On the derivation and mathematical analysis of some quantum–mechanical models accounting for Fokker–Planck type dissipation: Phase space, Schrödinger and hydrodynamic descriptions

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1. R. Alicki, K. Lendi, Quantum dynamical semigroups and applications, Springer, Berlin, (1987). Zbl0652.46055
2. V. Ambegaokar, Quantum brownian motion and its classical limit, Berichte der Bunsengesellschaft für Physikalische Chemie, 95, 400–404 (1991). 
3. M. G. Ancona, Density–gradient theory analysis of electron distributions in heterostructures, Superlattics and Microstructures, 7, 119–130 (1990). 
4. P. Antonelli, P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys. 287, 657–686 (2009). Zbl1177.82127
5. A. Arnold, Mathematical properties of quantum evolution equations. In: G. Allaire, A. Arnold, P. Degond, and T. Y. Hou. Quantum Transport–Modelling, Analysis and Asymptotics. Lect. Notes Math. 1946, 45–109 (2008). 
6. A. Arnold, J. A. Carrillo, I. Gamba, C.–W. Shu, Low and high field scaling limits for the Vlasov– and Wigner– Poisson–Fokker–Planck systems, Transp. Theory Stat. Phys. 100, 121–153 (2001). [Crossref] Zbl1106.82381
7. A. Arnold, E. Dhamo, C. Mancini, Dispersive effects in quantum kinetic equations, Indiana Univ. Math. J. 56, 1299–1332 (2007). Zbl1161.35040
8. A. Arnold, E. Dhamo, C. Mancini, The Wigner–Poisson–Fokker–Planck system: global–in–time solutions and dispersive effects. Annales de l’IHP (C)–Analyse non lineaire 24, 645–676 (2007). Zbl1121.82031
9. A. Arnold, F. Fagnola, L. Neumann, Quantum Fokker–Planck models: the Lindblad and Wigner approaches. Quantum Probability and related Topics, Proceedings of the 28th Conference (Series: QP–PQ: Quantum Probability and White Noise Analysis–Vol. 23), J.C. García, R. Quezada, S. B. Sontz (Eds.), World Scientific, (2008). Zbl1180.81089
10. A. Arnold, A. Jüngel, Multi–scale modeling of quantum semiconductor devices. In Analysis, modeling and simulation multi scale problems, A. Mielke ed., Springer, Berlin–Heidelberg, 331–363 (2006). 
11. A. Arnold, J. L. López, P. A. Markowich, J. Soler, An analysis of quantum Fokker–Planck models: a Wigner function approach, Rev. Mat. Iberoamericana 20, 771–814 (2004). [Crossref] Zbl1062.35097
12. G. Auberson, P. C. Sabatier, On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys. 35, 4028–4040 (1994). [Crossref] Zbl0813.35112
13. A. V. Avdeenkov, K. G. Zloshchastiev, Quantum Bose liquids with logarithmic nonlinearity: self–sustainability and emergence of spatial extent, J. Phys. B.: At. Mol. Opt. Phys. 44, 195303 (2011). [Crossref] 
14. P. Bechouche, F. Poupaud, J. Soler, Quantum transport and Boltzmann operators, J. Stat. Phys. 122, 417–436 (2006). [Crossref] Zbl1149.82338
15. F. Benatti, R. Floreanini, Open quantum dynamics: Complete positivity and entanglement, Int. J. Mod. Phys. B 19, 3063–3139 (2005). [Crossref] Zbl1152.81861
