# American Institute of Mathematical Sciences

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

 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.
Citation: Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic and 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), 74-101. doi: 10.1137/S003613999833294X. [2] G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices, Solid-State Electronics, 28 (1985), 407-416. doi: 10.1016/0038-1101(85)90100-5. [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), 511-525. doi: 10.1103/PhysRev.94.511. [4] S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297. [5] A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616. doi: 10.1016/0378-4371(83)90013-4. [6] P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. [7] P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X. [8] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, in "Quantum Transport" (eds. G. Allaire et al.), 111-168, Lecture Notes Math., 1946, Springer, Berlin, 2008. [9] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, in "Progress in Industrial Mathematics at ECMI 2006" (eds. L. Bonilla et al.), 106-122, Mathematics in Industry, 12, Springer, Berlin, 2008. [10] P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3. [11] P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525. [12] J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal., 73 (2010), 854-856. doi: 10.1016/j.na.2010.03.047. [13] W. Dreyer, Maximisation of the entropy in non-equilibrium, J. Phys. A, 20 (1987), 6505-6517. doi: 10.1088/0305-4470/20/18/047. [14] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. [15] C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425. [16] F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl., 12 (2011), 1733-1735. doi: 10.1016/j.nonrwa.2010.11.005. [17] A. Jüngel, "Transport Equations for Semiconductors," Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. [18] A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. [19] A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives, Nonlin. Anal., 74 (2011), 2813-2818. doi: 10.1016/j.na.2011.01.002. [20] A. Jüngel, Dissipative quantum fluid models, to appear in Revista Mat. Univ. Parma, 2011. [21] A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 85 (2005), 806-814. doi: 10.1002/zamm.200510232. [22] A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68. doi: 10.1137/050644823. [23] C. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. [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), 409-412. [25] F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565–-602. doi: 10.1007/s10955-010-0003-z.

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##### References:
 [1] M. Anile, V. Romano and G. Russo, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math., 61 (2000), 74-101. doi: 10.1137/S003613999833294X. [2] G. Baccarani and M. Wordeman, An investigation of steady-state velocity overshoot effects in Si and GaAs devices, Solid-State Electronics, 28 (1985), 407-416. doi: 10.1016/0038-1101(85)90100-5. [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), 511-525. doi: 10.1103/PhysRev.94.511. [4] S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297. [5] A. Caldeira and A. Leggett, Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616. doi: 10.1016/0378-4371(83)90013-4. [6] P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Commun. Math. Sci., 5 (2007), 887-908. [7] P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 6 (2007), 246-272. doi: 10.1137/06067153X. [8] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, in "Quantum Transport" (eds. G. Allaire et al.), 111-168, Lecture Notes Math., 1946, Springer, Berlin, 2008. [9] P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum diffusion models derived from the entropy principle, in "Progress in Industrial Mathematics at ECMI 2006" (eds. L. Bonilla et al.), 106-122, Mathematics in Industry, 12, Springer, Berlin, 2008. [10] P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3. [11] P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525. [12] J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal., 73 (2010), 854-856. doi: 10.1016/j.na.2010.03.047. [13] W. Dreyer, Maximisation of the entropy in non-equilibrium, J. Phys. A, 20 (1987), 6505-6517. doi: 10.1088/0305-4470/20/18/047. [14] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004. [15] C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425. [16] F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlin. Anal. Real World Appl., 12 (2011), 1733-1735. doi: 10.1016/j.nonrwa.2010.11.005. [17] A. Jüngel, "Transport Equations for Semiconductors," Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. [18] A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. [19] A. Jüngel, Effective velocity in compressible Navier-Stokes equations with third-order derivatives, Nonlin. Anal., 74 (2011), 2813-2818. doi: 10.1016/j.na.2011.01.002. [20] A. Jüngel, Dissipative quantum fluid models, to appear in Revista Mat. Univ. Parma, 2011. [21] A. Jüngel and D. Matthes, Derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 85 (2005), 806-814. doi: 10.1002/zamm.200510232. [22] A. Jüngel, D. Matthes and J.-P. Milišić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68. doi: 10.1137/050644823. [23] C. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. [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), 409-412. [25] F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys., 140 (2010), 565–-602. doi: 10.1007/s10955-010-0003-z.
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