本文对反常扩散理论进行了探讨，解析推导出了欠扩散的极限：局域化下粒子的经典和量子行为，并同弹道扩散（超扩散的极限）加以比较，揭示出这两类非各态历经的不同之处。 第二章讨论了各态历经的基本概念以及它的几种判据。在此基础上提出了非各态历经的外在表现：如果系统的渐进分布依赖于粒子坐标或速度的初始准备，则该系统是非各态历经的。 第三章从经典和量子两个方面，研究弹道扩散系统长时间后坐标和速度所表现出来的非各态历经行为。当给该系统加上简谐势后，它的各态历经性得到恢复，因此我们把这种非各态历经称为第一类非各态历经。　 　 第四章通过经典和量子两种情况的解析推导，得到了局域化时系统坐标的非各态历经行为。同弹道扩散相比，当给系统加上简谐势后，各态历经性并不能够得到恢复，所以本文称这种非各态历经为第二类非各态历经。第五章中，本文提出了两类非各态历经的内在条件：阻尼核函数的拉普拉斯变换在低频时分别正比、反比于频率。第六章是总结与展望。

In this thesis, we have given some discussion about the theory of anomalous diffusion. In the limit of sub-diffusion, namely localization, the particles's behavior in both classical and quantum case has been obtained from the analytic derivation. Also, it has been compared with ballistic diffusion--the limit ofsuper-diffusion. Finally, we have gotten the differences between the two kinds of nonergodicity.In Chapter 2, we discuss the basic concept of ergodicity and some of its criterions. Based on this, we introduce the extrinsic exhibition of nonergodicity. A system will be termed nonergodic if its long-time statistical distribution dependents on the initial preparation of coordinate or velocity. In Chapter 3, the system of ballistic diffusion is studied in both classical and quantum cases. The results show the nonergodic behavior of coordinate and velocity in the long-time. However, the ergodicity of the system can be resumed by an external harmonic potential. So we call this nonergodicity the first kind of nonergodicity.In Chapter 4, localization is studied in classical and quantum situations. The coordinate of the localization system exhibits nonergodic behavior, and different with ballistic diffusion, the ergodicity of this system can not be resumed by an external bound potential. In this paper, we call it the second kind of nonergodicity.In chapter 5, we present the intrinsic conditions of the two kindsof nonergodicity. The first kind is due to the Laplace transform of the damping kernel is proportional to the frequency at low frequencies while the second is inverse proportional to the frequency.Concluding remarks and future perspectives are presented in Chapter６.