The Statistical Mechanics and Combinatorics of Self-Avoiding Walk and Directed Path Models of Polymers

聚合物自回避行走和有向路径模型的统计力学和组合学

• 批准号: RGPIN-2014-04731
• 负责人: JansevanRensburg, Esaias
• 金额: $ 1.02万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587983
• 关键词: Statistical; Mechanics; Combinatorics; Self; Avoiding

The polymer entropy problem underlies a rich and varied mathematical world, including lattice models such as the self-avoiding walk and related models, directed path models in combinatorial mathematics, percolation, as well as numerical methods including Monte Carlo methods and transfer matrix approaches. The models are ubiquitous in statistical mechanics and in the theory of phase transitions, making this field one which straddles rigorous and applied statistical mechanics, combinatorial mathematics, probability theory, and mathematical physics. There is also a connection to experimental polymer physics because scaling and phase behaviour in the models are related to the physical properties of polymers, for example, the adsorption transition in a polymer can be modelled by an adsorbing self-avoiding walk model. My research program into aspects of the polymer entropy problem relates in particular to the rigorous analysis of lattice models, numerical simulation to examine phase behaviour in the models and to collect data on knot entropy by modelling knotted lattice polygons, and the exact solution and asymptotic analysis of directed lattice path models. I propose to continue my activity in each of these different areas. In numerical work our construction of GARM and GAS Monte Carlo algorithms gave us an efficient way to use microcanonical sampling. This sampling is effective in the knot entropy problem, and I will be continuing to work in this area. My immediate goals are the thermodynamic properties of lattice models of knotted ring polymers. The GARM and GAS algorithms are also efficient at sampling lattice walks and polygons, and we recently computed the entropic pressure field near a square lattice polygon, determining the scaling properties of the pressure field. It is an immediate goal to apply our methods to related models of walks, lattice animals and trees, in each case examining the pressure and its scaling properties. Recent work with collaborators on directed path models enabled us to determine the scaling of the pressure field near a directed lattice path. This work is ongoing, and we are working on more general models, including partially directed paths. These models require ever more sophisticated methods, and the kernel method for solving functional recurrences have proven very useful. My aim is to advance this field by applying the kernel method to more general models, perhaps including interacting directed vesicles, to gain insight in both the mathematical properties, phase diagrams, and scaling of directed models. Techniques such as atomic force microscopy makes it possible to subject adsorbed polymers to pulling forces. We have modelled this by subjected an adsorbing self-avoiding walk to an externally pulled force. We proved existence of a thermodynamic limit in the model, and by examining the asymptotic shape of a phase boundary separating a ballistic phase from an adsorbed phase, proved that the phenomenon of re-entrance occurs in this model. An immediate goal is to extend our results to models with forces pulling parallel or at an angle to the adsorbing surface. Overall my research program straddles areas of numerical work (Monte Carlo methods of discrete lattice objects), with exact combinatorial methods (variants of directed lattice path models and their exact solutions) and with rigorous approaches to interacting models related to the self-avoiding walk. An important additional interest is the knot entropy problem, in particular the entropy of knotted lattice polygons (these are models of knotted polymers, and are often related to knotting in DNA molecules).

聚合物的熵问题是一个丰富多彩的数学世界，包括格子模型，如自回避行走及其相关模型，组合数学中的有向路径模型，渗流，以及数值方法，包括蒙特卡罗方法和传递矩阵方法。这些模型在统计力学和相变理论中普遍存在，使这一领域成为一个跨越严谨和应用的统计力学、组合数学、概率论和数学物理的领域。这也与实验聚合物物理有关，因为模型中的比例和相行为与聚合物的物理性质有关，例如，聚合物中的吸附转变可以用吸附自回避行走模型来模拟。 我对聚合物熵问题各方面的研究特别涉及晶格模型的严格分析、检验模型中相行为的数值模拟以及通过模拟结晶格多边形来收集关于结点熵的数据，以及有向晶格路径模型的精确解和渐近分析。 我建议继续在这些不同领域中的每一个领域开展活动。在数值工作中，我们构造的GARM和GAS蒙特卡罗算法为我们提供了一种使用微正则抽样的有效方法。这种采样在节点熵问题中是有效的，我将继续在这一领域工作。我目前的目标是研究打结的环聚合物晶格模型的热力学性质。 GARM和GAS算法在采样格子游动和多边形方面也很有效，最近我们计算了正方形格子多边形附近的熵压力场，确定了压力场的标度性质。一个直接的目标是将我们的方法应用于步行、晶格动物和树木的相关模型，在每种情况下都检查压力及其标度性质。 最近与合作者在有向路径模型上的工作使我们能够确定有向晶格路径附近压力场的标度。这项工作正在进行中，我们正在研究更一般的模型，包括部分定向路径。这些模型需要更复杂的方法，而用于求解函数递归的内核方法已被证明非常有用。我的目标是通过将核方法应用于更一般的模型来推进这一领域，可能包括相互作用的定向囊泡，以获得对定向模型的数学属性、相图和比例的洞察。 原子力显微镜等技术使吸附的聚合物受到拉力成为可能。我们对此进行了建模，将一种吸附的自我回避行走置于外部拉力之下。我们证明了模型中热力学极限的存在性，并通过检验弹道相与吸附相之间的相界的渐近形状，证明了该模型中存在再入现象。一个直接的目标是将我们的结果扩展到与吸附表面平行或成一定角度拉力的模型。 总体而言，我的研究计划跨越了数值工作的领域(离散晶格对象的蒙特卡罗方法)，使用精确的组合方法(有向晶格路径模型的变体及其精确解)，以及与自我回避行走相关的模型交互的严格方法。另一个重要的额外兴趣是结熵问题，特别是打结晶格多边形的熵(这些是打结聚合物的模型，通常与DNA分子中的打结有关)。