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

[摘要]魏斯：本提案旨在研究动力系统在统计物理、几何、种群生物学/生态学等领域的应用。(1)压力和自由能是晶格自旋系统统计物理学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，自由能捕获的微观势的信息也很微妙，而且很难理解。PI已经启动了一个项目，研究某些一维晶格自旋系统的Holder连续势的自然类别是否或在多大程度上是由它们的自由能决定的。我们还计划研究自由能的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之处。（ii）对正弯曲流形上测地线流动的动力学知之甚少。计划继续研究正曲率与测地线流复杂动力学之间的关系。（iii） PI已经开始系统地研究非线性Leslie模型的全局动力学和分岔，其中生育率和生存概率作为人口规模的函数具有各种自然函数形式。(1)压力和自由能是晶格自旋系统统计物理学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个重要的和有启发性的模型家族。例如，相变对应于某些自由能导数的不可微性。然而，即使对于最简单的晶格自旋系统，自由能捕获的微观势的信息也很微妙，而且很难理解。PI已经启动了一个项目来研究一维晶格系统的势能是否或在多大程度上是由它们的自由能决定的。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。（ii）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莱斯利模型。许多人口生物学家、生态学家和人口统计学家现在都在寻找更准确的人口预测的非线性人口模型。PI已经启动了一个项目，系统地研究非线性Leslie模型的全局动态和分岔，其中生育率和生存概率具有各种自然函数形式，作为人口规模的函数。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者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.