Mathematical Sciences: Interacting Particle Systems

本研究涉及相互作用粒子系统在生物学上的应用。特别是，研究者将构建模型来解释生物现象，如麻疹的时间振荡和海洋中浮游生物的斑块。另一个有趣的领域是与非线性偏微分方程的联系。通过快速混合粒子和缩放空间，系统收敛到一个非线性偏微分方程。这使得推导出更精确地模拟生物系统的偏微分方程成为可能，也证明了关于快速混合粒子系统的结果。相互作用的粒子系统具有一种结构，使它们适合于在物理、化学和生物学的各种环境中使用。有一个站点网格，每个站点都可以有一组固定的值。例如，一个城市的人可能对麻疹易感、感染或免疫。每个站点以依赖于其相邻站点的值的速率改变其值，本研究感兴趣的是确定随着时间的推移系统会发生什么变化。这些模型最终将有助于更好地理解全球生态问题。

This research involves applications of interacting particle systems to biology. In particular, the investigator will construct models to explain biological phenomena such as the temporal oscillations of measles and the patchiness of plankton in the sea. Another area of interest is the connection with nonlinear partial differential equations. By mixing the particles at a fast rate and scaling space, the system converges to a nonlinear partial differential equation. This makes it possible to derive partial differential equations that more accurately model the biological systems and also to prove results about the particle systems with fast mixing. Interacting particle systems have a structure that makes them appropriate for use in a variety of contexts in physics, chemistry, and biology. There is a grid of sites, each of which can have a fixed set of values. For example, a city of individuals can be susceptible to, infected with, or immune to measles. Each site changes its value at a rate that depends on the values of its neighbors, and this research is interested in determining what happens to the system over time. These models should eventually provide a better understanding of global ecological problems.