Frontiers of Mathematics in China >
Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions
Received date: 04 May 2016
Accepted date: 03 Nov 2016
Published date: 06 Jul 2017
Copyright
We study the stationary Wigner equation on a bounded, onedimensional spatial domain with inflow boundary conditions by using the parity decomposition of L. Barletti and P. F. Zweifel [Transport Theory Statist. Phys., 2001, 30(4-6): 507–520]. The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd L2-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.
Ruo LI , Tiao LU , Zhangpeng SUN . Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions[J]. Frontiers of Mathematics in China, 2017 , 12(4) : 907 -919 . DOI: 10.1007/s11464-017-0612-9
1 |
ArnoldA, LangeH, ZweifelP F. A discrete-velocity, stationary Wigner equation. J Math Phys, 2000, 41(11): 7167–7180
|
2 |
BarlettiL. A mathematical introduction to the Wigner formulation of quantum mechanics. Boll Unione Mat Ital, 2003, 6-B(3): 693–716
|
3 |
BarlettiL, ZweifelP F. Parity-decomposition method for the stationary Wigner equation with inflow boundary conditions. Transport Theory Statist Phys, 2001, 30(4-6): 507–520
|
4 |
CaiZ, FanY, LiR, LuT, WangY. Quantum hydrodynamics models by moment closure of Wigner equation. J Math Phys, 2012, 53: 103503
|
5 |
CostolanskiA S, KelleyC T. Efficient solution of the Wigner-Poisson equations for modeling resonant tunneling diodes. IEEE Trans Nanotechnology, 2010, 9(6): 708–715
|
6 |
EvansL C. Partial Differential Equations. 2nd ed. Providence: Amer Math Soc, 2010
|
7 |
FerryD K, GoodnickS M. Transport in Nanostructures. Cambridge: Cambridge Univ Press, 1997
|
8 |
FrensleyW R. Wigner function model of a resonant-tunneling semiconductor device. Phys Rev B, 1987, 36: 1570–1580
|
9 |
GehringA, KosinaH. Wigner function-based simulation of quantum transport in scaled DG-MOSFETs using a Monte Carlo method. J Comput Electr, 2005, 4: 67–70
|
10 |
HilleryM, ÓConnellR F, ScullyM O, WignerE P. Distribution functions in physics: Fundamentals. Phys Rep, 1984, 106(3): 121–167
|
11 |
JensenK L, BuotF A. Numerical aspects on the simulation of I-V characteristics and switching times of resonant tunneling diodes. J Appl Phys, 1990, 67: 2153–2155
|
12 |
JiangH, CaiW, TsuR. Accuracy of the Frensley inflow boundary condition for Wigner equations in simulating resonant tunneling diodes. J Comput Phys, 2011, 230: 2031–2044
|
13 |
JiangH, LuT, CaiW. A device adaptive inflow boundary condition for Wigner equations of quantum transport. J Comput Phys, 2014, 248: 773–786
|
14 |
LiR, LuT, SunZ-P. Convergence of semi-discrete stationary Wigner equation with inflow boundary conditions. Preprint, 2014
|
15 |
LiR, LuT, SunZ-P. Stationary Wigner equation with inflow boundary conditions: Will a symmetric potential yield a symmetric solution? SIAM J Appl Math, 2014, 70(3): 885–897
|
16 |
QuerliozQ, Saint-MartinJ, DoV-N, BournelA, DollfusP. A study of quantum transport in end-of-roadmap DG-MOSFETs using a fully self-consistent Wigner Monte Carlo approach. IEEE Trans Nanotechnology, 2006, 5(6): 737–744
|
17 |
ShaoS, LuT, CaiW. Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport. Commun Comput Phys, 2011, 9: 711–739
|
18 |
ShihJ J, HuangH C, WuG Y. Effect of mass discontinuity in the Wigner theory of resonant-tunneling diodes. Phys Rev B, 1994, 50(4): 2399–2405
|
19 |
TsuchiyaH, OgawaM. Simulation of quantum transport in quantum device with spatially varying effective mass. IEEE Trans Electron Devices, 1991, 38(6): 1246–1252
|
20 |
WignerE. On the quantum correction for thermodynamic equilibrium. Phys Rev, 1932, 40(5): 749–759
|
/
〈 | 〉 |