16. I. Bialynicki–Birula, On the linearity of the Schrödinger equation, Brazilian J. Phys. 35, 211–215 (2005),. 
17. I. Bialynicki–Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys. 100, 62–93 (1976). [Crossref] 
18. I. Bialynicki–Birula, J. Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Physica Scripta 209, 539–544 (1979). Zbl1063.81528
19. D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. I and II, Phys. Rev. 85, 166–179 & 180–193 (1952). [Crossref] Zbl0046.21004
20. O. Bokanowski, J. L. López, J. Soler, On an exchange interaction model for quantum transport: the Schrödinger– Poisson–Slater system, Math. Models Meth. Appl. Sci. 13, 1–16 (2003). Zbl1073.35188
21. M. Bottiglieri, C. Godano, G. Lauro, Volcanic eruptions: Initial state of magma melt pulse unloading, Europhys. Lett. 97, 29001 (2012). [Crossref] 
22. L. Brüll, L. Lange, The Schrödinger–Langevin equation: Special solutions and nonexistence of solitary waves, J. Math. Phys. 25, 786–790 (1984). [Crossref] Zbl0555.35123
23. S. Brull, F. Méhats, Derivation of viscous correction terms for the isothermal quantum euler model, Z. Angew. Math. Mech., 90, 219–230 (2010). [Crossref] Zbl05681525
24. A. Budini, A. K. Chattah, M. O. Cáceres, On the quantum dissipative generator: weak–coupling approximation and stochastic approach, J. Phys. A: Math. Gen. 32, 631–646 (1999). [Crossref] Zbl0964.82033
25. M. O. Cáceres, A. K. Chattah, On the Schrödinger–Langevin picture and the master equation, Physica A 234, 322–340 (1996). 
26. M. O. Cáceres, A. K. Chattah, The quantum random walk within the Schrödinger–Langevin approach, J. Molecular Liquids 71, 187–194 (1997). 
27. A. O. Caldeira, A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121, 587–616 (1983). Zbl0585.60082
28. J. A. Cañizo, J. L. López, J. Nieto, Global L1 theory and regularity of the 3D nonlinear Wigner–Poisson–Fokker– Planck system, J. Diff. Equ. 198, 356–373 (2004). Zbl1039.35093
29. F. Castella, L. Erdös, F. Frommlet, P. A. Markowich, Fokker–Planck equations as scaling limits of reversible quantum systems, J. Stat. Phys. 100, 543–601 (2000). [Crossref] Zbl0988.82038
30. T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos 22, Rio de Janeiro, (1989). 
31. T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Analysis T. M. A. 7, 1127–1140 (1983). Zbl0529.35068
32. T. Cazenave, A. Haraux, Equations d’évolution avec non linéarité logarithmique, Annals Fac. Sci. Univ. Toulouse 2, 21–55 (1980). [Crossref] Zbl0411.35051
33. T. Cazenave, A. Haraux, Introduction aux problemes d’évolution semi–linéaires. Mathématiques et Applications #1. Ellipses, Paris, (1990). Zbl0786.35070
34. A. M. Chebotarev, F. Fagnola, Sufficient conditions for conservativity of quantum dynamical semigroups, J. Funct. Anal. 118, 131–153 (1993). [Crossref] Zbl0801.46083
35. C. Cid, J. Dolbeault, Defocusing nonlinear Schrödinger equation: confinement, stability and asymptotic stability, 2001 (preliminary version). 
36. N. Cufaro Petroni, S. De Martino, S. De Siena, F. Illuminati, Stochastic-hydrodynamic model of halo formation in charged particle beams, Phys. Rev. ST Accel. Beams 6, 034206 (2003). 
