U.S.-Brazil Cooperative Research on Interacting Particle Systems and Hydrodynamic Limits

该奖项支持以罗格斯大学Joel Lebowitz为首的一组美国科学家与五位巴西数学家在相互作用粒子系统的数学分析方面的合作研究：E.D.Andjel，P.A.Ferrari，圣保罗大学的J.A.Galves和R.H.Schonmann以及里约热内卢CNPq的Matematica Pura e Aplicada(IMPA)的M.A.瓦雷斯。研究人员将继续并加强长达五年的合作，分析相互作用的粒子系统及其流体动力学极限，这是数学和物理领域的一个活跃研究领域。这些限制将相互作用的粒子系统与流体力学偏微分方程式联系起来，对理解这些方程式是有用的。这次合作卓有成效，发表了至少11篇关于相互作用粒子系统的论文。这一领域的最初动机来自随机力学，其目标是使用概率模型来研究具有经典平衡态的系统的时间演化。这样的模型自然而然地出现了，例如，作为对吉布斯测量进行蒙特卡洛模拟的算法。很快就清楚了，同样的技术也可以应用于生物学、生态学和行为科学的模型。研究人员未来的工作将集中在获得关于相互作用系统的更多信息，包括将微观和宏观波动联系起来，将先前的结果扩展到预期具有不寻常标度的大偏差的情况，并将这些结果应用于更一般的设置，例如，粒子在确定性动力学和相互作用的布朗运动或随机游动下运动。

This award supports cooperative research in the mathematical analysis of interacting particle systems between a group of U.S. scientists headed by Joel Lebowitz of Rutgers University and five Brazilian mathematicians: E.D. Andjel, P.A. Ferrari, J.A. Galves and R.H. Schonmann of the Universidade de Sao Paulo and M.A. Vares of the Instituto de Matematica Pura e Aplicada (IMPA) of CNPq in Rio de Janeiro. The researchers will continue and strengthen a five year long collaboration in the analysis of interacting particle systems and their hydrodynamical limits, an active field of research in mathematics and physics. These latter limits relate interacting particle systems to hydrodynamical partial differential equations and are useful in understanding those equations. This collaboration has been very fruitful, resulting in the publication of at least eleven papers on interacting particle systems. The original motivation for this field arose from stochastic mechanics where the objective was to use probabilistic models in the study of the temporal evolution of systems with classical equilibrium states. Such models appear naturally, for example, as algorithms for making Monte Carlo simulations of Gibbs measures. It soon became clear that the same techniques could also be applied to models from biology, ecology, and the behavioral sciences. Future work by the researchers will center on obtaining more information about interacting systems, including relating microscopic and macroscopic fluctuations, extending previous results to the case where large deviations with unusual scalings are expected, and applying these results to more general settings, for example, particles moving under deterministic dynamics and interacting Brownian motions or random walks.