近年来，随着实验技术的发展，人们发现了大量表现出反常输运行为的系统。在研究此类系统的诸多理论和模型中，连续时间无规行走(CTRW)以其不涉及系统动力学行为而是基于观测过程，物理图像直观清楚，便于数值模拟的特点，非常适用于研究复杂外势中的粒子输运。本文提出了两种可用于研究任意外部势场中任意扩散指数系统输运行为的CTRW模型及其数值模拟算法，并利用它们研究了一些典型势场中粒子输运相关问题。第一章综述了反常输运现象，分析并比较了研究该现象的几种常用理论，同时对连续时间无规行走理论做了较深入的概述。第二章从连续时间无规行走理论出发，推导描述反常扩散自由粒子空间分布函数演化的分数阶扩散方程和描述外部势场中反常输运系统空间分布函数演化的分数阶Klein-Kramers方程和分数阶Fokker-Planck方程。本章还就上述方程与分数阶Langevin方程对反常扩散与反常输运的描述进行了比较分析。第三章建立了CTRW数值算法，分正常扩散、欠扩散和超扩散三种情况分别确定等待时间和跳跃距离的分布函数的选取及抽样算法。同时给出并分析两种可用于外部势场中粒子输运研究的CTRW模型：欠扩散系统固定小步长CTRW数值算法和任意功率指数系统的CTRW-Metropolis数值算法。第四章将我们发展的CTRW-Metropolis方案应用到三个具体的问题中，即倾斜线性势中的反常扩散、一维不对称周期势中的布朗马达和亚稳势中的粒子逃逸，得到了以下结论：倾斜线性势能够改变粒子的扩散行为，放大系统的扩散指数；闪烁棘轮势和摇摆棘轮势能够整流反常扩散产生定向流，并且闪烁布朗马达定向流的极大值出现在超扩散情况下，而摇摆布朗马达定向流的最大值出现在弹道扩散情况下；亚稳势中粒子稳定逃逸速率的极大值出现在超扩散条件下，并且存在“势垒相消”效应。同时，本章还将上述结论与理论推导以及其它数值算法得到的结果进行了比较。第五章提出了一种广义的CTRW模型，即环境依赖的连续时间无规行走模型，其特点在于将CTRW与分子动力学有机结合，弥补了以往CTRW模型只能模拟过阻尼系统的不足，将环境（阻尼）对输运系统的作用通过动力学的方式（即随机速度）引入到CTRW体系中来。同时，我们发现了“小概率大贡献”现象，通过这一现象，我们对超扩散（Levy飞行）的机制以及方均位移发散的原因提出了一种新的理解。第六章将环境依赖的连续时间无规行走模型用于研究一个对环境阻尼敏感依赖的问题：受外部周期信号驱动的阻尼谐振子的动力学共振现象。我们以系统总能量作为“共振探针”来研究系统受到的有效阻尼随反常指数的变化，得到当alpha<0.4时，系统总是处于有效欠阻尼状态的结论，并与其他数值算法得到的结果进行了比较。第七章分析了金融市场风险评估领域的VaR体系，初步建立了可用于预测资产价值变动的CTRW算法。第八章对本文的工作进行了总结，并将来拟开展的工作进行了展望。附录部分给出正文中涉及的部分数学公式的定义和推导，并予以详细的说明。

In recent years, with the devolopment of experimental technique, scientists have discovered numbers of systems that act anomalous tranport behaviors. Compared with the other models and theories used to study anomalous systems, the continuous time random walk (CTRW) theory dose not base on the dynamical behavior of the particles but on the process of observation, so it is convenient to be numerically simulated. Therefore, the CTRW model is especially suitable for investgating the transports in complex potentials. In this paper, two types of CTRW models and their numerically simulating methods are proposed, which are applied to studying questions about tranports in several typical potentials.In Chapter 1, we summarize the phenomenon of anomalous transport. As well, several theories applied to study this phenomenon is analysed and compared. In addition, the CTRW theory is particularly introduced. In Chapter 2, we derive the fractional diffusion equation that is used to describe the diffusion of force-free particles and the fractional Fokker-Planck equation that is used to describe the transport process in an external potential. The derivation is based on the idea of CTRW. Further more, the fractional Langevin equation is shown and compared with the fractional equations mentioned above.In Chapter 3, we establishe the numerical algorithm of the CTRW model. The distribution functions of the jumping distance and the waiting time and their sampling methods are decided individually based on the case of sub-diffusion, normal diffusion and super-diffusion. Two types of CTRW models that can be employed in the condition that an external potential exsists are introduced and analysed, which are the CTRW algorithm with fixed small step-length and the CTRW-Metropolis scheme.In Chapter 4, we apply the CTRW-Metropolis scheme to three specific problems, which are anomalous diffusion in sloping linear potential, the Brownian motor in asymmetric periodic potential and the escapement of particles in metastable potential. The following conclusions are obtained: the sloping linear potential can change the diffusion behavior of the system and enlarge the power index; the maximum of the directional current of the Brownian motor occurs in the case of super-diffusion in a flashing ratchet potential and in the case of ballistic diffusion in a rocking ratchet potential; The maximum of escape rate occurs in the case of super-diffusion and the cancellation phenomenon in the barrier region is discovered. These phenomena we discovered are compared with the theorical results as well.In Chapter 5, we propose a generalized CTRW model, i.e., the environment-dependent continuous time random walk model, which conbines the random flight with the molecular dynamics. The former CTRW models can simulate overdamped conditions only, however, the generalized CTRW model overcomes this shortage and truely introduces the influence of the damping into the CTRW model for the first time. Further more, the mechanism of super-diffusion is newly understood from the phenomenon named ``large contribution with small probability''.In Chapter 6, we apply the environment-dependent continuous time random walk model to study the effective damping, where the resonance behavior of a damped harmonic oscillator subjected to a periodic signal is regarded as a probe. The system is asserted to be in effectively underdamped region if the dynamical resonance occurs. A conclusion is obtained that the system is always in effectively underdamped condition when alpha is smaller than about 0.4.In Chapter 7, we analyse the VaR system in the domain of risk evoluation in financial market, and the CTRW algorithm used in forecasting the fluctuation of the price of asset is preliminarily established.In Chapter 8, we give concluding remarks and future perspectives for our work.Some mathematical definitions and derivations appearing in the thesis are presented in detail in the appendix.