37. M. P. Davidson, Comments on the nonlinear Schrödinger equation, Il Nuovo Cimento B V116B, 1291–1296 (2001). 
38. E. B. Davies, Markovian master equations, Commun. Math. Phys. 39, 91–110 (1974). [Crossref] Zbl0294.60080
39. E. B. Davies, Quantum Theory of Open Systems, Academic Press, New York, (1976). Zbl0388.46044
40. L. De Broglie, Wave mechanics and the atomic structure of matter and radiation, J. Phys. 8, 225–241 (1927). 
41. D. De Falco, D. Tamascelli, Quantum annealing and the Schrödinger–Langevin–Kostin equation. Phys. Rev. A 79, 012315 (2009). [Crossref] 
42. S. De Martino, M. Falanga, C. Godano, G. Lauro, Logarithmic Schrödinger–like equation as a model for magma transport, Europhys. Lett. 63, 472–475 (2003). [Crossref] 
43. S. De Martino, G. Lauro, Soliton–like solutions for a capillary fluid, Waves and Stability in Continuous Media, Proceedings of the 12th Conference on WASCOM, (2003). Zbl1066.35092
44. P. Degond, F. Méhats, C. Ringhofer, Quantum energy–transport and drift–diffusion models. J. Stat. Phys. 118, 625–665 (2005). [Crossref] Zbl1126.82314
45. P. Degond, C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys. 112, 587–628 (2003). [Crossref] Zbl1035.82028
46. H. Dekker, Quantization of the linearly damped harmonic oscillator, Phys. Rev. A 16, 2126–2134 (1977). [Crossref] 
47. H. Dekker, M. C. Valsakumar, A fundamental constraint on quantum mechanical diffusion coefficients, Phys. Lett. A 104, 67–71 (1984). [Crossref] 
48. L. Delle Site, I. Hamilton, R. Mosna, Variational approach to dequantization, J. Phys. A 39, L229–L235 (2006). Zbl1089.81031
49. B. Desjardins, C–K. Lin, On the semiclassical limit of the general modified NLS equation, J. Math. Anal. Appl. 260, 546–571 (2001). [Crossref] Zbl0980.35153
50. B. Desjardins, C–K. Lin , T.–C. Tso, Semiclassical limit of the derivative nonlinear Schrödinger equation, Math. Models Meth. Appl. Sci. 10, 261–285 (2000). Zbl1012.81016
51. L. Diósi, Caldeira–Leggett master equation and medium temperatures, Physica A 199, 517–526 (1993). 
52. L. Diósi, On high–temperature Markovian equation for quantum Brownian motion, Europhys. Lett. 22, 1–3 (1993). [Crossref] 
53. H. D. Doebner, G. A. Goldin, On a general nonlinear Schrödinger equation admitting diffusion currents, Phys. Lett. A 162, 397–401 (1992). [Crossref] 
54. H. D. Doebner, G. A. Goldin, Properties of nonlinear Schrödinger equations associated with diffeomorphism group representations, J. Phys. A: Math. Gen. 27, 1771–1780 (1994). [Crossref] Zbl0842.35113
55. H. D. Doebner, G. A. Goldin, Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations, Phys. Rev. A 54, 3764–3771 (1996). [Crossref] 
56. H. D. Doebner, G. A. Goldin, P. Nattermann, A family of nonlinear Schrödinger equations: linearizing transformations and resulting structure. In: Quantization, Coherent States and Complex Structures, J.–P. Antoine et al. (Eds.), 27–31, Plenum, (1996). 
57. I. Fényes,, Eine wahrscheinlichkeitstheoretische begrundung und interpretation der Quantenmechanik, Z. Phys. 132, 81–103 (1952). [Crossref] Zbl0048.44201
58. D. Ferry, J. R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling. Phys. Rev. B 48, 7944–7950 (1993). [Crossref] 
59. R. P. Feynman, F. L. Vernon, The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. 24, 118–173 (1963). [Crossref] 
60. W. R. Frensley, Boundary conditions for open quantum systems driven far from equilibrium, Rev. Mod. Phys. 62, 745–791 (1990). [Crossref] 
61. L. Fritsche, M. Haugk, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Ann. Phys. 12, 371–403 (2003). [Crossref] Zbl1037.81058
62. P. Fuchs, A. Jüngel, M. von Renesse, On the Lagrangian structure of quantum fluid models. Preprint, Vienna University of Technology, 2011. Online available at http://www.asc.tuwien.ac.at/index.php?id=132. Zbl1276.35130
63. S. Gao, Dissipative quantum dynamics with a Lindblad functional, Phys. Rev. Lett. 79, 3101–3104 (1997). [Crossref] Zbl0944.82018
64. P. Garbaczewski, Modular Schrödinger equation and dynamical duality, Phys. Rev. E 78, 031101 (2008). [Crossref] 
65. C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54, 409–427 (1994). [Crossref] Zbl0815.35111
66. P. Gérard, Remarques sur l’analyse semi–classique de l’équation de Schrödinger non linéaire, Séminaire EDP de l’École Politechnique, Palaiseau, France (1992–93), Lecture XIII. 
