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Dissipative Quantum Mechanics Using GENERIC

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Recent Trends in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 35))

Abstract

Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (Phys. Rev. A 82:052119(11), 2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition:

$$\displaystyle{\dot{q} = \mathbb{J}(q)\mathrm{D}\mathcal{E}(q) + \mathbb{K}(q)\mathrm{D}\mathcal{S}(q).}$$

One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo–Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare our gradient flow structure for quantum systems with the Wasserstein gradient flow for the Fokker–Planck equation and the entropy gradient flow for reversible Markov chains.

Dedicated to Jürgen Scheurle, who exited my love for geometric mechanics, on the occasion of his 60th birthday

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References

  1. Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin/Cummings Publishing Co., Advanced Book Program, Reading, MA (1978). Second edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman

    Google Scholar 

  2. Bhatia, R., Holbrook, J.A.R.: On the Clarkson-McCarthy inequalities. Math. Ann. 281(1), 7–12 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlen, E.A., Maas, J.: An analog of the 2-Wasserstein metric in non-commutative probability under which the Fermionic Fokker–Planck equation is gradient flow for the entropy. arXiv 1203.5377, to appear in Commun. Math. Physics (2013)

    Google Scholar 

  4. Erbar, M., Maas, J.: Ricci curvature of finite Markov chains via convexity of the entropy. arXiv: 1111.2687 (2011)

    Google Scholar 

  5. Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonself-adjoint Operators. AMS Transl. Mathem. Monographs (1969)

    Google Scholar 

  6. Grabert, H.: Nonlinear relaxation and fluctuations of damped quantum systems. Z. Phys. B 49(2), 161–172 (1982)

    Article  MathSciNet  Google Scholar 

  7. Grmela, M.: Why GENERIC? J. Non-Newtonian Fluid Mech. 165, 980–986 (2010)

    Article  MATH  Google Scholar 

  8. Grmela, M., Öttinger, H.C.: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E (3) 56(6), 6620–6632 (1997)

    Google Scholar 

  9. Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York/London (1974)

    MATH  Google Scholar 

  10. Jelić, A., Hütter, M., Öttinger, H.C.: Dissipative electromagnetism from a nonequilibrium thermodynamics perspective. Phys. Rev. E 74, 041126, 8 pp (2006)

    Google Scholar 

  11. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kubo, R., Toda, M., Hashitsume, N.: Nonequilibrium Statistical Mechanics, 2nd edn. Springer, New York (1991)

    MATH  Google Scholar 

  13. Kubo, R.: The fluctuation-dissipation theorem. Part I. Rep. Prog. Phys. 29, 255–284 (1966)

    Article  Google Scholar 

  14. Liero, M., Mielke, A.: Gradient structures and geodesic convexity for reaction-diffusion systems. Phil. Trans. Royal Soc. A (2013) To appear. WIAS preprint 1701 (April 2012)

    Google Scholar 

  15. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119–130 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lindblad, G.: Nonequilibrium Entropy and Irreversibility, vol. 5. D. Reidel Publishing, Dordrecht (1983)

    Book  Google Scholar 

  17. Maas, J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261, 2250–2292 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edn. Springer, New York (1999)

    Google Scholar 

  19. Mielke, A.: Formulation of thermoelastic dissipative material behavior using GENERIC. Contin. Mech. Thermodyn. 23(3), 233–256 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Part. Diff. Eqns. (2012). Online 10.1007/s00526-012-0538-8

    Google Scholar 

  21. Mielke, A., Neidhardt, H., Racec, P.: Coupling simple heat baths to quantum systems using GENERIC. In preparation (2013)

    Google Scholar 

  22. Mielke, A., Thomas, M.: GENERIC – a powerful tool for thermomechanical modeling. In preparation (2012)

    Google Scholar 

  23. Michor, P.W., Petz, D., Andai, A.: On the curvature of a certain Riemannian space of matrices. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3(2), 199–212 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Öttinger, H.C., Grmela, M.: Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys. Rev. E (3) 56(6), 6633–6655 (1997)

    Google Scholar 

  25. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Öttinger, H.C.: Beyond Equilibrium Thermodynamics. Wiley, New Jersey (2005)

    Book  Google Scholar 

  27. Öttinger, H.C.: The nonlinear thermodynamic quantum master equation. Phys. Rev. A 82, 052119(11) (2010)

    Article  Google Scholar 

  28. Öttinger, H.C.: The geometry and thermodynamics of dissipative quantum systems. Europhys. Lett. 94, 10006(6) (2011)

    Article  Google Scholar 

  29. Petz, D., Sudár, C.: Extending the Fisher metric to density matrices. In: Barndorff-Nielsen, O., Vendel Jensen, E. (eds.), Geometry in Present Day Science, pp. 21–34. World Scientific, Singapore (1999)

    Google Scholar 

  30. Petz, D.: Geometry of canonical correlation on the state space of a quantum system. J. Math. Phys. 35(2), 780–795 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Streater, R.F.: Information geormetry and reduced quantum description. Rep. Math. Phys. 38, 419–436 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Streater, R.F.: The analytic quantum information manifold. Canad. Math. Soc. Conf. Proc. 29, 603–611 (2000)

    MathSciNet  Google Scholar 

  33. Tanimura, Y.: Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems. J. Phys. Soc. Jpn. 75(8), 082001–39 (2006)

    Article  Google Scholar 

  34. von Renesse, M.-K.: An optimal transport view of Schrödinger equation. Canad. Math. Bull. 55, 858–869 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Weiss, U.: Quantum Dissipative Systems, 2nd edn. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

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Acknowledgements

The author is grateful for helpful and stimulation discussions with Hans-Christian Öttinger and for Hagen Neidhardt’s help in proving Proposition 21.2. He also thanks P.S. Krishnaprasad for discussion concerning \(\mathcal{C}_{\rho }\) as a non-commutative generalization of the Fisher information. The author thanks an unknown referee for many helpful remarks. The research was partially supported by the European Research Council under AnaMultiScale ERC-2010-AdG 267802.

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Authors and Affiliations

  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

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  1. Alexander Mielke
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Correspondence to Alexander Mielke .

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Editors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Andreas Johann

  2. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Hans-Peter Kruse

  3. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Florian Rupp

  4. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Stephan Schmitz

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Mielke, A. (2013). Dissipative Quantum Mechanics Using GENERIC. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_21

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