Stochastic Constitutive Models for Nano-Scale Heat Transport

在纳米力学尺度上，热传导过程表现出与宏观观察不同的各种现象。传统的模型，如傅立叶定律，是不充分的。此外，有压倒性的观察表明，热传导特性严重依赖于系统的大小和几何形状，这给数学模型的可移植性带来了独特的挑战。该项目旨在通过开发一种基于第一原则的方法来弥合建模差距，该方法导致了通用模型的层次结构。新模型有可能描述热传导过程的各种重要行为，包括波的传播特性、涨落、延迟等。提出的方法将从第一原理推导出新的本构模型，从数学上讲，这些模型是从微观多粒子描述中获得的，具有三个级别的约化：空间约化消除原子级细节并分离出平均能量的动力学，时间约化消除时间上的非局域性，以及统计约化有效地对尺度相关的统计噪声进行采样。这种基于第一原理的方法的好处在于它结合了数学上的透明性、稳健性和误差估计的适应性。与传统理论不同，目前的方法产生了一个随机本构模型。与能量平衡相结合，本构模型产生了随机偏微分方程式，其中有大量有趣的数学和计算问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

At the nano-mechanical scale, heat conduction processes exhibit a wide variety of phenomena that are different from macroscopic observations. Conventional models, such as the Fourier's law, are inadequate. In addition, there are overwhelming observations that suggest that heat conduction properties depend critically on the system size and geometry, creating a unique challenge for the portability of the mathematical models. This project aims to bridge the modeling gap by developing a first-principle based approach that leads to a hierarchy of generalized models. The new models have the potential capabilities to describe various important behavior of heat conduction processes, including the wave propagation characteristic, fluctuation, delay, and more. The proposed approach will derive new constitutive models from first-principle, and mathematically, the models are obtained from a microscopic many-particle description with three levels of reductions: A spatial reduction that eliminates atomic-level details and singles out the dynamics of the average energy, a temporal reduction that removes the nonlocality in time, and a statistical reduction that efficiently samples scale-dependent, statistical noises. Such a first-principle based approach derives its benefit from its combination of mathematical transparency, robustness, and amenability to error estimates. Unlike conventional theories, the current approach yields a stochastic constitutive model. Combined with energy balance, the constitutive models lead to stochastic partial differential equations, for which there is a wealth of interesting mathematical and computational issues.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.