理想量子氣體自由分子流噴流問題之解析解

Abstract

本文內容探討數個量子氣體自由分子流問題，主要為流體從熱平衡的容器中擴張到真空當中；從物理空間與速度空間的關係中，以統計力學的觀點對粒子的分佈函數做積分後，得到數密度(number density)與分子平均速度在流場的分佈。粒子的分佈函數使用了三種不同的分佈函數：費米-迪拉克(Fermi-Dirac)、玻色-愛因思坦(Bose-Einstein)、馬克士威爾-波茲曼(Maxwell-Boltzmann)分佈函數，分別代表了三種表現不同的粒子：費米子(Fermions)、玻色子(Bosons)幾及古典粒子；前兩種量子分佈函數在得到的解析解中表現出了量子效應，產生與古典粒子完全不同之行為；透過改變分佈函數中的逸度(fugacity)，我們觀察到不同程度的量子效應，而在系統趨近於古典極限，也就是逸度遠小於1時，兩種量子統計將會回復到古典統計，解析的結果顯示古典粒子的表現介於兩種量子粒子之間。數值模擬方面採用直接模擬蒙地卡羅法(direct simulation Monte Carlo)模擬稀薄氣體的流場，將粒子間假設為沒有碰撞，試著驗證自由分子流解析解的正確性，可以看出兩者的結果非常相近。

This paper concentrates on several collisionless quantum gas flow problems . The problems concern collisionless flow expanding into vacuum. From a relationship between particle position and velocity, we obtain the corresponding exact solutions for the number density and velocity distributions for the problems. We obtain the exact solutions with 3 distribution functions, which are Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann distribution function, which represent 3 kinds of particles, fermions, bosons and classical particles. Solutions obtained from quantum distribution functions show the differences with classical particles, the quantum effects are found by changing the fugacity of the quantum distribution. In the classical limit, the fugacity is less than 1, three distribution functions converge in the limit. In general the classical particles behave between two quantum particles. Numerical solution results obtained with the direct simulation Monte Carlo method in agreement with the analytical solutions. In general the comparisons between the exact solution and the numerical solutions are virtually identical.