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2013 | 2 | 49-80
Tytuł artykułu

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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This paper is intended to provide the reader with a review of the authors’ latest results dealing with the modeling of quantum dissipation/diffusion effects at the level of Schrödinger systems, in connection with the corresponding phase space and fluid formulations of such kind of phenomena, especially in what concerns the role of the Fokker–Planck mechanism in the description of open quantum systems and the macroscopic dynamics associated with some viscous hydrodynamic models of Euler and Navier–Stokes type.
Twórcy
  • Departamento de Matemática Aplicada,
    Facultad de Ciencias, Universidad de Granada,
    18071 Granada, Spain, jllopez@ugr.es
  • Departamento de Matemática Aplicada,
    Facultad de Ciencias, Universidad de Granada,
    18071 Granada, Spain, jmontejo@ugr.es
Bibliografia
  • R. Alicki, K. Lendi, Quantum dynamical semigroups and applications, Springer, Berlin, (1987).
  • V. Ambegaokar, Quantum brownian motion and its classical limit, Berichte der Bunsengesellschaft für PhysikalischeChemie, 95, 400–404 (1991).
  • M. G. Ancona, Density–gradient theory analysis of electron distributions in heterostructures, Superlattics andMicrostructures, 7, 119–130 (1990).
  • P. Antonelli, P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math.Phys. 287, 657–686 (2009).
  • 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).
  • 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]
  • A. Arnold, E. Dhamo, C. Mancini, Dispersive effects in quantum kinetic equations, Indiana Univ. Math. J. 56,1299–1332 (2007).
  • A. Arnold, E. Dhamo, C. Mancini, The Wigner–Poisson–Fokker–Planck system: global–in–time solutions anddispersive effects. Annales de l’IHP (C)–Analyse non lineaire 24, 645–676 (2007).
  • A. Arnold, F. Fagnola, L. Neumann, Quantum Fokker–Planck models: the Lindblad and Wigner approaches. QuantumProbability and related Topics, Proceedings of the 28th Conference (Series: QP–PQ: Quantum Probabilityand White Noise Analysis–Vol. 23), J.C. García, R. Quezada, S. B. Sontz (Eds.), World Scientific, (2008).
  • A. Arnold, A. Jüngel, Multi–scale modeling of quantum semiconductor devices. In Analysis, modeling and simulationmulti scale problems, A. Mielke ed., Springer, Berlin–Heidelberg, 331–363 (2006).
  • A. Arnold, J. L. López, P. A. Markowich, J. Soler, An analysis of quantum Fokker–Planck models: a Wigner functionapproach, Rev. Mat. Iberoamericana 20, 771–814 (2004).[Crossref]
  • G. Auberson, P. C. Sabatier, On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys. 35,4028–4040 (1994).[Crossref]
  • A. V. Avdeenkov, K. G. Zloshchastiev, Quantum Bose liquids with logarithmic nonlinearity: self–sustainability andemergence of spatial extent, J. Phys. B.: At. Mol. Opt. Phys. 44, 195303 (2011).[Crossref]
  • P. Bechouche, F. Poupaud, J. Soler, Quantum transport and Boltzmann operators, J. Stat. Phys. 122, 417–436(2006).[Crossref]
  • F. Benatti, R. Floreanini, Open quantum dynamics: Complete positivity and entanglement, Int. J. Mod. Phys. B 19,3063–3139 (2005).[Crossref]
  • I. Bialynicki–Birula, On the linearity of the Schrödinger equation, Brazilian J. Phys. 35, 211–215 (2005),.
  • I. Bialynicki–Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys. 100, 62–93 (1976).[Crossref]
  • I. Bialynicki–Birula, J. Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Physica Scripta209, 539–544 (1979).
  • 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]
  • 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).
