Interactions and Information Transmission in Diffusing Agents

本项目福尔斯EPSRC“数学科学”的研究领域。传播事件通过一对个体之间的短程相互作用事件发生在微观尺度上。在一个复杂的系统中有许多个体，这些传播事件导致宏观尺度上的模式出现。例如，在社会生活的动物中共享食物位置，在搜索机器人之间传递信息或在人群中传播传染性病原体。这些相互作用的位置和频率本质上是随机过程。因此，统计物理学领域非常适合研究这种系统。通过利用该领域的最新发展-解决格子随机行走（LRW）中的100年历史问题[1] -现在可以确定随机移动个体对之间的传输概率[2]。通过这些进展，现在知道了很多关于d维，超立方，有界晶格中随机移动和相互作用的行为。然而，在其他不太标准的晶格结构上，关于运动和相互作用动力学还有很多东西要学。这些包括具有多个异质内部状态的晶格，定向晶格和其他二维镶嵌晶格-三角形和六边形。目前，许多相互作用代理的研究，特别是那些在非标准晶格结构中的研究依赖于随机模拟[3]，这通常是非常昂贵的计算。通过在非标准有界晶格中的随机步行者的占用概率的分析解决方案的进步，这个博士学位的目的是扩大域的范围，其中相互作用传输动力学的研究能够得到分析。这可能是通过一些方法来实现的，比如部署一种技术来处理晶格中的不均匀位置，缺陷技术[4]和图像方法[5]，一种从静电学中借用的技术，用于约束空间动力学。目前我的注意力在于获得六边形和三角形晶格中的解析解。通过扩展用于获得有界欧氏格中表示格随机游走占据概率、传播子或格绿色函数的解析公式的最新结果的技术，我能够获得Hénical域中以前未知的解析传播子。我通过使用所谓的Her坐标系统[6]对动态进行建模来做到这一点。这不仅是一个重要的理论进步-已知的分析结果建模六边形LRW都依赖于映射到标准欧几里得晶格[7] -传播在真正的有界六边形空间是感兴趣的，因为以前的计算研究[8]表明，一个种群的气味标记领土动物的领土结构自然形成了一个镶嵌的六边形晶格结构。这是由于牛分枝杆菌的传播，牛结核病（bTB）的病原菌在欧洲獾（Meles meles）种群中的传播，这一问题每年花费英国纳税人数百万英镑[9]。因此，有了一个分析框架，阐明了非标准晶格中的相互作用和传输动力学，人们就能够从一个全新的、计算成本低廉的角度来解决这些问题。[1]L. Giuggioli; Phys. Rev. X; 2020[2] L. Giuggioli，S. Sarvaharman;提交; 2022[3] T. Stutz，et.等; PLoS One; 2021[4] V. M. Kenkre;在内存功能，投影运算符和缺陷技术; 2021[5] E。Montroll，B. J. West，《统计力学研究：涨落现象》，1979年[6] I. Her; Proc ASME DTC; 1992[7] E.数学物理; 1968年[8] A.海布卢姆罗夫莱斯湖Giuggioli; Phys. Rev. E; 2018[9]英国政府; 2018

This project falls within the EPSRC 'Mathematical Sciences' research area.Transmission events occur on the microscopic scale through short-range interaction events between a pair of individuals. With many individuals in a complex system, these transmission events lead to the emergence of patterns on the macroscopic scale. Examples include the sharing of food locations in socially living animals, the passing of information between search robots or the spreading of infectious pathogens through a population. The locations and frequency of these interactions are inherently stochastic processes. As such, the field of statistical physics lends itself perfectly to the study of such systems. By exploiting the most recent developments in the field - the resolution of a 100-year old problem in lattice random walks (LRW) [1] - it is now possible to determine the transmission probability between pairs of randomly moving individuals [2]. Through these advances, much is now known about the behaviour of randomly moving and interacting agents in d-dimensional, hypercubic, bounded lattices. However, there is still much to learn about both movement and interactions dynamics on other, less standard lattices structures. These include lattices with multiple heterogenous internal states, directed lattices and other 2-dimensionally tessellating lattices - triangular and hexagonal.Currently, many studies of interacting agents, especially those in non-standard lattice structures rely on stochastic simulations [3], which are often highly computationally expensive. Through advancements in analytic solutions of occupation probabilities of a random walker in non-standard bounded lattices, this PhD aims to broaden the scope of domains in which studies of interaction transmission dynamics are able to be obtained analytically. This is likely to be achieved through methods such as the deployment of a technique to deal with inhomogeneous sites in a lattice, the defect technique [4] and the method of images [5], a technique borrowed from electrostatics used to bound spatial dynamics.Currently my attention lies in obtaining analytical solutions in hexagonal and triangular lattices. By extending techniques deployed to obtain state-of-the-art results for analytic formulas representing the Lattice Random Walk occupation probability, the propagator or Lattice Green's Function, in bounded Euclidean Lattices [1], I am able to obtain previously unknown analytic propagators in Hexagonal domains. I do this by modelling the dynamics using the so called Her's Co-ordinate system [6]. Not only is this a significant theoretical advance - known analytical results modelling hexagonal LRWs all rely on mappings to standard Euclidean lattices [7] - propagators in true bounded hexagonal space are of interest as previous computational studies [8] show that the territorial structure of a population of scent-marking territorial animals naturally forms a tessellated hexagonal lattice structure. This is pertinent due to the spread of Mycobacterium bovis, the causative bacterium of bovine Tuberculosis (bTB) within the European Badger (Meles meles) population, an issue costing the UK taxpayer millions of pounds every year [9]. Hence, with an analytic framework that elucidates interaction and transmission dynamics in non-standard lattices, one is able to tackle such problems from a brand new, computationally inexpensive angle.[1] L. Giuggioli; Phys. Rev. X; 2020[2] L. Giuggioli, S. Sarvaharman; Submitted; 2022[3] T. Stutz, et. al.; PLoS One; 2021[4] V. M. Kenkre; In Memory Functions, Projection Operators, and the Defect Technique; 2021[5] E. Montroll, B. J. West; In Studies in Statistical Mechanics: Fluctuation Phenomena, 1979[6] I. Her; Proc ASME DTC; 1992[7] E. Montroll; Math. Phys.; 1968 [8] A. Heiblum Robles, L. Giuggioli; Phys. Rev. E; 2018[9] United Kingdom Government; 2018