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April  2014, 34(4): 1375-1396. doi: 10.3934/dcds.2014.34.1375

On the Lagrangian structure of quantum fluid models

Philipp Fuchs 1, , Ansgar Jüngel 1, and  Max von Renesse 2,

1. 

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

Institute for Mathematics, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

Received  January 2013 Revised  May 2013 Published  October 2013

Some quantum fluid models are written as the Lagrangian flow of mass distributions and their geometric properties are explored. The first model includes magnetic effects and leads, via the Madelung transform, to the electromagnetic Schrödinger equation in the Madelung representation. It is shown that the Madelung transform is a symplectic map between Hamiltonian systems. The second model is obtained from the Euler-Lagrange equations with friction induced from a quadratic dissipative potential. This model corresponds to the quantum Navier-Stokes equations with density-dependent viscosity. The fact that this model possesses two different energy-dissipation identities is explained by the definition of the Noether currents.
Keywords: magnetic Schrödinger equation, symplectic form, Noether theory, quantum Navier-Stokes equations, osmotic velocity., Geometric mechanics.
Mathematics Subject Classification: Primary: 35Q40, 37J10, 58D3.
Citation: Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Second edition, (2008). Google Scholar

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A. Arnold, Mathematical properties of quantum evolution equations,, in, 1946 (2008), 45. doi: 10.1007/978-3-540-79574-2_2. Google Scholar

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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

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B. van Brunt, "The Calculus of Variations,'', Universitext, (2004). Google Scholar

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P. Dirac, The Lagrangian in quantum mechanics,, Phys. Z. Sowjet., 3 (1933), 64. Google Scholar

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Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics,, Acta Mech., 23 (1975), 17. Google Scholar

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

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J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws,, J. Math. Pures Appl. (9), 97 (2012), 318. doi: 10.1016/j.matpur.2011.11.004. Google Scholar

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L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'', World Scientific Publishing Co. Pte. Ltd., (2005). doi: 10.1142/9789812567635. Google Scholar

[12]

T. Frankel, "The Geometry of Physics. An Introduction,'', Cambridge University Press, (1997). Google Scholar

[13]

G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control,, Intern. J. Tomogr. Stat., 5 (2007), 109. Google Scholar

[14]

J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems,, Phys. Lett. A, 317 (2003), 255. doi: 10.1016/j.physleta.2003.08.028. Google Scholar

[15]

A. Jüngel, "Transport Equations for Semiconductors,'', Lecture Notes in Physics, 773 (2009). doi: 10.1007/978-3-540-89526-8. Google Scholar

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

[17]

A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach,, J. Stat. Phys., 145 (2011), 1661. doi: 10.1007/s10955-011-0388-3. Google Scholar

[18]

A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution,, Kinetic Related Models, 4 (2011), 785. doi: 10.3934/krm.2011.4.785. Google Scholar

[19]

J. Lafferty, The density manifold and configuration space quantization,, Trans. Amer. Math. Soc., 305 (1988), 699. doi: 10.1090/S0002-9947-1988-0924776-9. Google Scholar

[20]

J. Lott, Some geometric calculations on Wasserstein space,, Commun. Math. Phys., 277 (2008), 423. doi: 10.1007/s00220-007-0367-3. Google Scholar

[21]

E. Madelung, Quantentheorie in hydrodynamischer Form,, Z. Phys., 40 (1926), 322. doi: 10.1007/BF01400372. Google Scholar

[22]

P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit,, J. Funct. Anal., 259 (2010), 1542. doi: 10.1016/j.jfa.2010.05.013. Google Scholar

[23]

R. McCann, Polar factorization of maps on Riemannian manifolds,, GAFA Geom. Funct. Anal., 11 (2001), 589. doi: 10.1007/PL00001679. Google Scholar

[24]

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

[25]

F. Otto, The geometry of dissipative evolution equations: The porous-medium equation,, Commun. Part. Diff. Eqs., 26 (2001), 101. doi: 10.1081/PDE-100002243. Google Scholar

[26]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,, J. Funct. Anal., 173 (2000), 361. doi: 10.1006/jfan.1999.3557. Google Scholar

