Mathematical Sciences: Studies of Exactly Solvable Models in Statistical Mechanics

9307816 PERK这是一个继续研究手征POTS模型、广义手征POTS模型和多组分手征POTS模型中许多未解决问题的研究项目。这些数学模型描述了可实验实现的体系中的相变，如分子在薄膜、多层体系、膜上的吸附等。尽管到目前为止在计算手征Potts模型的各种物理量方面取得了许多成功，但这些结果还没有得到充分的探索和利用。这个项目的目的是将准确的结果翻译成对物理学家更透明的语言，以便蒙特卡罗计算、高温和低温展开以及各种其他方法的专家可以使用它们。然后，准确的结果可以用来适当地衡量近似方法的误差，并解决长期存在的争议。将与我们的俄罗斯同事Yu.G.Stroganov合作，尝试获得新的三维可解模型。这项工作的进展可能会带来数学和物理中许多其他研究领域的相互作用，如量子群和纽结理论、单极子和弦理论、凝聚态物质中的相变以及生物物理学中的生物膜。！！A(Times New Roman Symbol&amp；Arial“h；%；%/Terri L.Hicks Terri L.Hicks

9307816 Perk This is a research project for the continuation of the study of many of the unsolved problems in the chiral Potts model, generalized chiral Potts models, and multi-component chiral Potts models. These mathematical models describe phase transitions in experimentally realizable systems such as adsorption of molecules on thin films, multi-layered systems, membranes, etc. In spite of many successes so far in calculating various physical quantities of the chiral Potts model, these results have not yet been fully explored and made use of. It is intended in this project to translate the exact results into language more transparent to physicists, so that experts in Monte Carlo calculations, high- and low-temperature expansions, and various other methods, can use them. The exact results can be then used to properly gauge the errors of the approximate methods and solve long-standing controversies. Attempts to obtain new solvable models in three dimensions will be made in collaboration with our Russian colleague Yu.G. Stroganov. Progress in this work may bring forth interaction between many other research areas in mathematics and physics such as quantum groups and knot theory, monopoles and string theory, phase transitions in condensed matter, and biomembranes in biophysics. ! ! A ( Times New Roman Symbol & Arial " h ; % ; % / terri l. hicks terri l. hicks