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September  2012, 5(3): 517-536. doi: 10.3934/krm.2012.5.517

On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models

José Luis López 1, and  Jesús Montejo-Gámez 1,

1. 

Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, Spain

Received  December 2011 Revised  May 2012 Published  August 2012

The aim of this paper is to derive the quantum hydrodynamic system associated with the most general class of nonlinear Schrödinger equations accounting for Fokker--Planck type diffusion of the probability density, called of Doebner--Goldin. This 'Doebner--Goldin hydrodynamic system' is shown to be reduced in most cases to a simpler one of quantum Euler type by means of the introduction of a nonlinear gauge transformation that changes the fluid mean velocity into a new effective velocity corrected by an osmotic contribution. Finally, we also discuss some particular situations of especial interest and compare the structure of the resulting fluid systems with that of the viscous quantum hydrodynamic and the quantum Navier--Stokes equations stemming from maximization of the quantum entropy for Wigner--BGK models.
Keywords: quantum viscous hydrodynamic equation, Doebner-Goldin equation, Quantum hydrodynamics, quantum Navier-Stokes equation., Schrödinger description of quantum mechanics.
Mathematics Subject Classification: Primary: 81Q05, 76Y05, 35Q30; Secondary: 81S30, 78M0.
Citation: José Luis López, Jesús Montejo-Gámez. On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models. Kinetic and Related Models, 2012, 5 (3) : 517-536. doi: 10.3934/krm.2012.5.517
References:
[1]

M. G. Ancona, Density-gradient theory analysis of electron distributions in heterostructures, Superlattics and Microstructures, 7 (1990), 119-130. doi: 10.1016/0749-6036(90)90124-P.

[2]

A. Arnold, J. L. López, P. Markowich and J. Soler, An analysis of quantum Fokker-Planck models: a Wigner function approach, Rev. Mat. Iberoamericana, 20 (2004), 771-814. doi: 10.4171/RMI/407.

[3]

G. Auberson and P. C. Sabatier, On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys., 35 (1994), 4028-4040. doi: 10.1063/1.530840.

[4]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93. doi: 10.1016/0003-4916(76)90057-9.

[5]

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.

[6]

S. Burger, F. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. Chiofalo and M.Tosi, Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic optimal potential, Phys. Rev. Lett., 86 (2001), 4447-4450. doi: 10.1103/PhysRevLett.86.4447.

[7]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

[8]

H. D. Doebner and G. A. Goldin, On a general nonlinear Schrödinger equation admitting diffusioncurrents, Phys. Lett., A, 162 (1992), 397-401. doi: 10.1016/0375-9601(92)90061-P.

[9]

H. D. Doebner and G. A. Goldin, Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations, J. Phys. A: Math. Gen., 27 (1994), 1771-1780. doi: 10.1088/0305-4470/27/5/036.

[10]

H. D. Doebner and G. A. Goldin, Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations, Phys. Rev. A (3), 54 (1996), 3764-3771. doi: 10.1103/PhysRevA.54.3764.

[11]

H. D. Doebner, G. A. Goldin and P. Nattermann, A family of nonlinear Schrödinger equations: linearizing transformations and resulting structure, in "Quantization, Coherent States and Complex Structures" (eds. J.-P. Antoine et al.), Plenum, (1996), 27-31.

[12]

P. Frampton, "Gauge Field Theories," Wiley-VCH, Berlin, 2008. doi: 10.1002/9783527623358.

[13]

L. Fritsche and M. Haugk, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Ann. Phys., 12 (2003), 371-403. doi: 10.1002/andp.200310017.

[14]

P. Garbaczewski, Modular Schrödinger equation and dynamical duality, Phys. Rev. E (3), 78 (2008), 031101. doi: 10.1103/PhysRevE.78.031101.

[15]

M. P. Gualdani and A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Europ. J. Appl. Math., 15 (2004), 577-595. doi: 10.1017/S0956792504005686.

[16]

P. Guerrero, J. L. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Analysis R. W. A., 11 (2010), 79-87. doi: 10.1016/j.nonrwa.2008.10.017.

[17]

P. Guerrero, J. L. López, J. Montejo-Gámez, J. Nieto, Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation, preprint.

[18]

R. Harvey, Navier-Stokes analog of quantum mechanics, Phys. Rev., 152 (1966), 1115. doi: 10.1103/PhysRev.152.1115.

[19]

A. Jüngel, "Transport Equations for Semiconductors," Lect. Notes Phys., 773, Springer, Berlin, 2009.

[20]

A. Jüngel, Effective velocity in compressible Navier-Stokes equations wirth third-order derivatives, Nonlinear Analysis, 74 (2011), 2813-2818. doi: 10.1016/j.na.2011.01.002.