67. P. Gérard, P. A. Markowich, N. J. Mauser, F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50, 321–377 (1997). Zbl0881.35099
68. N. Gisin, Microscopic derivation of a class of non–linear dissipative Schrödinger–like equations, Physica A 111, 364–370 (1982). [Crossref] 
69. V. Gorini, A. Kossakowski, E. C. G. Sudarshan, Completely positive dynamical semigroups of N–level systems, J. Math. Phys. 17, 821–825 (1976). [Crossref] 
70. E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proceedings of Amer. Math. Soc., 126, 523–530 (1998). Zbl0910.35115
71. M. P. Gualdani, A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Europ. J. Appl. Math., 15, 577–595 (2004). 
72. M. P. Gualdani, A. Jüngel, G. Toscani, Exponential decay in time of solutions of the viscous quantum hydrodynamic equations, Appl. Math. Lett. 16, 1273–1278 (2003). [Crossref] Zbl1042.82047
73. F. Guerra, Structural aspects of stochastic mechanics and stochastic field theory, Phys. Rep. 77, 263–312 (1981). [Crossref] 
74. P. Guerrero, Analysis of dissipation and diffusion mechanisms modeled by nonlinear PDEs in developmental biology and quantum mechanics, PhD Thesis, (2010). 
75. P. Guerrero, J. L. López, J. Montejo–Gámez, A wavefunction description of quantum Fokker–Planck dissipation: derivation, stationary dynamics and numerical approximation, in progress. Zbl1293.82018
76. P. Guerrero, J. L. López, J. Nieto, Global H1 solvability of the 3D logarithmic Schrödinger equation, Nonlinear Analysis: Real World Applications 11, 79–87 (2012). Zbl1180.81071
77. P. Guerrero, J. L. López, J. Montejo–Gámez, J. Nieto, Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner–Goldin type modeling quantum dissipation, J. Nonlinear Sci. 22, 631–663 (2012). [Crossref] Zbl1258.35003
78. R. Harvey, Navier–Stokes analog of quantum mechanics, Phys. Rev. 152, 1115 (1966). [Crossref] 
79. E. F. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomegeneous term to nuclear physics, Phys. Rev. A 32, 1201–1204 (1985). [PubMed][Crossref] 
80. B. L. Hu, J. P. Paz, Y. Zhang, Quantum Brownian motion in a general environment: exact master equation with nonlocal dissipation and colored noise, Phys. Rev. D 45, 2843–2861 (1992). [Crossref] Zbl1232.82011
81. G. Iche, P. Nozieres, Quantum brownian motion of a heavy particle: an adiabatic expansion, Physica A 91, 485–506 (1978). [Crossref] 
82. A. Isar, A. Sandulescu, W. Scheid, Purity and decoherence in the theory of a damped harmonic oscillator, Phys. Rev. E 60, 6371–6381 (1999). [Crossref] Zbl1062.82505
83. A. Jüngel, Transport Equations for Semiconductors, Lect. Notes Phys. 773, Springer, Berlin, (2009). 
84. A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal., 42, 1025–1045 (2010). [Crossref] Zbl1228.35083
85. A. Jüngel, Dissipative quantum fluid models, Riv. Mat. Univ. Parma, 3, 217–290 (2012). Zbl1257.82085
86. A. Jüngel, D. Mathes, A derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech 85, 806–814 (2005). [Crossref] Zbl1077.76074
87. A. Jüngel, D. Matthes, J.–P. Milisic, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math. 67, 46–68 (2006),. [Crossref] Zbl1121.35117
88. A. Jüngel, M. C. Mariani, D. Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Meth. Appl. Sci. 12, 485–495 (2002). Zbl1215.81031
89. A. Jüngel, J.–P. Milisic, Full compressible Navier–Stokes equations for quantum fluids: Derivation and numerical solution equations, Kinetic and related models 4, 785–807 (2011). Zbl1231.35148
90. A. Jüngel, J. L. López, J. Montejo–Gámez, A new derivation of the quantum Navier–Stokes equations in the Wigner– Fokker–Planck approach, J. Stat. Phys. 145, 1661–1673 (2011). [Crossref] Zbl1238.82024
91. G. Kaniadakis, A. M. Scarfone, Nonlinear transformation for a class of gauged Schrödinger equations with complex nonlinearities, Reports on Math. Phys., 48, 115–121 (2001). Zbl1078.35114
92. A. Kossakowski, On necessary and sufficient conditions for a generator of a quantum dynamical semi–group, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astr. Phys. 20, 1021–1025 (1972). 
