with a mean-field approximation (the self-consistent Born approximation) which is well controlled in the weak-disorder regime for dimension d>2. The eigenvalue density in the complex plane is nonzero within a wedge that encloses the negative real axis. Particle motion is diffusive at long times, but for short times we find a novel time dependence of the mean-square displacement, <r(2)> approximately t(2/d) in dimension d>2, associated with the imaginary parts of eigenvalues.
我们研究了福克 - 普朗克算子的谱性质，该算子描述了粒子在与时间无关的随机速度场中通过扩散和平流的组合进行运动，详细介绍了在其他地方概述的工作[J. T. 查尔克和Z. J. 王，《物理评论快报》79, 1797 (1997)]。我们使用为非厄米随机算子的预解式开发的图解展开，以及在维度d > 2的弱无序区域中得到良好控制的平均场近似（自洽玻恩近似），解析地计算了该福克 - 普朗克算子的系综平均单粒子格林函数和本征值密度。复平面中的本征值密度在包围负实轴的一个楔内非零。粒子运动在长时间是扩散性的，但在短时间内，我们发现在维度d > 2中，均方位移<r(2)>约为t(2/d)，具有与本征值虚部相关的新颖时间依赖性。
