近些年，伴随着物理，生物等学科实验技术的发展，越来越多的反常扩散现象在生物和物理系统中被发现。在这些现象背后的随机动力学中，广义郎之万方程由于其形式的简洁与物理图像的直观而得到广泛应用。在本文中，我们通过广义郎之万方程研究了不同复杂势场中的扩散现象并重点研究其时间平均方均位移与系综平均方均位移的关系。
第一章阐述了关于广义郎之万方程及反常扩散理论的一些基本内容，讨论了时间平均与各态历经破缺的关系。
第二章中我们首先推导出了噪声的功率谱与反常扩散类型之间的内在联系，指出了长时间后系统的扩散行为实际上取决于功率谱函数零频时的值，并用推导出的关系讨论了限带白噪声、简谐噪声、简谐速度噪声的扩散情况。
第三章中我们通过模拟白噪声情况下谐振子势、对数势、周期势、四次方势中的郎之万方程，得到了不同势场下的时间平均方均位移位移、系综平均方均位移以及两者比值的关系并进行了讨论。
第四章中我们模拟了在倾斜周期势中由白噪声和简谐速度噪声驱动的广义
郎之万方程，得到了不同噪声情况下系统的时间平均方均位移和系综平均方均位移以及两者比值的关系，观察到了不同倾斜力下系统有效扩散系数被放大的反常现象。并对不同倾斜力对其产生的影响进行了讨论。
第五章对本文的研究内容进行了系统的总结，并讨论了对本文内容进一步深入研究的相关方向。

In recent years, the advancement of experimental techniques in physics, biology,and other disciplines has led to the discovery of an increasing number of anomalous diffusion phenomena in biological and physical systems. The generalized Langevin equation is widely employed in the stochastic dynamics underlying these phenomena due to its simplicity and intuitive physical images. This paper utilizes the generalized Langevin equation to investigate diffusion phenomena within various complex potential fields, with a focus on examining the relationship between time and ensemble averaged mean square displacement.
In Chapter 1, fundamental concepts regarding the generalized Langevin equation and anomalous diffusion theory are introduced, along with a discussion on the relationship between time average and ergodic breaking.
Chapter 2 establishes an intrinsic connection between the power spectrum of noise and abnormal diffusion types. It highlights that long-term system diffusion behavior is determined by the value of the power spectrum function at zero frequency. The derived
relation is then used to analyze band-limited white noise, quasi-monochromatic noise,and harmonic velocity noise..
Chapter 3 involves simulating the Langevin equation within different potential fields (harmonic oscillator potential, logarithmic potential, periodic potential, fourth power potential) under white noise conditions. This simulation yields insights about relationships among time and ensemble averaged mean square displacement and their ratios.
In chapter 4, we simulate generalized Langevin equations driven by white noise and harmonic velocity noise in tilted period potential, obtain the time and ensemble averaged mean square displacement and their ratios of the system under different noise
conditions. We observe the anomalous phenomenon that the effective diffusion coefficient is amplified under different tilt forces, and discuss the influence of differenttilt forces on it.

The fifth chapter provides a systematic summary of this paper's research content and discussing future directions for related research topics.