  • M. Bottiglieri, C. Godano, G. Lauro, Volcanic eruptions: Initial state of magma melt pulse unloading, Europhys.Lett. 97, 29001 (2012).[Crossref]
  • L. Brüll, L. Lange, The Schrödinger–Langevin equation: Special solutions and nonexistence of solitary waves, J.Math. Phys. 25, 786–790 (1984).[Crossref]
  • S. Brull, F. Méhats, Derivation of viscous correction terms for the isothermal quantum euler model, Z. Angew.Math. Mech., 90, 219–230 (2010).[Crossref]
  • A. Budini, A. K. Chattah, M. O. Cáceres, On the quantum dissipative generator: weak–coupling approximationand stochastic approach, J. Phys. A: Math. Gen. 32, 631–646 (1999).[Crossref]
  • M. O. Cáceres, A. K. Chattah, On the Schrödinger–Langevin picture and the master equation, Physica A 234,322–340 (1996).
  • M. O. Cáceres, A. K. Chattah, The quantum random walk within the Schrödinger–Langevin approach, J. MolecularLiquids 71, 187–194 (1997).
  • A. O. Caldeira, A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121, 587–616 (1983).
  • 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).
  • F. Castella, L. Erdös, F. Frommlet, P. A. Markowich, Fokker–Planck equations as scaling limits of reversiblequantum systems, J. Stat. Phys. 100, 543–601 (2000).[Crossref]
  • T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos 22, Rio deJaneiro, (1989).
  • T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Analysis T. M. A. 7, 1127–1140(1983).
  • T. Cazenave, A. Haraux, Equations d’évolution avec non linéarité logarithmique, Annals Fac. Sci. Univ. Toulouse2, 21–55 (1980).[Crossref]
  • T. Cazenave, A. Haraux, Introduction aux problemes d’évolution semi–linéaires. Mathématiques et Applications#1. Ellipses, Paris, (1990).
  • A. M. Chebotarev, F. Fagnola, Sufficient conditions for conservativity of quantum dynamical semigroups, J. Funct.Anal. 118, 131–153 (1993).[Crossref]
  • C. Cid, J. Dolbeault, Defocusing nonlinear Schrödinger equation: confinement, stability and asymptotic stability,2001 (preliminary version).
  • N. Cufaro Petroni, S. De Martino, S. De Siena, F. Illuminati, Stochastic-hydrodynamic model of halo formation incharged particle beams, Phys. Rev. ST Accel. Beams 6, 034206 (2003).
  • M. P. Davidson, Comments on the nonlinear Schrödinger equation, Il Nuovo Cimento B V116B, 1291–1296 (2001).
  • E. B. Davies, Markovian master equations, Commun. Math. Phys. 39, 91–110 (1974).[Crossref]
  • E. B. Davies, Quantum Theory of Open Systems, Academic Press, New York, (1976).
  • L. De Broglie, Wave mechanics and the atomic structure of matter and radiation, J. Phys. 8, 225–241 (1927).
  • D. De Falco, D. Tamascelli, Quantum annealing and the Schrödinger–Langevin–Kostin equation. Phys. Rev. A 79,012315 (2009).[Crossref]
  • S. De Martino, M. Falanga, C. Godano, G. Lauro, Logarithmic Schrödinger–like equation as a model for magmatransport, Europhys. Lett. 63, 472–475 (2003).[Crossref]
  • 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).
  • P. Degond, F. Méhats, C. Ringhofer, Quantum energy–transport and drift–diffusion models. J. Stat. Phys. 118,625–665 (2005).[Crossref]
  • P. Degond, C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys. 112, 587–628(2003).[Crossref]
  • H. Dekker, Quantization of the linearly damped harmonic oscillator, Phys. Rev. A 16, 2126–2134 (1977).[Crossref]
  • H. Dekker, M. C. Valsakumar, A fundamental constraint on quantum mechanical diffusion coefficients, Phys. Lett.A 104, 67–71 (1984).[Crossref]
  • L. Delle Site, I. Hamilton, R. Mosna, Variational approach to dequantization, J. Phys. A 39, L229–L235 (2006).
  • B. Desjardins, C–K. Lin, On the semiclassical limit of the general modified NLS equation, J. Math. Anal. Appl.260, 546–571 (2001).[Crossref]
  • 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).
  • L. Diósi, Caldeira–Leggett master equation and medium temperatures, Physica A 199, 517–526 (1993).
