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September  2011, 4(3): 785-807. doi: 10.3934/krm.2011.4.785

Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution

Ansgar Jüngel 1, and  Josipa-Pina Milišić 2,

1. 

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien, Austria
2. 

Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb Unska 3, 10000 Zagreb, Croatia

Received  January 2011 Revised  March 2011 Published  August 2011

Navier-Stokes equations for compressible quantum fluids, including the energy equation, are derived from a collisional Wigner equation, using the quantum entropy maximization method of Degond and Ringhofer. The viscous corrections are obtained from a Chapman-Enskog expansion around the quantum equilibrium distribution and correspond to the classical viscous stress tensor with particular viscosity coefficients depending on the particle density and temperature. The energy and entropy dissipations are computed and discussed. Numerical simulations of a one-dimensional tunneling diode show the stabilizing effect of the viscous correction and the impact of the relaxation terms on the current-voltage charcteristics.
Keywords: tunneling diode, finite differences, Chapman-Enskog expansion, semiconductors., density-dependent viscosity, Quantum Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q30, 76Y05; Secondary: 35Q40, 82D3.
Citation: Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785
References:
[1]

M. Anile, V. Romano and G. Russo, Extended hydrodynamical model of carrier transport in semiconductors,, SIAM J. Appl. Math., 61 (2000), 74.  doi: 10.1137/S003613999833294X.  Google Scholar

[2]

G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices,, Solid-State Electronics, 28 (1985), 407.  doi: 10.1016/0038-1101(85)90100-5.  Google Scholar

[3]

P. Bhatnagar, E. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,, Phys. Rev., 94 (1954), 511.  doi: 10.1103/PhysRev.94.511.  Google Scholar

[4]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model,, Z. Angew. Math. Mech., 90 (2010), 219.  doi: 10.1002/zamm.200900297.  Google Scholar

[5]

A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion,, Physica A, 121 (1983), 587.  doi: 10.1016/0378-4371(83)90013-4.  Google Scholar

[6]

P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models,, Commun. Math. Sci., 5 (2007), 887.   Google Scholar

[7]

P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation,, Multiscale Model. Simul., 6 (2007), 246.  doi: 10.1137/06067153X.  Google Scholar

[8]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle,, in, 1946 (2008), 111.   Google Scholar

[9]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle,, in, 12 (2006), 106.   Google Scholar

[10]

P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625.  doi: 10.1007/s10955-004-8823-3.  Google Scholar

[11]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle,, J. Stat. Phys., 112 (2003), 587.  doi: 10.1023/A:1023824008525.  Google Scholar

[12]

J. Dong, A note on barotropic compressible quantum Navier-Stokes equations,, Nonlin. Anal., 73 (2010), 854.  doi: 10.1016/j.na.2010.03.047.  Google Scholar

[13]

W. Dreyer, Maximisation of the entropy in non-equilibrium,, J. Phys. A, 20 (1987), 6505.  doi: 10.1088/0305-4470/20/18/047.  Google Scholar

[14]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Series in Mathematics and its Applications, 26 (2004).   Google Scholar

[15]

C. Gardner, The quantum hydrodynamic model for semiconductor devices,, SIAM J. Appl. Math., (1994), 409.  doi: 10.1137/S0036139992240425.  Google Scholar

[16]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations,, Nonlin. Anal. Real World Appl., 12 (2011), 1733.  doi: 10.1016/j.nonrwa.2010.11.005.  Google Scholar

[17]

A. Jüngel, "Transport Equations for Semiconductors,", Lecture Notes in Physics, 773 (2009).   Google Scholar

[18]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, SIAM J. Math. Anal., 42 (2010), 1025.   Google Scholar

[19]

A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives,, Nonlin. Anal., 74 (2011), 2813.  doi: 10.1016/j.na.2011.01.002.  Google Scholar

[20]

A. Jüngel, Dissipative quantum fluid models,, to appear in Revista Mat. Univ. Parma, (2011).   Google Scholar

[21]

A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization,, Z. Angew. Math. Mech., 85 (2005), 806.  doi: 10.1002/zamm.200510232.  Google Scholar

[22]

A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization,, SIAM J. Appl. Math., 67 (2006), 46.  doi: 10.1137/050644823.  Google Scholar

[23]

C. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021.  doi: 10.1007/BF02179552.  Google Scholar

[24]

T. Li, Z. Yu, Y. Wang, L. Huang and C. Xiang, "Numerical Simulation of Negative Differential Resistance Characteristics in Si/Si$_1-x$Ge$_x$ RTD at Room Temperature,", Proceedings of the 2005 IEEE Conference on Electron Devices and Solid-State Circuits, (2005), 409.   Google Scholar

[25]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics,, J. Stat. Phys., 140 (2010).  doi: 10.1007/s10955-010-0003-z.  Google Scholar

show all references

References:
[1]

M. Anile, V. Romano and G. Russo, Extended hydrodynamical model of carrier transport in semiconductors,, SIAM J. Appl. Math., 61 (2000), 74.  doi: 10.1137/S003613999833294X.  Google Scholar

[2]

G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices,, Solid-State Electronics, 28 (1985), 407.  doi: 10.1016/0038-1101(85)90100-5.  Google Scholar

[3]

P. Bhatnagar, E. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,, Phys. Rev., 94 (1954), 511.  doi: 10.1103/PhysRev.94.511.  Google Scholar

[4]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model,, Z. Angew. Math. Mech., 90 (2010), 219.  doi: 10.1002/zamm.200900297.  Google Scholar

[5]

A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion,, Physica A, 121 (1983), 587.  doi: 10.1016/0378-4371(83)90013-4.  Google Scholar

[6]

P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models,, Commun. Math. Sci., 5 (2007), 887.   Google Scholar

[7]

P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation,, Multiscale Model. Simul., 6 (2007), 246.  doi: 10.1137/06067153X.  Google Scholar

[8]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle,, in, 1946 (2008), 111.   Google Scholar

[9]

P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle,, in, 12 (2006), 106.   Google Scholar

[10]

P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625.  doi: 10.1007/s10955-004-8823-3.  Google Scholar

[11]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle,, J. Stat. Phys., 112 (2003), 587.  doi: 10.1023/A:1023824008525.  Google Scholar

[12]

J. Dong, A note on barotropic compressible quantum Navier-Stokes equations,, Nonlin. Anal., 73 (2010), 854.  doi: 10.1016/j.na.2010.03.047.  Google Scholar

[13]

W. Dreyer, Maximisation of the entropy in non-equilibrium,, J. Phys. A, 20 (1987), 6505.  doi: 10.1088/0305-4470/20/18/047.  Google Scholar

[14]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford Series in Mathematics and its Applications, 26 (2004).   Google Scholar

[15]

C. Gardner, The quantum hydrodynamic model for semiconductor devices,, SIAM J. Appl. Math., (1994), 409.  doi: 10.1137/S0036139992240425.  Google Scholar

[16]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations,, Nonlin. Anal. Real World Appl., 12 (2011), 1733.  doi: 10.1016/j.nonrwa.2010.11.005.  Google Scholar

[17]

A. Jüngel, "Transport Equations for Semiconductors,", Lecture Notes in Physics, 773 (2009).   Google Scholar

[18]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, SIAM J. Math. Anal., 42 (2010), 1025.   Google Scholar

[19]

A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives,, Nonlin. Anal., 74 (2011), 2813.  doi: 10.1016/j.na.2011.01.002.  Google Scholar

[20]

A. Jüngel, Dissipative quantum fluid models,, to appear in Revista Mat. Univ. Parma, (2011).   Google Scholar

[21]

A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization,, Z. Angew. Math. Mech., 85 (2005), 806.  doi: 10.1002/zamm.200510232.  Google Scholar

[22]

A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization,, SIAM J. Appl. Math., 67 (2006), 46.  doi: 10.1137/050644823.  Google Scholar

[23]

C. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021.  doi: 10.1007/BF02179552.  Google Scholar

[24]

T. Li, Z. Yu, Y. Wang, L. Huang and C. Xiang, "Numerical Simulation of Negative Differential Resistance Characteristics in Si/Si$_1-x$Ge$_x$ RTD at Room Temperature,", Proceedings of the 2005 IEEE Conference on Electron Devices and Solid-State Circuits, (2005), 409.   Google Scholar

[25]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics,, J. Stat. Phys., 140 (2010).  doi: 10.1007/s10955-010-0003-z.  Google Scholar

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