Wang Yingxi, Zhao Nanrong, Yan YiJing
College of Chemistry, Sichuan University, Chengdu 610064, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Apr;85(4 Pt 1):041142. doi: 10.1103/PhysRevE.85.041142. Epub 2012 Apr 26.
We analyze the dissipative dynamics of a particle governed by a two-dimensional generalized Langevin equation with coupled fractional Gaussian noise and white noise in its respective coordinates, assuming the lowest-order coupling form. Two situations are studied: In the first the particle is free from external force and in the second the particle is subject to a two-dimensional harmonic potential. We derive the general expressions for the mean values, variances, and velocity autocorrelation function and evaluate their temporal evolutions via the numerical Laplace inversion technique. Through the analytical results of the short-time and long-time behaviors, we also explicitly elucidate the effects of fluctuation correlation coupling and interoscillator coupling on the dynamic behaviors of the particle. It is shown that in both situations the couplings do not affect the short-time behavior of self-diffusions in each coordinate, and the subdiffusive and normal diffusive features of these processes resemble those in a one-dimensional system with fractional Gaussian noise and white noise, respectively. However, over a long time period, the fluctuation correlation extends the characteristic time scales for the self-diffusions of a free particle; while only the interoscillator coupling induces a retardation of the relaxation processes of a bounded particle toward equilibrium. Moreover, both couplings generate a cross diffusion, whose long-time approximation has two possible forms, the selection of which depends on the relevant time scales of self-diffusions in each coordinate.
我们分析了一个粒子的耗散动力学，该粒子由二维广义朗之万方程描述，其各自坐标中存在耦合分数高斯噪声和白噪声，假设为最低阶耦合形式。研究了两种情况：第一种情况是粒子不受外力作用，第二种情况是粒子受到二维谐振势的作用。我们推导了平均值、方差和速度自相关函数的一般表达式，并通过数值拉普拉斯反演技术评估它们的时间演化。通过对短时和长时行为的解析结果，我们还明确阐明了涨落关联耦合和振子间耦合对粒子动力学行为的影响。结果表明，在两种情况下，耦合均不影响各坐标中自扩散的短时行为，且这些过程的亚扩散和正常扩散特征分别类似于具有分数高斯噪声和白噪声的一维系统中的情况。然而，在长时间内，涨落关联扩展了自由粒子自扩散的特征时间尺度；而只有振子间耦合会导致有界粒子向平衡态弛豫过程的延迟。此外，两种耦合都会产生交叉扩散，其长时间近似有两种可能形式，具体选择取决于各坐标中自扩散的相关时间尺度。