  • L. Diósi, On high–temperature Markovian equation for quantum Brownian motion, Europhys. Lett. 22, 1–3 (1993).[Crossref]
  • H. D. Doebner, G. A. Goldin, On a general nonlinear Schrödinger equation admitting diffusion currents, Phys.Lett. A 162, 397–401 (1992).[Crossref]
  • H. D. Doebner, G. A. Goldin, Properties of nonlinear Schrödinger equations associated with diffeomorphism grouprepresentations, J. Phys. A: Math. Gen. 27, 1771–1780 (1994).[Crossref]
  • H. D. Doebner, G. A. Goldin, Introducing nonlinear gauge transformations in a family of nonlinear Schrödingerequations, Phys. Rev. A 54, 3764–3771 (1996).[Crossref]
  • H. D. Doebner, G. A. Goldin, P. Nattermann, A family of nonlinear Schrödinger equations: linearizing transformationsand resulting structure. In: Quantization, Coherent States and Complex Structures, J.–P. Antoine et al.(Eds.), 27–31, Plenum, (1996).
  • I. Fényes,, Eine wahrscheinlichkeitstheoretische begrundung und interpretation der Quantenmechanik, Z. Phys.132, 81–103 (1952).[Crossref]
  • D. Ferry, J. R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor devicemodeling. Phys. Rev. B 48, 7944–7950 (1993).[Crossref]
  • 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]
  • W. R. Frensley, Boundary conditions for open quantum systems driven far from equilibrium, Rev. Mod. Phys. 62,745–791 (1990).[Crossref]
  • L. Fritsche, M. Haugk, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Ann.Phys. 12, 371–403 (2003).[Crossref]
  • P. Fuchs, A. Jüngel, M. von Renesse, On the Lagrangian structure of quantum fluid models. Preprint, ViennaUniversity of Technology, 2011. Online available at http://www.asc.tuwien.ac.at/index.php?id=132.
  • S. Gao, Dissipative quantum dynamics with a Lindblad functional, Phys. Rev. Lett. 79, 3101–3104 (1997).[Crossref]
  • P. Garbaczewski, Modular Schrödinger equation and dynamical duality, Phys. Rev. E 78, 031101 (2008).[Crossref]
  • C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54, 409–427(1994).[Crossref]
  • P. Gérard, Remarques sur l’analyse semi–classique de l’équation de Schrödinger non linéaire, Séminaire EDP del’École Politechnique, Palaiseau, France (1992–93), Lecture XIII.
  • P. Gérard, P. A. Markowich, N. J. Mauser, F. Poupaud, Homogenization limits and Wigner transforms, Comm. PureAppl. Math. 50, 321–377 (1997).
  • N. Gisin, Microscopic derivation of a class of non–linear dissipative Schrödinger–like equations, Physica A 111,364–370 (1982).[Crossref]
  • V. Gorini, A. Kossakowski, E. C. G. Sudarshan, Completely positive dynamical semigroups of N–level systems, J.Math. Phys. 17, 821–825 (1976).[Crossref]
  • E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proceedings of Amer. Math.Soc., 126, 523–530 (1998).
  • M. P. Gualdani, A. Jüngel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, Europ. J.Appl. Math., 15, 577–595 (2004).
  • M. P. Gualdani, A. Jüngel, G. Toscani, Exponential decay in time of solutions of the viscous quantum hydrodynamicequations, Appl. Math. Lett. 16, 1273–1278 (2003).[Crossref]
  • F. Guerra, Structural aspects of stochastic mechanics and stochastic field theory, Phys. Rep. 77, 263–312 (1981).[Crossref]
  • P. Guerrero, Analysis of dissipation and diffusion mechanisms modeled by nonlinear PDEs in developmental biologyand quantum mechanics, PhD Thesis, (2010).
  • P. Guerrero, J. L. López, J. Montejo–Gámez, A wavefunction description of quantum Fokker–Planck dissipation:derivation, stationary dynamics and numerical approximation, in progress.
  • P. Guerrero, J. L. López, J. Nieto, Global H1 solvability of the 3D logarithmic Schrödinger equation, NonlinearAnalysis: Real World Applications 11, 79–87 (2012).
