Diffusion and Kinetics of Reaction Models: Theory and Selected Applications

本文将研究扩散限制反应模型和我们最初介绍的处理聚并问题的空区间法A + A-A。 我们把区间方法推广到湮灭、A + A - 0、分支湮灭和双分支湮灭过程以及T=0极限下的q态Potts模型。不同的研究小组最近采用了区间方法来分析不适合于自由Majorana费米子互补方法的模型。研究将进行到几个抽象的开放性问题，包括具有反常亚扩散输运的反应动力学;精确可解模型的非普遍动力学相变;和屏蔽效应的聚结，作为更精确的解决方案和新的近似方法的基础。还将对我们迄今为止的工作所激发的一些实际问题进行研究。 其中包括：（1）Fokas提出的求解凸域上任意边界条件下偏微分方程的新方法;（2）模拟反应扩散系统的快速计算机算法;（3）网络（如Internet）在面对随机故障或协同攻击时的稳定性;（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。 该项目特别适合教育目的。一个学生已经完成了物理学硕士学位，另一个即将获得博士学位。一些本科生已经参与其中（主要是计算机模拟），他们的努力已经在参考期刊上发表了至少三篇论文。 我们期待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.