模拟Bose-Fermi混合气体在绝对零度温度下动力学的一种时间分裂有限差分法
A Splitting Finite Difference Method for Simulating the Dynamics of Mixture of Bose-Fermi Gas at Zero Temperature
摘要: 在耦合Gross-Pitaevskii方程组数值解的基础上，我们探讨在绝对零度下Bose-Fermi混合物的动力学规律。为了计算简便，我们首先把最初的耦合Gross-Pitaevskii方程组转化为无量纲形式方程组，然后把无量纲三维形式方程组简化成二维形式再进一步简化为一维形式，并证明了与耦合Gross-Pitaevskii方程组相关的守恒律——模量守恒以及能量守恒。其次，为了探讨Bose-Fermi混合物的动力学规律，我们提出了一种高效的数值方法——时间分裂差分法来求解耦合Gross-Pitaevskii方程组。并证明了该数值方法具有无条件稳定性以及保持耦合Gross-Pitaevskii方程组守恒律等优点。最后利用该方法来数值模拟Bose-Fermi混合物的动力学规律。
Abstract: In this paper, based on the numerical solutions of coupled Gross-Pitaevskii equations, we investigate the dynamics of Bose-Fermi mixtures at zero temperature. Firstly, to simplify the numerical computation, we reformulate the equations into three-dimensional dimensionless ones, which are also further reduced to two-dimensional ones and one-dimensional ones. Secondly, to compute the numerical solutions, we present a high-efficiency method for the equations, the time-splitting difference method. We have proved that the method is unconditionally stable and keeps well the related conservation laws. Lastly we apply the method into studying the dynamics of Bose-Fermi mixtures.
文章引用：张志红, 高文, 王伟民. 模拟Bose-Fermi混合气体在绝对零度温度下动力学的一种时间分裂有限差分法[J]. 应用数学进展, 2018, 7(4): 401-412. https://doi.org/10.12677/AAM.2018.74050

参考文献

[1] Modugno, G., Roati, G., Riboli, F., et al. (2002) Collapse of a Degenerate Fermi Gas. Science, 297, 2240-2243. [Google Scholar] [CrossRef] [PubMed]
[2] Mølmer, K. (1998) Bose Condensates and Fermi Gases at Zero Temperature. Physical Review Letters, 80, 1804-1807. [Google Scholar] [CrossRef] 
[3] Truscott, A.G., Strecker, K.E., Mcalexander, W.I., et al. (2001) Observation of Fermi Pressure in a Gas of Trapped Atoms. Science, 291, 2570-2572. [Google Scholar] [CrossRef] [PubMed]
[4] Georges, A., Kotliar, G., Krauth, W., et al. (1996) Dynamical Mean-Field Theory of Strongly Correlated Fermion Systems and the Limit of Infinite Dimensions. Review of Modern Physics, 68, 13. [Google Scholar] [CrossRef] 
[5] Roth, R. and Feldmeier, H. (2002) Mean-Field Instability of Trapped Dilute Boson-Fermion Mixtures. Physical Review A, 65, 275-280.
[6] Schreck, F., Khaykovich, L., Corwin, K.L., et al. (2001) Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea. Physical Review Letters, 87, 080403.
[7] Hadzibabic, Z., Stan, C.A., Dieckmann, K., et al. (2002) Two-Species Mixture of Quantum Degenerate Bose and Fermi Gases. Physical Review Letters, 88, 160401. [Google Scholar] [CrossRef] 
[8] Modugno, M., Ferlaino, F., Riboli, F., et al. (2012) Mean-Field Analysis of the Stability of a K-Rb Fermi-Bose Mixture. Physical Review A, 68, 043626.
[9] Goldwin, J., Papp, S.B., Demarco, B., et al. (2002) Two-Species Magneto-Optical Trap with 40K, and 87Rb. Physical Review A, 65, 021402. [Google Scholar] [CrossRef] 
[10] Roati, G., Riboli, F., Modugno, G., et al. (2002) Fermi-Bose Quantum Degenerate 40K-87Rb Mixture with Attractive Interaction. Physical Review Letters, 89, 150403. [Google Scholar] [CrossRef] 
[11] Ospelkaus, C., Ospelkaus, S., Sengstock, K., et al. (2006) Interaction-Driven Dynamics of 40K-87Rb Fermion-Boson Gas Mixtures in the Large-Particle-Number Limit. Physical Review Letters, 96, Article ID: 020401. [Google Scholar] [CrossRef] 
[12] Roth, R. (2002) Structure and Stability of Trapped Atomic Boson-Fermion Mixtures. Physical Review A, 66, 144-144. [Google Scholar] [CrossRef] 
[13] Tsurumi, T. and Wadati, M. (2000) Dynamics of Magnetically Trapped Boson-Fermion Mixtures. Journal of the Physical Society of Japan, 69, 97-103. [Google Scholar] [CrossRef] 
[14] Capuzzi, P., Minguzzi, A. and Tosi, M.P. (2003) Collective Excitations in Trapped Boson-Fermion Mixtures: From Demixing to Collapse. Physical Review A, 68, Article ID: 033605.
[15] Capuzzi, P., Minguzzi, A. and Tosi, M.P. (2004) Collisional Oscillations of Trapped Boson-Fermion Mixtures in the Approach to the Collapse Instability. Physical Review A, 69, 521-524. [Google Scholar] [CrossRef] 
[16] Adhikari, S.K. (2004) Mean-Field Description of Dynamical Collapse of a Fermionic Condensate in a Trapped Boson-Fermion Mixture. Physical Review A, 70, 628-628. [Google Scholar] [CrossRef] 
[17] Adhikari, S.K. and Salasnich, L. (2007) Mixing-Demixing Transition and Collapse of a Vortex State in a Quasi-Two-Dimensional Boson-Fermion Mixture. Physical Review A, 75, 497-500. [Google Scholar] [CrossRef] 
[18] Bao, W., Cai, Y. and Wang, H. (2010) Efficient Numerical Methods for Computing Ground States and Dynamics of Dipolar Bose-Einstein Condensates. Journal of Computational Physics, 229, 7874-7892. [Google Scholar] [CrossRef] 
[19] Bao, W. and Wang, H. (2006) An Efficient and Spectrally Accurate Numerical Method for Computing Dynamics of Rotating Bose-Einstein Condensates. Journal of Computational Physics, 217, 612-626. [Google Scholar] [CrossRef] 
[20] Bao, W., Wang, H. and Markowich, P.A. (2005) Ground, Symmetric and Central Vortex States in Rotating Bose-Einstein Condensates. Communications in Mathematical Sciences, 3, 57-88. [Google Scholar] [CrossRef] 
[21] Bao, W. and Wang, H. (2007) A Mass and Magnetization Conservative and Energy-Diminishing Numerical Method for Computing Ground State of Spin-1 Bose-Einstein Condensates. Siam Journal on Numerical Analysis, 45, 2177-2200. [Google Scholar] [CrossRef] 
[22] 刘云龙, 陆瑞征. 普朗克常数[J]. 自然杂志, 1988(10): 46-50.
[23] 刘锋, 贾多杰, 李晓礼, 等. 无量纲化的方法[J]. 安顺学院学报, 2008, 10(3): 78-80.

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