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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611211
• 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而多样化的数学世界的基础，包括诸如自我避免行走和相关模型之类的晶格模型，组合数学中的定向路径模型，渗透以及包括蒙特卡洛方法和传递矩阵方法在内的数值方法。这些模型在统计力学和相变理论中无处不在，这使该领域跨越了严格和应用的统计力学，组合数学，概率理论和数学物理学。与实验聚合物物理学有联系，因为模型中的缩放和相行为与聚合物的物理特性有关，例如，聚合物中的吸附转变可以通过吸附的自我避免自我的步行模型来建模。 我在聚合物熵问题方面进行的研究计划特别与晶格模型的严格分析，数值模拟有关，以检查模型中的相位行为，并通过对结的晶格多边形进行建模以及精确的解决方案以及对定向晶格路径模型的渐近解决方案进行建模。 我建议在每个不同领域中继续进行活动。在数值工作中，我们的GARM和GAL MONTE CARLO算法的构建为我们提供了一种使用微型人体采样的有效方法。这种采样在结熵问题中有效，我将继续在这方面工作。我的直接目标是打结的环聚合物晶格模型的热力学特性。 GARM和气体算法在采样晶格步道和多边形方面也有效，我们最近计算了方形晶格多边形附近的熵压力场，确定了压力场的缩放特性。在每种情况下，将我们的方法应用于相关的步行，晶格动物和树木的相关模型，以检查压力及其缩放特性，这是一个直接的目标。 与合作者在定向路径模型上的最新工作使我们能够确定定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，包括部分定向的路径。这些模型需要越来越复杂的方法，并且证明求解功能复发的内核方法非常有用。我的目的是通过将内核方法应用于更通用的模型，包括相互作用的有向囊泡，以洞悉定向模型的数学属性，相图和缩放，以提高这一领域。 诸如原子力显微镜之类的技术使吸附聚合物的拉力使得有可能。我们通过对外部拉动力进行了吸附的自我避免自我的步行来对此进行建模。我们证明了模型中的热力学极限，并通过检查将弹道相与吸附相分开的相边界的渐近形状，证明了重新进入的现象发生在该模型中。一个直接的目标是将我们的结果扩展到具有平行力或与吸附表面一定角度的力的模型。 总体而言，我的研究计划跨越了数值工作的领域（蒙特卡洛方法的离散晶格对象方法），具有精确的组合方法（定向晶格路径模型及其精确解决方案的变体）以及与自我避免步行有关的相互作用模型的严格方法。一个重要的额外兴趣是结熵问题，特别是打结的格子多边形的熵（这些是打结的聚合物的模型，并且通常与DNA分子中的打结有关）。