Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions

Abstract

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.

Key words： Stationary Wigner equation; inflow boundary conditions; wellposedness; parity-decomposition; moment analysis

References

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