[21]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.

[22]

A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, To appear in J. Stat. Phys., 145 (2011), 1661-1673.

[23]

A. Jüngel and J.-P. Milisić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution equations, Kinetic and related models, 4 (2011), 785-807.

[24]

A. Jüngel, M. C. Mariani and D. Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Meth. Appl. Sci., 12 (2002), 485-495. doi: 10.1142/S0218202502001751.

[25]

G. Kaniadakis and A. M. Scarfone, Nonlinear transformation for a class of gauged Schrödinger equations with complex nonlinearities, Reports on Math. Phys., 48 (2001), 115-121. doi: 10.1016/S0034-4877(01)80070-7.

[26]

J. L. López, Nonlinear Ginzburg-Landau-type approach to quantum dissipation, Phys. Rev. E., 69 (2004), 026110. doi: 10.1103/PhysRevE.69.026110.

[27]

J. L. López, J. Montejo-Gámez, A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics, Physica D, 238 (2009), 622-644.

[28]

R. Mosna, I. Hamilton and L. Delle Site, Variational approach to dequantization, J. Phys. A, 39 (2006), L229-L235. doi: 10.1088/0305-4470/39/14/L03.

[29]

P. Nattermann and W. Scherer, Nonlinear gauge transformations and exact solutions of the Doebner-Goldin equation, in "Nonlinear, Deformed and Irreversible Quantum Systems" (eds. Doebner et al.), World Scientific, (1995), 188-199.

[30]

P. Nattermann and R. Zhdanov, On integrable Doebner-Goldin equations, J. Phys. A, 29 (1996), 2869-2886. doi: 10.1088/0305-4470/29/11/021.

[31]

E. Nelson, Derivation of the Schrödinger equation from Newtonian Mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.

[32]

C. Sabatier, Multidimensional nonlinear Schrödinger equations with exponentially confined solutions, Inverse Problems, 6 (1990), L47-L53. doi: 10.1088/0266-5611/6/5/002.

[33]

A. Scarfone, Gauge equivalence among quantum nonlinear many body systems, Act. Appl. Math., 102 (2008), 179-217. doi: 10.1007/s10440-008-9213-7.

[34]

A. G. Ushveridze, Dissipative quantum mechanics. A special Doebner-Goldin equation, its properties and exact solutions, Phys. Lett. A, 185 (1994), 123-127. doi: 10.1016/0375-9601(94)90834-6.

[35]

A. G. Ushveridze, The special Doebner-Goldin equation as a fundamental equation of dissipative quantum mechanics, Phys. Lett. A, 185 (1994), 128-132. doi: 10.1016/0375-9601(94)90835-4.

show all references

References:
[1]

M. G. Ancona, Density-gradient theory analysis of electron distributions in heterostructures, Superlattics and Microstructures, 7 (1990), 119-130. doi: 10.1016/0749-6036(90)90124-P.

[2]

A. Arnold, J. L. López, P. Markowich and J. Soler, An analysis of quantum Fokker-Planck models: a Wigner function approach, Rev. Mat. Iberoamericana, 20 (2004), 771-814. doi: 10.4171/RMI/407.

[3]

G. Auberson and P. C. Sabatier, On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys., 35 (1994), 4028-4040. doi: 10.1063/1.530840.

[4]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93. doi: 10.1016/0003-4916(76)90057-9.

[5]

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.

[6]

S. Burger, F. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. Chiofalo and M.Tosi, Superfluid and dissipative dynamics of a Bose-Einstein condensate in a periodic optimal potential, Phys. Rev. Lett., 86 (2001), 4447-4450. doi: 10.1103/PhysRevLett.86.4447.

[7]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

[8]

H. D. Doebner and G. A. Goldin, On a general nonlinear Schrödinger equation admitting diffusioncurrents, Phys. Lett., A, 162 (1992), 397-401. doi: 10.1016/0375-9601(92)90061-P.

[9]

H. D. Doebner and G. A. Goldin, Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations, J. Phys. A: Math. Gen., 27 (1994), 1771-1780. doi: 10.1088/0305-4470/27/5/036.

[10]

H. D. Doebner and G. A. Goldin, Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations, Phys. Rev. A (3), 54 (1996), 3764-3771. doi: 10.1103/PhysRevA.54.3764.

