Applications of Dynamical Systems to Statistical Physics, Geometry, and Population Biology/Demography

这项建议是研究动力系统在统计物理、几何和种群生物学/生态学中的几个应用。(I)压强和自由能是晶格自旋系统统计物理的两个基本研究对象。然而，即使对于最简单的晶格自旋系统，关于自由能捕获的微观势能的信息也是微妙的，人们对此知之甚少。PI已经启动了一项计划，研究某些一维晶格自旋系统的Holder连续势的自然类是否或在多大程度上由它们的自由能决定。我们还计划研究自由能刚性与谱几何和数论中令人着迷的刚性问题之间惊人的相似性。(Ii)对正曲线流形上的测地线流的动力学知之甚少。PI计划继续研究测地线流的正曲率和复杂动力学之间的关系。(Iii)PI已经启动了一个程序来系统地研究非线性Leslie模型的全局动力学和分叉，其中生育率和存活概率具有作为种群大小的函数的各种自然函数形式。(I)压力和自由能是晶格自旋系统统计物理中的两个基本研究对象。晶格自旋系统在统计物理、凝聚态物理和化学中提供了一系列重要且具有启发性的模型。例如，相变对应于自由能的某些导数的不可微性。然而，即使对于最简单的晶格自旋系统，关于自由能捕获的微观势能的信息也是微妙的，人们对此知之甚少。PI已经启动了一项计划，研究一维晶格系统的势是否或在多大程度上由它们的自由能决定。我们希望这项工作将为这个重要而神秘的数量提供新的见解。(2)目前使用的所有人口和动物种群模型基本上都是基于线性Leslie模型。许多人口生物学家、生态学家和人口学家现在都在寻找非线性人口模型来进行更准确的人口预测。PI已经启动了一个项目来系统地研究非线性Leslie模型的全局动力学和分支，其中生育率和生存概率具有各种自然函数形式作为种群大小的函数。我们的最终目标之一是创建一个“种群建模工具箱”，它可以被广泛的种群建模人员用来更准确地预测动物种群。

ABSTRACTWeissThis proposal is to study several applications of dynamical systems tostatistical physics, geometry, and population biology/ecology. (i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, natural classes of Holder continuous potentials for certain one-dimensional lattice spin systems are determined by their free energy. We also plan to investigate striking similarities between the rigidity of free energy and fascinating rigidity problems in spectral geometry and number theory. (ii) Little is known about thedynamics of the geodesic flow on positively curved manifolds. The PIplans to continue studying the relations between positive curvature andcomplicated dynamics of the geodesic flow. (iii) The PI has started aprogram to systematically study the global dynamics and bifurcations fornonlinear Leslie models where the fertility rates and survivalprobabilities have various natural functional forms as functions of thepopulation size.(i) The pressure and free energy are the two fundamental objects of study in the statistical physics of lattice spin systems. Lattice spin systems provide an important and illuminating family of models in statistical physics, condensed matter physics, and chemistry. For instance, phase transitions correspond to non-differentiability for some derivative of free energy. However, even for the simplest lattice spin systems, the information about the microscopic potential that the free energy captures is subtle and poorly understood. The PI has started a program to study whether, or to what extent, potentials forone-dimensional lattice systems are determined by their free energy. Wehope this work will provide new insights into this important, yetmysterious, quantity. (ii) Essentially all demographic and animalpopulation models in current use are based on the linear Leslie model. Many population biologists, ecologists, and demographers are now lookingto nonlinear population models for more accurate population forecasting. The PI has started a program to systematically study the global dynamicsand bifurcations for nonlinear Leslie models where the fertility rates andsurvival probabilities have various natural functional forms as functionsof the population size. One of our ultimate goals is to create a``population modeling toolbox'' which could be used by a wide range ofpopulation modelers to more accurately predict animal populations.