  • P. Guerrero, J. L. López, J. Montejo–Gámez, J. Nieto, Wellposedness of a nonlinear, logarithmic Schrödinger equationof Doebner–Goldin type modeling quantum dissipation, J. Nonlinear Sci. 22, 631–663 (2012).[Crossref]
  • R. Harvey, Navier–Stokes analog of quantum mechanics, Phys. Rev. 152, 1115 (1966).[Crossref]
  • E. F. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomegeneous term to nuclearphysics, Phys. Rev. A 32, 1201–1204 (1985).[PubMed][Crossref]
  • B. L. Hu, J. P. Paz, Y. Zhang, Quantum Brownian motion in a general environment: exact master equation withnonlocal dissipation and colored noise, Phys. Rev. D 45, 2843–2861 (1992).[Crossref]
  • G. Iche, P. Nozieres, Quantum brownian motion of a heavy particle: an adiabatic expansion, Physica A 91, 485–506(1978).[Crossref]
  • 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]
  • A. Jüngel, Transport Equations for Semiconductors, Lect. Notes Phys. 773, Springer, Berlin, (2009).
  • A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal.,42, 1025–1045 (2010).[Crossref]
  • A. Jüngel, Dissipative quantum fluid models, Riv. Mat. Univ. Parma, 3, 217–290 (2012).
  • A. Jüngel, D. Mathes, A derivation of the isothermal quantum hydrodynamic equations using entropy minimization,Z. Angew. Math. Mech 85, 806–814 (2005).[Crossref]
  • 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]
  • 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).
  • A. Jüngel, J.–P. Milisic, Full compressible Navier–Stokes equations for quantum fluids: Derivation and numericalsolution equations, Kinetic and related models 4, 785–807 (2011).
  • 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]
  • G. Kaniadakis, A. M. Scarfone, Nonlinear transformation for a class of gauged Schrödinger equations with complexnonlinearities, Reports on Math. Phys., 48, 115–121 (2001).
  • 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).
  • M. D. Kostin, On the Schrödinger–Langevin equation, J. Chem. Phys. 57, 3589–3591 (1972).[Crossref]
  • M. D. Kostin, Friction and dissipative phenomena in quantum mechanics, J. Stat. Phys. 12, 145–151 (1975).[Crossref]
  • 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).
  • N. A. Lemos, Dissipative forces and the algebra of operators in stochastic quantum mechanics, Phys. Lett. A 78,239–241 (1980).[Crossref]
  • H. Li, C–K. Lin, Semiclassical limit and well–posedness of nonlinear Schrödinger–Poisson, EJDE 93, 1–17 (2003).
  • G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48, 119–130 (1976).[Crossref]
  • P. L. Lions, T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9, 553–618 (1993).[Crossref]
  • H. Liu, E. Tadmor, Semiclassical limit of the nonlinear Schrödinger–Poisson equation with subcritical initial data,Methods Appl. Anal. 9, 517–532 (2002).
  • J. L. López, Nonlinear Ginzburg–Landau–type approach to quantum dissipation, Phys. Rev. E 69, 026110 (2004).[Crossref]
  • J. L. López, J. Montejo–Gámez, A hydrodynamic approach to multidimensional dissipation–based Schrödinger modelsfrom quantum Fokker–Planck dynamics, Physica D 238, 622–644 (2009).
  • J. L. López, J. Montejo–Gámez, On a rigorous interpretation of the quantum Schrödinger–Langevin operator inbounded domains, J. Math. Anal. Appl. 383, 365–378 (2011).[Crossref]
  • 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).
  • 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).
  • E. Madelung, Quantentheorie in hydrodynamischer form, Z. Physik 40, 322–326 (1927).[Crossref]
  • G. Manfredi, H. Haas, Self–consistent fluid model for a quantum electron gas, Phys. Rev. B 64, 075316 (2001).[Crossref]
  • P. A. Markowich, C. A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, New York, (1990).