[11]

H. D. Doebner, G. A. Goldin and P. Nattermann, A family of nonlinear Schrödinger equations: linearizing transformations and resulting structure, in "Quantization, Coherent States and Complex Structures" (eds. J.-P. Antoine et al.), Plenum, (1996), 27-31.

[12]

P. Frampton, "Gauge Field Theories," Wiley-VCH, Berlin, 2008. doi: 10.1002/9783527623358.

[13]

L. Fritsche and M. Haugk, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Ann. Phys., 12 (2003), 371-403. doi: 10.1002/andp.200310017.

[14]

P. Garbaczewski, Modular Schrödinger equation and dynamical duality, Phys. Rev. E (3), 78 (2008), 031101. doi: 10.1103/PhysRevE.78.031101.

[15]

M. P. Gualdani and A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Europ. J. Appl. Math., 15 (2004), 577-595. doi: 10.1017/S0956792504005686.

[16]

P. Guerrero, J. L. López and J. Nieto, Global $H^1$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Analysis R. W. A., 11 (2010), 79-87. doi: 10.1016/j.nonrwa.2008.10.017.

[17]

P. Guerrero, J. L. López, J. Montejo-Gámez, J. Nieto, Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation, preprint.

[18]

R. Harvey, Navier-Stokes analog of quantum mechanics, Phys. Rev., 152 (1966), 1115. doi: 10.1103/PhysRev.152.1115.

[19]

A. Jüngel, "Transport Equations for Semiconductors," Lect. Notes Phys., 773, Springer, Berlin, 2009.

[20]

A. Jüngel, Effective velocity in compressible Navier-Stokes equations wirth third-order derivatives, Nonlinear Analysis, 74 (2011), 2813-2818. doi: 10.1016/j.na.2011.01.002.

[21]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.

[22]

A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, To appear in J. Stat. Phys., 145 (2011), 1661-1673.

[23]

A. Jüngel and J.-P. Milisić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution equations, Kinetic and related models, 4 (2011), 785-807.

[24]

A. Jüngel, M. C. Mariani and D. Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Meth. Appl. Sci., 12 (2002), 485-495. doi: 10.1142/S0218202502001751.

[25]

G. Kaniadakis and A. M. Scarfone, Nonlinear transformation for a class of gauged Schrödinger equations with complex nonlinearities, Reports on Math. Phys., 48 (2001), 115-121. doi: 10.1016/S0034-4877(01)80070-7.

[26]

J. L. López, Nonlinear Ginzburg-Landau-type approach to quantum dissipation, Phys. Rev. E., 69 (2004), 026110. doi: 10.1103/PhysRevE.69.026110.

[27]

J. L. López, J. Montejo-Gámez, A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics, Physica D, 238 (2009), 622-644.

[28]

R. Mosna, I. Hamilton and L. Delle Site, Variational approach to dequantization, J. Phys. A, 39 (2006), L229-L235. doi: 10.1088/0305-4470/39/14/L03.

[29]

P. Nattermann and W. Scherer, Nonlinear gauge transformations and exact solutions of the Doebner-Goldin equation, in "Nonlinear, Deformed and Irreversible Quantum Systems" (eds. Doebner et al.), World Scientific, (1995), 188-199.

[30]

P. Nattermann and R. Zhdanov, On integrable Doebner-Goldin equations, J. Phys. A, 29 (1996), 2869-2886. doi: 10.1088/0305-4470/29/11/021.

[31]

E. Nelson, Derivation of the Schrödinger equation from Newtonian Mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.

[32]

C. Sabatier, Multidimensional nonlinear Schrödinger equations with exponentially confined solutions, Inverse Problems, 6 (1990), L47-L53. doi: 10.1088/0266-5611/6/5/002.

[33]

A. Scarfone, Gauge equivalence among quantum nonlinear many body systems, Act. Appl. Math., 102 (2008), 179-217. doi: 10.1007/s10440-008-9213-7.

[34]

A. G. Ushveridze, Dissipative quantum mechanics. A special Doebner-Goldin equation, its properties and exact solutions, Phys. Lett. A, 185 (1994), 123-127. doi: 10.1016/0375-9601(94)90834-6.

[35]

A. G. Ushveridze, The special Doebner-Goldin equation as a fundamental equation of dissipative quantum mechanics, Phys. Lett. A, 185 (1994), 128-132. doi: 10.1016/0375-9601(94)90835-4.

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