RUI: Extremal Paths in Complex Systems

本本科生研究基金为随机介质中定向聚合物（DPRM）的统计力学和含杂质磁体中的畴壁的理论研究提供支持。由于常见的技术方面，DPRM已被命名为自旋玻璃问题的婴儿版本。因此，T=0 DPRM是一个全局优化问题，表现出一种超测量树结构，让人想起那些在河流三角洲盆地、毛细血管网络和大脑中的神经元阵列中发现的结构。Hopf-Cole变换将DPRM映射到kardar - paris - zhang （KPZ）随机增长模型的远离平衡动力学。最后的结果来自KPZ和噪声Burgers方程的血缘关系，后者与驱动晶格气体的非平衡动力学有关。当前的一个重要目标是进一步探索丰富的KPZ三要素：DPRM、动力学粗化和驱动晶格气体。然而，随着DPRM成为非凝聚态物质的经典问题，未来的研究将集中在几何复杂性的各种一般问题上，包括地貌学、进化生物学和其他多学科统计力学背景下的极端轨迹和网络的缩放。本本科生研究基金（RUI）为随机介质中定向聚合物（DPRM）的统计力学和含杂质磁铁中的畴壁的理论研究提供支持。这些统计模型除了其内在的根本利益外，还具有深远的应用。这项研究涉及广泛的大学生参与

0083204Halpin-HealyThis grant for Research at an Undergraduate Institution (RUI) provides support for theoretical research on the statistical mechanics of directed polymers in random media (DPRM) and domain-walls in impurity-stricken magnets. Because of common technical aspects, the DPRM has been christened a baby-version of the spin-glass problem. As such, the T=0 DPRM is a matter of global optimization, exhibiting an ultrametric tree structure reminiscent of those found in river delta basins, capillary blood vessel networks, and neuronal arrays in the brain. A Hopf-Cole transformation maps the DPRM to the far-from-equilibrium dynamics of Kardar-Parisi-Zhang (KPZ) stochastic growth models. A final payoff follows from the blood relation of KPZ and noisy Burgers equations, the latter relevant to nonequilibrium kinetics of driven lattice gases. An important goal at hand is further exploration of this rich KPZ triumvirate: DPRM, kinetic roughening, and driven lattice gases. With the DPRM becoming a classic problem of ill-condensed matter, however, future research will focus on diverse, general issues concerning geometric complexity, including scaling of extremal trajectories and networks in geomorphology, evolutionary biology, and other multidisciplinary statistical mechanics contexts.%%%This grant for Research at an Undergraduate Institution (RUI) provides support for theoretical research on the statistical mechanics of directed polymers in random media (DPRM) and domain-walls in impurity-stricken magnets. These statistical models have far-reaching applications in addition to their intrinsic fundamental interest. The research involves extensive undergraduate participation.***