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，如自避免行走和相关模型，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无处不在，使得这一领域跨越了严格的应用统计力学、组合数学、概率论和数学物理。 也有一个连接到实验聚合物物理学，因为模型中的缩放和相行为与聚合物的物理性质，例如，在聚合物中的吸附过渡可以模拟的吸附自避免行走模型。 我的研究计划到方面的聚合物熵的问题，特别是涉及到严格的分析晶格模型，数值模拟研究相行为的模型和收集数据的结熵模型打结晶格多边形，和精确的解决方案和渐近分析定向晶格路径模型。 我建议继续在这些不同领域开展活动。 在数值计算中，我们构造的GARM和GAS蒙特卡罗算法为我们提供了一种有效的方法来使用微正则采样。 这种采样在纽结熵问题中是有效的，我将继续在这一领域工作。 我的近期目标是打结环聚合物晶格模型的热力学性质。 GARM和GAS算法在采样网格行走和多边形方面也很有效，我们最近计算了正方形网格多边形附近的熵压力场，确定了压力场的缩放特性。 这是一个直接的目标，我们的方法应用到相关模型的步行，格子动物和树木，在每种情况下检查的压力和它的缩放特性。 最近的工作与合作者对有向路径模型使我们能够确定附近的压力场的缩放定向lattice路径。这项工作正在进行中，我们正在研究更一般的模型，包括部分有向路径。 这些模型需要更复杂的方法，和核方法解决功能递归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。 诸如原子力显微镜之类的技术使吸附的聚合物受到拉力成为可能。 我们已经模拟了这一点受到吸附自我回避步行到外部拉力。 我们证明了在该模型中存在的热力学极限，并通过检查分离的弹道相从吸附相的相边界的渐近形状，证明了再进入的现象发生在这个模型中。 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角度的力的模型。 总的来说，我的研究计划跨越了数值工作（离散晶格对象的蒙特卡罗方法），精确组合方法（定向晶格路径模型及其精确解的变体）以及与自回避行走相关的交互模型的严格方法。 一个重要的额外兴趣是结熵问题，特别是打结的晶格多边形的熵（这些是打结聚合物的模型，通常与DNA分子中的打结有关）。