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: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566706
• 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).

聚合物熵问题是一个丰富多样的数学世界的基础，包括晶格模型，如自回避行走和相关模型，组合数学中的定向路径模型，渗透，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无处不在，使其成为一个跨越严格和应用统计力学、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学有关，因为模型中的缩放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通过吸附自我避免行走模型来模拟。我的研究项目涉及到聚合物熵问题的各个方面，特别是与晶格模型的严格分析，数值模拟来检查模型中的相行为，并通过模拟打结的晶格多边形来收集结熵的数据，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开展活动。在数值计算中，GARM和GAS蒙特卡罗算法的构建为我们提供了一种使用微规范采样的有效方法。这种抽样在结熵问题中是有效的，我将继续在这个领域工作。我的直接目标是研究结环聚合物晶格模型的热力学性质。GARM和GAS算法在采样点阵行走和多边形方面也很有效，我们最近计算了正方形点阵多边形附近的熵压场，确定了压力场的缩放特性。将我们的方法应用于步行、格子动物和树木的相关模型是一个直接的目标，在每种情况下检查压力及其缩放特性。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缩放。这项工作正在进行中，我们正在研究更通用的模型，包括部分定向路径。这些模型需要更复杂的方法，而解函数递归的核方法已被证明是非常有用的。我的目标是通过将核方法应用于更一般的模型（可能包括相互作用的定向囊泡）来推进这一领域，以深入了解定向模型的数学性质、相图和缩放。原子力显微镜等技术使使吸附聚合物受到拉力的作用成为可能。我们通过使吸附式自我避免行走受到外部拉力来模拟这一点。我们证明了模型中热力学极限的存在性，并通过检验分离弹道相与吸附相的相边界的渐近形状，证明了模型中存在重入现象。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的力的模型。总体而言，我的研究项目跨越数值工作领域（离散晶格对象的蒙特卡罗方法），精确组合方法（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交互模型的严格方法。一个重要的附加兴趣是结熵问题，特别是结晶格多边形的熵（这些是结聚合物的模型，并且通常与DNA分子中的结有关）。