93. M. D. Kostin, On the Schrödinger–Langevin equation, J. Chem. Phys. 57, 3589–3591 (1972). [Crossref] 
94. M. D. Kostin, Friction and dissipative phenomena in quantum mechanics, J. Stat. Phys. 12, 145–151 (1975). [Crossref] 
95. G. Lauro, A note on a Korteweg fluid and the hydrodynamic form of the logarithmic Schrödinger equation, Geophys. and Astrophys. Fluid Dynamics 102, 373–380 (2008). 
96. N. A. Lemos, Dissipative forces and the algebra of operators in stochastic quantum mechanics, Phys. Lett. A 78, 239–241 (1980). [Crossref] 
97. H. Li, C–K. Lin, Semiclassical limit and well–posedness of nonlinear Schrödinger–Poisson, EJDE 93, 1–17 (2003). Zbl1055.35111
98. G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48, 119–130 (1976). [Crossref] Zbl0343.47031
99. P. L. Lions, T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9, 553–618 (1993). [Crossref] 
100. H. Liu, E. Tadmor, Semiclassical limit of the nonlinear Schrödinger–Poisson equation with subcritical initial data, Methods Appl. Anal. 9, 517–532 (2002). Zbl1166.35374
101. J. L. López, Nonlinear Ginzburg–Landau–type approach to quantum dissipation, Phys. Rev. E 69, 026110 (2004). [Crossref] 
102. J. L. López, J. Montejo–Gámez, A hydrodynamic approach to multidimensional dissipation–based Schrödinger models from quantum Fokker–Planck dynamics, Physica D 238, 622–644 (2009). Zbl1160.37430
103. J. L. López, J. Montejo–Gámez, On a rigorous interpretation of the quantum Schrödinger–Langevin operator in bounded domains, J. Math. Anal. Appl. 383, 365–378 (2011). [Crossref] Zbl1223.81126
104. J. L. López, J. Montejo–Gámez, On viscous quantum hydrodynamics associated with nonlinear Schrödinger– Doebner–Goldin models, Kinetic and related models 5, 517–536 (2012). Zbl1261.81065
105. J. L. López, J. Nieto, Global solutions of the mean–field, very high temperature Caldeira–Leggett master equation, Quart. Appl. Math. 64, 189–199 (2006). Zbl1115.82025
106. E. Madelung, Quantentheorie in hydrodynamischer form, Z. Physik 40, 322–326 (1927). [Crossref] Zbl52.0969.06
107. G. Manfredi, H. Haas, Self–consistent fluid model for a quantum electron gas, Phys. Rev. B 64, 075316 (2001). [Crossref] 
108. P. A. Markowich, C. A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, New York, (1990). Zbl0765.35001
109. F. Méhats, O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys. 140, 565–602 (2010). [Crossref] Zbl1201.82025
110. J. Montejo–Gámez, Estudio de fenómenos cuánticos disipativos mediante ecuaciones en derivadas parciales en la formulación de Schrödinger, PhD Thesis, (2011). 
