Diffusion and Kinetics of Reaction Models: Theory and Selected Applications

我们将对我们最初引入的处理合并的扩散限制反应模型和空区间方法A+A-A进行研究。我们将区间方法推广到湮灭，A+A-0，分支和双分支-湮没过程，以及T=0极限下的Q态Potts模型。该方法提供了详细的信息：我们导出了所有n的合并和湮灭的n点关联函数的显式表达式。最近，不同的研究小组采用区间法来分析不符合Free Majorana费米子补充方法的模型。研究将对几个抽象的开放问题进行研究，包括具有反常的次扩散输运的反应动力学；非普遍动力学相变的精确可解模型；以及合并中的屏蔽效应，作为更精确解和新的近似方法的基础。到目前为止，还将对我们的工作所激发的一些实际问题进行研究。其中包括：(1)基于Fokas的求解具有任意边界条件的凸域中的偏微分方程组的新方法；(2)用于模拟反应扩散系统的快速计算机算法；(3)面对随机故障或协同攻击时网络(如因特网)的稳定性；(4)真核细胞中的信号转导机制；(5)当代化学中各种实验的统计物理解释。该项目特别适合用于教育目的。一名学生获得了理科硕士学位。一个是物理学专业，另一个是博士学位毕业。一些本科生已经参与了(主要是计算机模拟)，他们的努力已经导致至少三篇论文发表在被引用的期刊上。我们预计会有更多的机会让学生参与进来。

Research will be conducted into models of diffusion-limited reactions andthe method of empty intervals which we introduced originally for the treatment of coalescence, A + A - A. We have extended the method of intervals to annihilation, A + A - 0, the branching- and double branching-annihilating process, and the q-state Potts model in the T=0 limit. The method yields detailed information: we derived explicit expressions for the n-point correlation functions of coalescence and annihilation, for all n.Different research groups have recently adopted the method of intervals for the analysis of models not amenable to the complementary method of freeMajorana fermions.Research will be conducted into several abstract open questions, includingthe kinetics of reactions with anomalous, subdiffusive transport; exactly solvable models of nonuniversal kinetic phase transitions; and the shielding effect in coalescence, as a basis for more exact solutions and for novel approximation methods. Research will be conducted also into a number of practical questions motivated by our work so far. These include: (1) A new method, due to Fokas, forsolving PDEs in convex domains with arbitrary boundary conditions; (2) Speedy computer algorithms for the simulation of reaction-diffusion systems; (3) Stability of networks, such as the Internet, in the face of random failure or a coordinated attack; (4) Mechanisms of signal transduction in eukaryotic cells; and (5) Statistical physics interpretation of various experiments in contemporary chemistry. The project is particularly well-suited for educational purposes. One student has completed a M.Sc. in physics and another is graduating with a Ph.D.degree. Several undergraduate students have already been involved (mainly with computer simulations), and their effort has resulted in at least threepublications in refereed journals. We forsee many more opportunities for students involvement.