非均质非齐次系统的反常扩散与非遍历行为分析 其他题名 Analyses of anomalous diffusion and nonergodic behavior of heterogeneous and inhomogeneous systems 学位类型 博士 导师 邓伟华 2020-05-17 学位授予单位 兰州大学 学位授予地点 兰州 学位名称 理学博士 学位专业 计算数学 关键词 布朗运动 非遍历行为 Feynman-Kac 方程 双状态过程 中文摘要 输运过程, 特别是扩散过程, 在物理和生物系统中都是普遍存在的. 在相当长的时间里, 布朗运动(正常扩散)在扩散领域中占领着独一无二的主导地位. 然而, 在最近几十年里, 非布朗运动的扩散(反常扩散)在实验中被大量地发现, 并越来越强烈地撼动了布朗运动的绝对统治地位. 布朗运动具有遍历的性质, 即时间平均等于系综平均. 而人们发现的大量区别于布朗运动的反常扩散过程, 往往是非遍历的. 进一步地, 人们在研究反常非遍历的扩散现象的内在机制过程中发现, 粒子的运动方式往往不像布朗运动那样简单, 它们往往受到非均质的环境的影响或者其运动模式是非齐次的(具有不同的状态或阶段). 本文将针对这一类随机过程, 讨论它们的反常扩散及非遍历行为, 以及它们的应用, 如 Feynman-Kac 方程, 占据时间, 首次通过时间等. 本文的具体内容如下： 在第一章中, 我们简要的介绍了反常扩散与非遍历行为的研究背景. 紧接着, 我们介绍了本文的结构, 从研究对象、研究方法和创新点的角度简要分析了本文每章的内容. 最后, 为了本文讨论问题的方便, 我们对正文中所涉及到的几个基本的反常扩散模型和研究方法做了一定程度的阐述. 在第二章中, 我们用一个过阻尼的朗之万方程来刻画受外部力影响的非均质扩散过程, 推导其相应的 Feynman-Kac 方程, 即该随机过程的路径泛函的概率密度函数所满足的方程. 粒子路径的泛函在物理、数学、水文、经济学等领域有着广泛的应用. 在连续时间随机游走的框架下, 各类正常扩散、反常扩散、甚至带有化学反应的随机过程的 Feynman-Kac 方程已被导出. 但是, 很多物理和化学中的随机过程可以很自然地用朗之万方程来描述. 朗之万方程在研究外力场作用下的动力学和涨落环境中噪声的影响具有显著的优势. 我们将基于朗之万方程, 运用傅里叶变换的技巧, 推导向前和向后两个版本的 Feynman-Kac 方程. 根据新建立的方程, 我们计算了占据时间和轨迹曲线下面积的分布, 并通过数值试验验证了结果. 在第三章中, 我们仍然考虑第二章中受外部力影响的非均质扩散过程, 但是重点讨论该过程的遍历与非遍历行为. 因为我们主要关心的是长时间远距离的行为, 我们假设乘性噪声系数和外部势均为幂率形式, 即：D(x) = D0|x|\alpha 和 U(x) = U0|x|\beta. 基于D(x) 和U(x) 之间的竞争关系, 我们主要讨论\beta&gt\alpha, \beta=\alpha 和\beta&lt\alpha 三种情况. 第一种情况中, 系统是遍历的, 即时间平均等于系综平均. 相反的是, 对于第三种情况, 系统不遍历, 我们利用无限遍历理论求解出时间平均与系综平均的关系. 而对第二种情况, 具体的遍历性质依赖于系数D0 和U0.我们同时还推导了时间平均的分布, 并用数值模拟验证了结果. 在第四章中, 我们主要讨论一类非齐次的双状态随机过程的反常非遍历行为. 随着单状态随机过程的动力学研究的发展, 双状态随机过程在复杂系统及动物觅食行为等诸多领域中被广泛地观察到, 现在受到越来越多的关注. 列维行走和布朗运动交替进行的双状态过程已经被证实是一种有效的间歇性搜索过程, 而本章将建立该过程的理论基础及其反常非遍历行为. 我们假设两个状态的逗留时间均为幂律分布, 指数为 \alpha_\pm\in(0,2). 我们主要分析弱老化和强老化情况下的系综平均和时间平均均方位移, 来说明该过程的反常非遍历的动力学行为. 我们发现 \alpha_\pm 的大小关系决定两个状态在长时间之后的占据比例, 这对均方位移起着至关重要的作用. 根据均方位移的一般表达式, 我们解释了双状态随机过程的固有特性, 并通过六种不同的情况具体分析. 在第五章中, 我们仍然讨论上一章的非齐次双状态过程, 但是重点在于它们的强反常扩散行为. 在复杂的物理和生物系统中人们可以经常观测到强反常扩散现象, 该现象的特征是绝对值的分数阶矩&lt|x|q&gt具有非线性的谱 q\nu(q). 我们仍然假设两个状态为列维行走和布朗运动, 且其逗留时间均为幂律分布, 指数为 \alpha_\pm\in(0,2). 通过细致的尺度分析, 我们发现该系统存在三种不同的尺度. 与单状态的列维行走不同, 在双状态过程中, 其指数 \alpha_+&lt1 也会产生强反常扩散现象. 当 1&lt\alpha_+&lt2 时, 中间部分的概率密度函数由于布朗阶段停留时间较长, 变成了拉伸的列维分布和高斯分布的组合, 而尾部(弹道尺度)的概率密度函数仍然由列维行走的无限密度所主导. 在第六章中, 我们对本文的研究内容和主要结论做了一个简要的总结, 并对毕业后将要研究的问题进行展望. 英文摘要 Transport processes, especially diffusion processes, are ubiquitous in both physical and biological systems. Brownian motion (normal diffusion) has been playing the dominating role in the field of diffusion for long times. In recent decades, however, non-Brownian motion (anomalous diffusion) was found in great quantities in experiments, and increasingly shook the absolute dominance of the Brownian motion. Brownian motion has the property of ergodicity which implies that the time average is equal to the ensemble average. However, a large number of anomalous diffusion processes are found to be non-ergodic, different from Brownian motion. Further, in studying the intrinsic mechanism of anomalous and non-ergodic diffusion, it is found that the motion of particles is not as simple as described by Brownian motion. In stead, they are often influenced by heterogeneous environments or their motion patterns are inhomogeneous (having different states or phases). In this dissertation, we will discuss the anomalous diffusion and non-ergodic behavior of these stochastic processes and their applications, such as Feynman-Kac equation, occupation time and first-passage time. The specific contents of this dissertation are as follows: In chapter one, we briefly introduce the research background of anomalous diffusion and non-ergodic behavior. Then we introduce the structure of this paper and briefly analyze the contents of each chapter from the perspective of research object, research method and innovation. Finally, for the convenience of discussing the problems in this dissertation, we make a certain degree of elaboration on some basic anomalous diffusion models and research methods involved in this dissertation. In chapter two, we describe the heterogenous diffusive process under the effect of an external force by an overdamped Langevin equation, and derive the corresponding Feynman-Kac equation, which governs the probability density function (PDF) of the path functional. The functionals have diverse applications in physics, mathematics, hydrology, economics, and other fields. Under the framework of a continuous-time random walk, the Feynman-Kac equations, including those of the