111. A. B. Nassar, Fluid formulation of a generalised Schrödinger–Langevin equation, J. Phys. A: Math. Gen 18, L509– L511 (1985). [Crossref] 
112. P. Nattermann, W. Scherer, Nonlinear gauge transformations and exact solutions of the Doebner–Goldin equation. In: Nonlinear, Deformed and Irreversible Quantum Systems, Doebner et al. (Eds.), 188–199, World Scientific, (1995). Zbl0942.81546
113. P. Nattermann, R. Zhdanov, On integrable Doebner–Goldin equations, J. Phys. A 29, 2869–2886 (1996). 
114. E. Nelson, Derivation of the Schrödinger equation from Newtonian Mechanics, Phys. Rev. 150, 1079–1085 (1966). [Crossref] 
115. T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45, 137–163 (1996). Zbl0859.35117
116. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, (1983). Zbl0516.47023
117. E. Pollak, S. Zhang, Quantum dynamics for dissipative systems: A numerical study of the Wigner–Fokker–Planck equation, J. Chem. Phys. 118, 4357–4364 (2003). 
118. P. C. Sabatier, Multidimensional nonlinear Schrödinger equations with exponentially confined solutions, Inverse Problems 6, L47–L53 (1990). Zbl0725.35100
119. O. Sánchez, J. Soler, Long–time dynamics of the Schrödinger–Poisson–Slater system, J. Stat. Phys. 114, 179–204 (2004). [Crossref] Zbl1060.82039
120. A. L. Sanin, A. A. Smirnovsky, Oscillatory motion in confined potential systems with dissipation in the context of the Schrödinger–Langevin–Kostin equation, Phys. Lett. A 372, 21–27 (2007). [Crossref] Zbl1217.81070
121. A. Scarfone, Gauge equivalence among quantum nonlinear many body systems, Act. Appl. Math. 102, 179–217 (2008). [Crossref] Zbl1178.35350
122. B.-S. K. Skagerstam, Stochastic mechanics and dissipative forces, J. Math. Phys. 18, 308–311 (1977). [Crossref] 
123. C. Sparber, J. A. Carrillo, J. Dolbeault, P. A. Markowich, On the long time behavior of the quantum Fokker–Planck equation. Monatsh. Math. 141, 237–257 (2004). [Crossref] Zbl1061.35095
124. Y. Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems, J. Phys. Soc. Japan 75, 082001 (2006). [Crossref] 
125. H. Teissmann, The Cauchy problem for the Doebner–Goldin equation, Physical applications and mathematical aspects of geometry, groups and algebras, H. D. Doebner, P. Nattermann and W. Scherer (Eds.), World Scientific, 433–438 (1997). 
126. W. G. Unruh, W. H. Zurek, Reduction of a wave packet in quantum Brownian motion, Phys. Rev. D 40, 1071–1094 (1989). [Crossref] 
127. A. G. Ushveridze, Dissipative quantum mechanics. A special Doebner–Goldin equation, its properties and exact solutions, Phys. Lett. A 185, 123–127 (1994). Zbl0959.81523
128. A. G. Ushveridze, The special Doebner–Goldin equation as a fundamental equation of dissipative quantum mechanics, Phys. Lett. A 185, 128–132 (1994). Zbl0959.81524
129. B. Vacchini, Completely positive quantum dissipation, Phys. Rev. Lett. 84, 1374–1377 (2000). [PubMed][Crossref] Zbl1042.82590
130. P. Ván, T. Fülöp, Stability of stationary solutions of the Schrödinger–Langevin equation, Phys. Lett. A 323, 374–381 (2004). [Crossref] Zbl1118.81450
131. N. G. Van Kampen, Stochastic process in Physics and Chemistry (2nd edition), North–Holland, Amsterdam , (1992). 
132. T. C. Wallstrom, Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations, Phys. Rev. A 49, 1613–1617 (1994). [PubMed][Crossref] 
133. U. Weiss, Quantum dissipative systems, World Scientific, Singapore, (1993). Zbl1137.81300
134. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749–759 (1932). [Crossref] Zbl58.0948.07
135. Y. Yan, Quantum Fokker–Planck theory in a non–Gaussian–Markovian medium, Phys. Rev. A 58, 2721–2732 (1998). [Crossref] 
136. K. Yasue, A note on the derivation of the Schrödinger–Langevin equation, J. Stat. Phys. 16, 113–116 (1977). [Crossref] 

1. Paolo Di Sia, Overview of Drude-Lorentz type models and their applications