  • F. Méhats, O. Pinaud, An inverse problem in quantum statistical physics, J. Stat. Phys. 140, 565–602 (2010).[Crossref]
  • J. Montejo–Gámez, Estudio de fenómenos cuánticos disipativos mediante ecuaciones en derivadas parciales en laformulación de Schrödinger, PhD Thesis, (2011).
  • A. B. Nassar, Fluid formulation of a generalised Schrödinger–Langevin equation, J. Phys. A: Math. Gen 18, L509–L511 (1985).[Crossref]
  • 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).
  • P. Nattermann, R. Zhdanov, On integrable Doebner–Goldin equations, J. Phys. A 29, 2869–2886 (1996).
  • E. Nelson, Derivation of the Schrödinger equation from Newtonian Mechanics, Phys. Rev. 150, 1079–1085 (1966).[Crossref]
  • T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45, 137–163 (1996).
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, NewYork, (1983).
  • E. Pollak, S. Zhang, Quantum dynamics for dissipative systems: A numerical study of the Wigner–Fokker–Planckequation, J. Chem. Phys. 118, 4357–4364 (2003).
  • P. C. Sabatier, Multidimensional nonlinear Schrödinger equations with exponentially confined solutions, InverseProblems 6, L47–L53 (1990).
  • O. Sánchez, J. Soler, Long–time dynamics of the Schrödinger–Poisson–Slater system, J. Stat. Phys. 114, 179–204(2004).[Crossref]
  • A. L. Sanin, A. A. Smirnovsky, Oscillatory motion in confined potential systems with dissipation in the context ofthe Schrödinger–Langevin–Kostin equation, Phys. Lett. A 372, 21–27 (2007).[Crossref]
  • A. Scarfone, Gauge equivalence among quantum nonlinear many body systems, Act. Appl. Math. 102, 179–217(2008).[Crossref]
  • B.-S. K. Skagerstam, Stochastic mechanics and dissipative forces, J. Math. Phys. 18, 308–311 (1977).[Crossref]
  • C. Sparber, J. A. Carrillo, J. Dolbeault, P. A. Markowich, On the long time behavior of the quantum Fokker–Planckequation. Monatsh. Math. 141, 237–257 (2004).[Crossref]
  • Y. Tanimura, Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipativesystems, J. Phys. Soc. Japan 75, 082001 (2006).[Crossref]
  • H. Teissmann, The Cauchy problem for the Doebner–Goldin equation, Physical applications and mathematicalaspects of geometry, groups and algebras, H. D. Doebner, P. Nattermann and W. Scherer (Eds.), World Scientific,433–438 (1997).
  • W. G. Unruh, W. H. Zurek, Reduction of a wave packet in quantum Brownian motion, Phys. Rev. D 40, 1071–1094(1989).[Crossref]
  • A. G. Ushveridze, Dissipative quantum mechanics. A special Doebner–Goldin equation, its properties and exactsolutions, Phys. Lett. A 185, 123–127 (1994).
  • A. G. Ushveridze, The special Doebner–Goldin equation as a fundamental equation of dissipative quantum mechanics,Phys. Lett. A 185, 128–132 (1994).
  • B. Vacchini, Completely positive quantum dissipation, Phys. Rev. Lett. 84, 1374–1377 (2000).[PubMed][Crossref]
  • P. Ván, T. Fülöp, Stability of stationary solutions of the Schrödinger–Langevin equation, Phys. Lett. A 323, 374–381(2004).[Crossref]
  • N. G. Van Kampen, Stochastic process in Physics and Chemistry (2nd edition), North–Holland, Amsterdam , (1992).
  • T. C. Wallstrom, Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations,Phys. Rev. A 49, 1613–1617 (1994).[PubMed][Crossref]
  • U. Weiss, Quantum dissipative systems, World Scientific, Singapore, (1993).
  • E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749–759 (1932).[Crossref]
  • Y. Yan, Quantum Fokker–Planck theory in a non–Gaussian–Markovian medium, Phys. Rev. A 58, 2721–2732 (1998).[Crossref]
  • K. Yasue, A note on the derivation of the Schrödinger–Langevin equation, J. Stat. Phys. 16, 113–116 (1977).[Crossref]
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