Scale Gap, Scale Invariance and Multiphysical Modeling

In this talk, I would like to introduce our recent advances in multi-scale analysis. They are the following: (1) Size effect, scale invariance and mathematical modeling at nano-scale; (2) Scale gap between quantum mechanics and classical mechanics. First, I will discuss the size effects and the scale invariance of some physical parameters such as elastic constants, thermal conductivity, dielectric constants and so forth. It can be proven that the elastic constants and the dielectric constants are approximate scale invariants for a nanocrystal in the framework of the local density approximation of density functional theory(DFT-LDA). From this, we will give some mathematical models at nano-scale. Some typical applications of the results in composite materials, CMOS transistor at nano-scale, photonic crystals and electrostatic forces of proteins in solution are presented. Second, I will introduce some new results on the scale gap between quantum mechanics and classical mechanics, and thermodynamic limits of Thomas-Fermi and Hartree-Fock type models. These questions are, of course, of great mathematical and physical interest. It can be proven that the size of the quantum region in the weak conjecture is of the order of a few Fermi wave-lengths, i.e. typically the inter-atomic distance, on the basis of DFT-LDA. It is a rigorous mathematical justification on the principle of “nearsightedness” and order N methods in quantum mechanics, which is the basis of some multiscale methods such as QM/MM and QM/MD/FEM in a sense. This talk is based on the joint work with Prof. Jun-Zhi Cui, Prof. Chong-Yu Wang, Prof. Jian-Lan Luo . This work is supported by National Natural Science Foundation of China (grant # 90405016) and Special Funds for Major State Basic Research Projects (grant # 2006CB605102).