[27]

M.-K. von Renesse, On optimal transport view on Schrödinger's equation,, Canad. Math. Bull., 55 (2012), 858. doi: 10.4153/CMB-2011-121-9. Google Scholar

[28]

W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics,, SIAM Review, 23 (1981), 467. doi: 10.1137/1023098. Google Scholar

[29]

R. Talman, "Geometric Mechanics,'', Wiley, (2000). Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Second edition, (2008). Google Scholar

[2]

A. Arnold, Mathematical properties of quantum evolution equations,, in, 1946 (2008), 45. doi: 10.1007/978-3-540-79574-2_2. Google Scholar

[3]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319. doi: 10.5802/aif.233. Google Scholar

[4]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2000), 375. doi: 10.1007/s002110050002. Google Scholar

[5]

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

[6]

B. van Brunt, "The Calculus of Variations,'', Universitext, (2004). Google Scholar

[7]

P. Dirac, The Lagrangian in quantum mechanics,, Phys. Z. Sowjet., 3 (1933), 64. Google Scholar

[8]

Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics,, Acta Mech., 23 (1975), 17. Google Scholar

[9]

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

[10]

J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws,, J. Math. Pures Appl. (9), 97 (2012), 318. doi: 10.1016/j.matpur.2011.11.004. Google Scholar

[11]

L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'', World Scientific Publishing Co. Pte. Ltd., (2005). doi: 10.1142/9789812567635. Google Scholar

[12]

T. Frankel, "The Geometry of Physics. An Introduction,'', Cambridge University Press, (1997). Google Scholar

[13]

G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control,, Intern. J. Tomogr. Stat., 5 (2007), 109. Google Scholar

[14]

J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems,, Phys. Lett. A, 317 (2003), 255. doi: 10.1016/j.physleta.2003.08.028. Google Scholar

[15]

A. Jüngel, "Transport Equations for Semiconductors,'', Lecture Notes in Physics, 773 (2009). doi: 10.1007/978-3-540-89526-8. Google Scholar

[16]

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

[17]

A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach,, J. Stat. Phys., 145 (2011), 1661. doi: 10.1007/s10955-011-0388-3. Google Scholar

[18]

A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution,, Kinetic Related Models, 4 (2011), 785. doi: 10.3934/krm.2011.4.785. Google Scholar

[19]

J. Lafferty, The density manifold and configuration space quantization,, Trans. Amer. Math. Soc., 305 (1988), 699. doi: 10.1090/S0002-9947-1988-0924776-9. Google Scholar

[20]

J. Lott, Some geometric calculations on Wasserstein space,, Commun. Math. Phys., 277 (2008), 423. doi: 10.1007/s00220-007-0367-3. Google Scholar

[21]

E. Madelung, Quantentheorie in hydrodynamischer Form,, Z. Phys., 40 (1926), 322. doi: 10.1007/BF01400372. Google Scholar

[22]

P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit,, J. Funct. Anal., 259 (2010), 1542. doi: 10.1016/j.jfa.2010.05.013. Google Scholar

[23]

R. McCann, Polar factorization of maps on Riemannian manifolds,, GAFA Geom. Funct. Anal., 11 (2001), 589. doi: 10.1007/PL00001679. Google Scholar

[24]

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

[25]

F. Otto, The geometry of dissipative evolution equations: The porous-medium equation,, Commun. Part. Diff. Eqs., 26 (2001), 101. doi: 10.1081/PDE-100002243. Google Scholar

[26]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality,, J. Funct. Anal., 173 (2000), 361. doi: 10.1006/jfan.1999.3557. Google Scholar

[27]

M.-K. von Renesse, On optimal transport view on Schrödinger's equation,, Canad. Math. Bull., 55 (2012), 858. doi: 10.4153/CMB-2011-121-9. Google Scholar

[28]

W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics,, SIAM Review, 23 (1981), 467. doi: 10.1137/1023098. Google Scholar

[29]

R. Talman, "Geometric Mechanics,'', Wiley, (2000). Google Scholar

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