RUI: Extremal Paths in Complex Systems

0083204 Halpin-Healy本科研究机构（RUI）的研究资助为随机介质（DPRM）中定向聚合物的统计力学和杂质冲击磁体中的畴壁的理论研究提供支持。 由于共同的技术方面，DPRM已被命名为一个婴儿版的自旋玻璃问题。 因此，T=0 DPRM是一个全局优化的问题，表现出超度量树结构，让人想起那些发现在河流三角洲盆地，毛细血管网络，在大脑中的神经元阵列。 Hopf-Cole变换将DPRM映射到Kardar-Parisi-Zhang（KPZ）随机增长模型的远离平衡态动力学。 最后的回报如下从血液关系的KPZ和嘈杂的伯格斯方程，后者有关的非平衡动力学驱动的晶格气体。 目前的一个重要目标是进一步探索这种丰富的KPZ三巨头：DPRM，动力学粗糙化和驱动晶格气体。 然而，随着DPRM成为一个经典的非凝聚态问题，未来的研究将集中在各种各样的，关于几何复杂性的一般问题上，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。该研究在本科院校（RUI）的补助金提供了对定向聚合物在随机介质（DPRM）和畴壁在杂质打击磁铁的统计力学的理论研究的支持。 这些统计模型有深远的应用，除了其内在的根本利益。 这项研究涉及广泛的本科生参与 *。

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.***