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Chin. Phys. Lett.  2017, Vol. 34 Issue (2): 020502    DOI: 10.1088/0256-307X/34/2/020502
Note on Divergence of the Chapman–Enskog Expansion for Solving Boltzmann Equation
Nan-Xian Chen1**, Bo-Hua Sun2
1State Key Laboratory of Low-dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084
2Department of Mechanical Engineering, Cape Peninsula University of Technology, Cape Town, South Africa
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Nan-Xian Chen, Bo-Hua Sun 2017 Chin. Phys. Lett. 34 020502
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Abstract Within about a year (1916–1917) Chapman and Enskog independently proposed an important expansion for solving the Boltzmann equation. However, the expansion is divergent or indeterminant in the case of relaxation time $\tau \geq 1$. Since then, this divergence problem has puzzled researchers for a century. Using a modified Möbius series inversion formula, we propose a modified Chapman–Enskog expansion with a variable upper limit of the summation. The new expansion can give not only a convergent summation but also the best-so-far explanation on some unbelievable scenarios occurring in previous practice.
Received: 16 January 2017      Published: 25 January 2017
PACS:  05.20.Dd (Kinetic theory)  
  05.70.Ln (Nonequilibrium and irreversible thermodynamics)  
  02.30.Mv (Approximations and expansions)  

http://cpl.iphy.ac.cn/10.1088/0256-307X/34/2/020502       OR      http://cpl.iphy.ac.cn/Y2017/V34/I2/020502
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Nan-Xian Chen
Bo-Hua Sun
[1]Pitaevskii L P and Lifshitz E M 1981 Physical Kinetics (Oxford: Butterworth-Heinemann)
[2]Cercignani C 1988 The Boltzmann Equation and Its Applications (New York: Springer-Verlag)
[3]Chapman S and Cowling T G 1990 The Mathematical Theory of Non-uniform Gases (Cambridge: Cambridge University Press)
[4]Bhatnagar P L, Gross E P and Krook M 1954 Phys. Rev. 94 511
[5]McLannan A 1965 Phys. Fluids 8 1580
[6]Santos A, Brey J J and Dufty J W 1986 Phys. Rev. Lett. 56 1571
[7]Hardy G H and Wright E M 1979 An Introduction to the Theory of Numbers (Oxford: Clarendon)
[8]Chen N X 1990 Phys. Rev. Lett. 64 1193
[9]Maddox J 1990 Nature 344 29
[10]Chen N X 2010 Möbius Inversion in Physics (Singapore: World Scientific)
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