扩散理论是研究扩散现象的重要理论，可划分为正常扩散理论和反常扩散理论.正常扩散的研究先于反常扩散，已经发展出一套成熟的理论框架.近年来,随着实验技术的发展和理论研究的深入，不同领域的研究者发现了大量表现出反常扩散行为的系统(Lévy飞行就是其中重要的一类,它的数学基础在很早以前就被数学家提出,只是较晚才发现其应用的领域).所有这些扩散理论不仅可以描述许多的实际现象,例如非晶半导体中的载流子输运、分形几何结构中的输运、单分子分光计、细菌运动、重离子碰撞导致的核熔合问题等,而且在统计物理、生物、材料模拟计算中具有基础意义的高维分布算法也可以在扩散理论的框架下得以概括和研究.本文从事了三方面的工作：(1)针对高维分布抽样算法提出了势分解策略，这一策略可结合任意抽样算法使用,有利于布朗粒子在高位垒多维等效势中的扩散从而使其更有效地遍历整个相空间；(2)利用解析和数值相结合的手段研究了一类强束缚单稳和双稳势中的Lévy飞行，发现其稳态分布呈现从单模向双模和从双模向三模的转换,其中三模稳态分布的发现还未见有相应的文献报导,这是一类强烈的非吉布斯-玻尔兹曼统计现象；(3)利用三维变系数耦合朗之万方程的模拟和常系数二次曲面近似下通过几率解析公式的推导，研究了核熔合问题的复合核形成阶段,提出了一种对熔合障碍形成机制的新理解,在进行了大量计算的前提下通过朗之万模拟验证了这一理解.第一章综述了本文工作所需的基础背景，对相应方向的一些进展做了阐述.第二章提出了一种势分解策略，其有利于高位垒多维势中的粒子扩散. 通过更有效地遍历相应的相空间,在原来抽样算法中不能观察到的态转换现象在结合了势分解策略后就能实现.这一策略可以结合马尔可夫性或非马尔可夫性的抽样算法一同使用,促进在多种领域中必不可少的模拟手段的实施.第三章利用Lévy白噪声驱动的过阻尼朗之万方程和空间变量具有分数阶导数的福克-普朗克方程等解析手段和一定的数值计算、蒙特卡罗模拟实现技术，研究了一类强束缚单稳和双稳势中Lévy飞行的稳态分布，发现随着势参数的变化稳态分布实现了从单模分布向双模分布及双模分布向三模分布的转换并通过数值计算给出了临界的参数值.上述发现是一类强烈的非吉布斯-玻尔兹曼统计现象，具有一定的理论意义.第四章以三维曲柄模型为基础，给出了复合原子核在相应形变参量空间的形变势能、惯性张量和粘滞张量的表达式，讨论了相应的细节,为第五章核熔合问题复合核形成阶段的基于三维变系数耦合朗之万方程的模拟研究奠定了基础.第五章在三维变系数耦合朗之万方程的基础上研究了核熔合反应的第二个阶段(复合核形成阶段),首次得到了常系数假设下三维二次曲面解析通过几率,通过数值模拟定量考察了常系数二次曲面近似下解析理论的近似程度,同时通过对势能曲面的分析提出了一种对熔合障碍形成机制的新解释并通过朗之万模拟(通过并行机进行了大量计算)验证了其正确性.第六章是对本文工作的总结及作者今后研究方向的展望.

Diffusion theory is an elegant theory on diffusion phenomenon. It can be roughly classified into normal diffusion theory and anomalous diffusion theory. The investigation on normal diffusion goes ahead before that on anomalous diffusion and has formed a mature theoretical framework. By recently, researchers in diverse fields have found anomalous diffusion phenomenon in many systems based on more advanced experimental technology and theoretical analysis(Among these phenomenon is the Lévy flight, whose mathematical foundation has been proposed many years ago, but the application in other fields has been reported rather lately). All these theories can not only describe many physical phenomenon but also generalize the sampling algorithms for multi-dimensional distribution which are elementary for simulations in stochastic physics,biology,material computing,and so on.In this dissertation, three works in the diffusion theory field are reported. One is about the potential-decomposition strategy in Markov Chain Monte Carlo sampling algorithms. This strategy can be combined with many sampling algorithms to accelerate the diffusion of Brownian particles in multi-dimensional potential. So the phase space is explored more effectively. The second work is the research on the stationary distribution of the Lévy flight in some steep single- or double wells potential. By theoretical analysis and numerical simulations we find the stationary distribution transforms from single model to double model and/or from double model to tri-model.The tri-model stationary distribution has not been reported in other literatures. The third work is the utilization of three-dimensional coupled Langevin equations and the analytical formula of the probability passing over the saddle with the assumption of constant coefficient and quadratic surface in the study of the compound nucleus formation. A new interpretation of the formation mechanism of hindrance to fusion is proposed and verified by the Langevin simulation. In Chapter 1, necessary backgrounds for understanding our works are displayed. Some new developments in correlative research fields are described. In Chapter 2, a potential-decomposition strategy in Markov Chain Monte Carlo sampling algorithms is proposed to accelarate the diffusion in multi-dimensional potential with high barriers. Switch between different states can only be realized with the strategy. Furthermore, this strategy can be combined with Markov or non-Markov sampling algorithms to promote the simulation artifice's application in more fields. In Chapter 3, the overdamped Langevin equation with white Lévy noise, the fractional Fokker-Planck equation and some numerical methods are used to search the stationary distribution of Lévy flights in steep potential with single- or double wells. It is found that with changes of potential parameters, the stationary distribution transforms from single model to double model and/or from double model to tri-model. The critical parameter is calculated by numerical technology. This phenomenon is a strong non-Gibbs-Boltzmann statistics and has a definite theoretical value. In Chapter 4, calculations, by using one kind of shape parametrization, of the potential energy surafce,inertia tensors and friction tensors corresponding to these shape parameters are described. Some details about these quantities are discussed. These calculations are the basis of the three-dimensional coupled Langevin simulations in Chapter 5. In Chapter 5, based on the results in Chapter 4,the three-dimensional coupled Langevin simulations and the analysis of potential energy surface verify a new understanding of the formation of hindarance to fusion. The analytical formula of the probability passing over the saddle with the assumption of constant coefficient and quadratic surface is proposed and compared with the simulation results. These works will improve the development of diffusion model theory about fusion reaction of two nuclei.In Chapter 6, concludind remarks and future perspectives are given.