paths of stochastic processes of normal diffusion, anomalous diffusion, and even diffusion with reaction, have been derived. Sometimes the stochastic processes in physics and chemistry are naturally described by Langevin equations. The Langevin picture has the significant advantage of studying the dynamics with an external force field and analyzing the effect of noise resulting from a fluctuating environment. Based on the Langevin equation and the technique of Fourier transform, we derive the forward and backward Feynman-Kac equations. For the newly built equations, their applications in solving the PDFs of the occupation time and area under the trajectory curve are provided, and the results are confirmed by simulations. In chapter three, we still consider the heterogeneous diffusion process affected by external forces in last chapter, but focus on the ergodic and non-ergodic behavior of the process. Since our main concern is the large-x behavior for long times, the diffusivity and potential are, respectively, assumed as the power-law forms D(x) = D0|x|\alpha and U(x) = U0|x|\beta for simplicity. Based on the competition roles played by D(x) and U(x), three different cases, \beta&gt\alpha, \beta=\alpha, and \beta&lt\alpha, are discussed. The system is ergodic for the first case \beta&gt\alpha, where the time average agrees with the ensemble average. By contrast, the system is non-ergodic for \beta&lt\alpha, where the relation between time average and ensemble average is uncovered by infinite-ergodic theory. For the middle case \beta=\alpha, the ergodic property depends on the prefactors D0 and U0, becoming more delicate. The PDFs of the time averaged occupation time for three different cases are also evaluated and confirmed by simulations. In chapter four, we mainly discuss the anomalous and non-ergodic behaviors of a class of inhomogeneous two-state stochastic processes. With the rich dynamics studies of single-state processes, the two-state processes are attracting more interest, since they are widely observed in complex system and have effective applications in diverse fields, such as foraging behavior of animals. This chapter builds the theoretical foundation of the process with two states: L\'{e}vy walk and Brownian motion, which have been proved to be an efficient intermittent search process. The sojourn time distributions in two states are both assumed to be heavy-tailed with exponents \alpha_\pm\in(0,2). The dynamical behaviors of this two-state process are obtained through analyzing the ensemble-averaged and time-averaged mean-squared displacements (MSDs) in weak and strong aging cases. It is discovered that the magnitude relationship of \alpha_\pm decides the fraction of two states for long times, playing a crucial role in these MSDs. According to the generic expressions of MSDs, some inherent characteristics of the two-state process are detected and discussed in detail for six different cases. In chapter five, we continue to discuss the inhomogeneous two-state processes of the previous chapter, but focus on their strong anomalous diffusion behaviors, which are often observed in complex physical and biological systems, and characterized by the nonlinear spectrum of exponents q\nu(q) by measuring the absolute q-th moment &lt|x|q&gt. We still assume that the two states are L\'{e}vy walk and Brownian motion with their sojourn times obeying the power law distributions with exponents \alpha_\pm\in(0,2). Detailed scaling analyzes are performed for the coexistence of three kinds of scalings in this system. Different from the pure L\'{e}vy walk, the phenomenon of strong anomalous diffusion can be observed for this two-state process even when the exponent of power law distribution at L\'{e}vy walk phase satisfies \alpha_+&lt1. When 1&lt\alpha_+&lt2, the PDF in the central part becomes a combination of stretched L\'{e}vy distribution and Gaussian distribution due to the long sojourn time in Brownian phase, while the PDF in the tail part (in the ballistic scaling) is still dominated by the infinite density of L\'{e}vy walk. In chapter six, we make a brief summary of the research contents and main conclusions of this dissertation. A prospect of the problems to be studied after graduation are provided finally. 页数 157 URL 查看原文 语种 中文 文献类型 学位论文 条目标识符 https://ir.lzu.edu.cn/handle/262010/463621 Collection 数学与统计学院
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