典型异常扩散网格蒙特卡洛仿真算法及精度分析

Anomalous diffusion observed in numerous physical, chemical and biological systems in recent years turns out to be quite ubiquitous which is characterized by a nonlinear behavior for the mean square displacement as a function of time. For anomalous diffusion described by fractional partial differential equation, lattice Monte Carlo (LMC) simulation is an important and effective method when it is difficult to get analytical solutions or necessary to track the trajectory of particles. This proposal suggests to study three typical anomalous diffusions which are widely used in real applications, i.e., the Galilei-invariant fractional diffusion-drift equation, the Galilei-variant fractional diffusion-drift equation, and the modified fractional diffusion equation with two time scales. The first task is to derive the analytical solutions with different initial and boundary conditions, the first passage time distributions (FPT) and the corresponding Laplace transforms. Then the LMC simulation algorithms can be designed and developed based on the theory of continuous time random walk (CTRW). The study attempts to determine if there exists a separable CTRW model with a fixed lattice step in its structure function that is equivalent to the anomalous diffusion distribution in the sense of finite order moments or discrete integral transformations. Both the macroscopic and the microscopic accuracy of the LMC simulation algorithm will be analyzed and verified quantitatively by means of the difference between the higher order moments and the distribution functions respectively with the help of stochastic simulation and numerical calculation. The goal is to reveal partly the microscopic mathematical and physical mechanism and the very nature of the typical anomalous diffusion as well as to provide rigorous mathematical theory and core algorithm for the application of stochastic simulation with high accuracy.

近年来在众多物理、化学、生物系统中观察到的异常扩散已相当普遍，其特征是一个随时间非线性变化的均方位移。对于分数偏微分方程描述的异常扩散，当难以解析求解或明确需要进行粒子追踪时，网格蒙特卡洛仿真就成了一种重要而有效的方法。项目以Galilei可变分数扩散-漂移方程、Galilei不变分数扩散-漂移方程、双时间尺度改进分数扩散方程三种应用较为广泛的典型异常扩散为研究对象，推导不同初始边界条件下的解析解和首次通过时间分布及Laplace变换，基于连续时间随机游走理论设计网格蒙特卡洛仿真算法，探索解决与异常扩散分布的有限阶矩或离散积分变换等价的具有固定网格步长结构函数的可分连续时间随机游走模型的存在判定问题，结合随机仿真和数值计算，通过高阶矩和分布函数差异从宏观和微观上定量分析验证算法精度，从一定程度上揭示典型异常扩散的微观数学物理机制和规律，为高精度随机仿真应用提供严格数学理论和核心算法支持。

本项目以三种典型异常扩散的网格蒙特卡洛仿真算法设计和精度分析为目标，具体开展异常有偏扩散首次通过时间分布的Laplace变换和解析解、基于Galilei分数扩散-漂移方程的异常扩散网格蒙特卡洛仿真算法设计、基于五转移概率的固定时间网格蒙特卡洛算法设计、网格蒙特卡洛仿真算法精度、以及异常扩散网格蒙特卡洛算法在核生化扩散中的应用等几个方面的研究工作。.采用分离变量法和H函数推导得出基于Galilei分数扩散-反映方程的异常扩散首次通过时间分布的Laplace变换和解析解。研究发现单边和双边吸收边界条件下异常扩散的首次通过时间分布之间的关系决定了首次通过时间条件分布。而且，首次通过时间条件分布的比例取决于Péclet数。根据求出的异常扩散首次通过时间分布Laplace变换和解析解，设计了网格蒙特卡洛仿真算法，很好地复现粒子扩散的平均位移和均方位移。研究发现二项运动位移导致的误差呈现一种幂律分布的变化形式。为提高网格蒙特卡洛仿真算法精度，设计了基于五个转移概率的网格蒙特卡洛仿真算法，推导出了转移概率的解析表达式。研究发现五个转移概率的网格蒙特卡洛仿真算法能够准确地复现扩散的前三阶矩。提出了使用偏度来测量网格蒙特卡洛仿真算法精度的方法。研究发现偏度与仿真步长的平方根成反比，并且第一个步长误差取决于Péclet数。.项目研究形成的算法和模型应用于核生化扩散仿真中，有效支撑核生化扩散应急管理与控制研究。

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