半古典晶格波滋曼方法

Abstract

本論文提出了半古典晶格波滋曼法，此方法是在波滋曼法的基礎下，使用量子統計(Bose-Einstein Statistics 和 Fermi-Dirac Statistics)取代古典統計(Maxwell-Boltzmann Statistics)後展開所得的方法。此方法可驗證在古典極限(Classical Limit)下可回復原本古典晶格波滋曼法，亦可由數值驗證得知古典極限下可得傳統晶格波滋曼法之結果。本文以此一新提出之模擬方法研究量子氣體運動問題，模擬之問題忽略粒子間交互作用，但因為使用量子統計之故，粒子在量子統計下的特性如庖立不相容原理(Pauli Exclusion Principle)、巨觀上傳輸係數之修正等等均有考慮在內。本文探討一維量子氣體之震波管問題、二維圓柱流問題、二維微管問題以及三維頂蓋流問題作為驗證半古典晶格波滋曼法之方式，模擬結果指出了量子統計和古典統計結果的主要差異。此外，本文亦提出了另一種雙分布函數熱晶格波滋曼法，可得出一般雙分布函數熱晶格波滋曼法所得結果，也可作為未來更進一步推廣半古典晶格波滋曼法之方向。 關鍵詞:晶格波滋曼法、量子統計、波滋曼方程式

Unlike describing the physical phenomenon in coordinate or momentum spaces in quantum mechanics, semiclassical Boltzmann equation treats the system in phase space, and it is much easier to describe the dynamics of quantum gases. In this thesis, a class of semiclassical lattice Boltzmann methods is developed for solving quantum hydrodynamics and beyond. The present method is directly derived by projecting the Uehling-Uhlenbeck Boltzmann-BGK equations onto the tensor Hermite polynomials following Grad's moment expansion method. The intrinsic discrete nodes of the Gauss-Hermite quadrature provide the natural lattice velocities for the semiclassical lattice Boltzmann method. Formulations for the second-order and third order expansion of the semiclassical equilibrium distribution functions are derived and their corresponding hydrodynamics are studied. Gases of particles of arbitrary statistics can be considered. Simulations of one-dimensional compressible gas flow by using D1Q5 lattices, two dimensional microchannel flow, two dimensional flow over cylinder by using D2Q9 lattices and three dimensional lid driven cavity flow by using D3Q19 lattices are provided for validating this method. It is shown that the classical flow patterns such like vortex and vortices shedding in flow over cylinder simulations, temperature and pressure contours together with streamline patterns could be produced from the present method in classical limit. The results also indicate the distinct characteristics of the effects of quantum statistics when they are compared with fluid phenomena in classical statistics. Keywords: Lattice Boltzmann Method, Semiclassical, Quantum.