Diffusion and Kinetics of Reaction Models in Contemporary Statistical Physics

反应-扩散系统是多种自然和人为现象的基础，从自组织和模式形成（胚胎形态、动物皮毛上的条纹和点等）到化学工业中的异质催化、半导体中的电子-空穴重组、反应的异常动力学和活细胞内的信号转导、观点的传播或疾病在人群中的传播。物种间的生态竞争引起了当代统计物理学的极大兴趣。扩散限制制度具有挑战性，因为它主要是由所有长度尺度上的波动决定的，而且没有有效处理它的综合办法。提出了扩散控制过程的研究，如在线上的扩散限制聚并、A+A - A，并引入了空区间法对其进行分析。长期以来，聚结一直是一种易于分析的模型，它展示了许多非平衡系统的特征，包括异常动力学、有序和自组织、费雪锋面的传播和动力学相变。我们进一步将区间方法推广到零温度极限下的湮灭、分支湮灭行走和q态波茨模型。从那以后，它被其他研究小组所接受，用于研究不能用互补技术分析的模型，比如自由马约拉纳费米子的方法。将对几个相关问题进行研究，包括定向渗透的精确解，非平衡动力学相变的主要代表类；区间法在新反应方案中的进一步推广屏蔽效应及其在求解无穷阶跃PDE系统中的应用；反应锋面波动的作用及反应扩散平均场速率方程的适用性极端统计的恶性步行者；一种新的依赖历史的随机漫步模型也可以进行精确分析，具有许多潜在的应用。研究还将在许多实际应用中进行，例如用于模拟反应扩散系统的快速计算机算法，颗粒气体的动力学和线性混合下的演化，纳米孔通道中的扩散和缓慢的药物释放，大型复杂网络（互联网，社交网络等）的结构和功能，以及大脑区域之间的结构和连接的建模。提议的工作特别适合教育目的。PI在与研究生和本科生合作和出版方面有着良好的记录。在过去的四年中，有9名本科生和4名研究生参与了相关项目。所要求的大部分资金都专门用于培训他们。拟议的工作涉及与克拉克森大学、波士顿大学、洛斯阿拉莫斯国家实验室、圣地亚哥大学、格拉纳达大学和加泰罗尼亚政治技术大学（西班牙）、米纳斯吉拉斯州（巴西）、巴伊兰大学（以色列）、法国师范学院（法国）的同事以及他们的许多学生和博士后的国内和国际合作。私家侦探和他的合作者和学生在许多研讨会和会议上展示了他们的研究成果。我们最近的发现已经在IoP精选和物理网、物理评论焦点、自然科学更新、新闻简报和科学广播节目中出现。

Reaction-diffusion systems underlie diverse natural and man-made phenomena from self-organization and pattern formation (embryo morphology, stripes and dots in the coats of animals, etc.), to heterogeneous catalysis in the chemical industry, to electron-hole recombination in semiconductors, to the anomalous kinetics of reactions and signal transduction inside living cells, to the spread of opinions, or diseases in populations, to ecological competition between species and attract considerable interest in contemporary statistical physics. The diffusion-limited regime is challenging, as it is dominated by fluctuations at all length scales, and there exists no comprehensive approach to deal with it effectively. Research is proposed into diffusion-controlled processes, such as diffusion-limited coalescence on the line, A+A - A, and the method of empty intervals that we introduced for its analysis. Coalescence has long been an example of an analytically tractable model that exhibits many of the hallmarks of nonequilibrium systems, including anomalous kinetics, ordering and self-organization, propagation of Fisher fronts, and kinetic phase transitions. We have further extended the method of intervals to annihilation, branching annihilating walks, and the q-state Potts model in the zero-temperature limit. It has since been embraced by other research groups, for the study of models not amenable to analysis by complementary techniques, such as the method of free Majorana fermions. Research will be conducted into several related problems, including a chance at an exact solution of directed percolation, the main representative class of nonequilibrium kinetic phase transitions; further generalizations of the method of intervals to new reaction schemes; the shielding effect and its use for solving infinite hierarchical PDE systems; the role of fluctuations in reaction fronts, and the applicability of reaction-diffusion mean-field rate equations; extreme statistics of vicious walkers; and a new model of history-dependent random walks that can too be analyzed exactly, with numerous potential applications. Research will also be conducted into a number of practical applications, such as fast computer algorithms for the simulation of reaction-diffusion systems, kinetics of granular gases and evolution under linear mixing, diffusion in nanopore channels and slow drug release, structure and function of large complex networks (the Internet, social nets, etc.), and modeling the architecture and connections between regions of the brain. The proposed work is particularly well suited for educational purposes. The PI has a strong record of working and publishing with graduate and undergraduate students. Nine undergraduate and four graduate students have been involved in related projects in the past four years. The bulk of the requested funding is earmarked to their training. The proposed work involves national and international collaborations with colleagues from Clarkson University, Boston University, Los Alamos National Lab, the University of San Diego, the University of Granada and Universitat Polit`ecnica de Catalunya (Spain), Minas Gerais (Brazil), Bar-Ilan University (Israel), Ecole Normale (France), and their many students and postdocs. The PI and his collaborators and students have presented their results in numerous seminars and meetings. Our recent findings have been featured in IoP Select and Physics Web, Physical Review Focus, Nature Science Update, News in Brief, and in a science radio show.