Kinetic and Related Models doi:10.3934/krm.2012.5.517 c American Institute of Mathematical Sciences Volume 5, Number 3, September 2012 pp. 517{536 ON VISCOUS QUANTUM HYDRODYNAMICS ASSOCIATED WITH NONLINEAR SCHRODINGER-DOEBNER-GOLDIN¨ MODELS Jose´ Luis Lopez´ and Jesus´ Montejo{Gamez´ Departamento de Matem´aticaAplicada Facultad de Ciencias, Universidad de Granada 18071 Granada, Spain (Communicated by Ansgar J¨ungel) Abstract. The aim of this paper is to derive the quantum hydrodynamic sys- tem associated with the most general class of nonlinear Schr¨odingerequations 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 contribu- tion. 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. 1. Introduction and setting of the problem. Quantum fluids have proved for many years to be quite a powerful tool for the analysis of quantum dissipative processes. Nowadays, quantum hydrodynamic modeling is a field of increasing interest due to its remarkable role in potential applications within various areas of current experimental and theoretical research such as quantum computing, Bose{ Einstein condensates, or numerical simulations of nanoscopic semiconductor devices (see for instance [6, 19]). Among the several advantages exhibited by this kind of fluid–mechanical models, it is especially noticeable the fact that they make it possible to carry out an efficient numerical treatment of the solutions as well as allow that macroscopic boundary conditions can be imposed. The aim of this paper is to model quantum viscosity effects in the macroscopic picture of fluid mechanics. Since the seminal paper by Harvey [18], this methodology has been vastly analyzed so far when starting from the Wigner equation endowed with a collision operator (see for example [19] for an exhaustive review). In this case, the moment method along with the maximum entropy principle are the key ingredients to be employed in order to derive a family of hydrodynamic systems 2000 Mathematics Subject Classification. Primary: 81Q05, 76Y05, 35Q30; Secondary: 81S30, 78M05. Key words and phrases. Quantum hydrodynamics, Schr¨odingerdescription of quantum me- chanics, Doebner-Goldin equation, quantum viscous hydrodynamic equation, quantum Navier- Stokes equation. The authors are partially supported by MEC (Spain), Project MTM2008{05271 and Junta de Andaluc´ıa,Project FQME{1268. 517 518 JOSE´ LUIS LOPEZ´ AND JESUS´ MONTEJO-GAMEZ´ based on a Chapman{Enskog expansion around the quantum equilibrium distribu- tion (see [7] for details). This procedure was recently applied by Brull and M´ehats [5] to the Wigner equation with a relaxation kernel of BGK type, leading to non- local quantum equations of Navier{Stokes type. On the contrary, local equations are realized by approximating the quantum equilibrium up to order O(~4), which yields a density{dependent viscosity within the stress tensor (the full compressible model, including the energy equation, was recently derived in [23]): @tn + rx · (nu) = 0 ; (1) 1 @ (nu) + div (nu ⊗ u + P ) + nr V = −2λnu + div (S) ; (2) t x NS m x x with 2 1 P = − ~ nr2 log(n) + nT ; S = 2αnT S(u) − r · u ; (3) NS 12m2 x I 3 x I 1 where m is the effective mass of the particles, 2λ is the relaxation time of the system, V (t; x) holds for an arbitrary external (or self{induced) potential, ~ is the Planck constant, I the (three{dimensional) identity matrix, α the perturbation parameter of the Chapman{Enskog expansion around the quantum Maxwellian, T = T (t; x) a Lagrange multiplier associated with the corresponding optimization problem for the quantum entropy, and where 1 S(u) = r u + r u> (4) 2 x x stands for the symmetric velocity gradient. Furthermore, the macroscopic magni- tudes n(t; x) and u(t; x) are linked with the Wigner function w(t; x; p) through its velocity moments, say Z (n; nu) = 1; pw dp : 3 R In what concerns quantum interactions of Fokker{Planck type (cf. Eq. (44) below), the moment method joint with the maximum entropy principle (where no Chapman{Enskog expansion of the Wigner function is now involved) give rise to the following viscous quantum hydrodynamic system (first studied in [15] for the constant temperature case) @t(nu) + rx · (nu) = Dqq∆xn ; (5) 1 @ (nu) + div nu ⊗ u + P + nr V = −2λnu + D ∆ (nu) ; (6) t x V m x qq x with the quantum stress tensor given now by 2D P = P + pq n : (7) V NS m I Beyond its own formulation, there is a structural connection between the models (1){(3) and (5){(7) by means of the so{called osmotic velocity uo = Drx log(n) introduced by Nelson in [31], via the relation v = u − Drx log(n) ; (8) where D plays the role of a diffusion coefficient. Indeed, by introducing this new velocity the system (5){(6) can be reformulated as (1){(2) whenever ~ > D (see [21]). It is also significant that the osmotic velocity plays a crucial role in the theory of dequantization [28], as well as the fact that it admits an interpretation in the ambit of Gauge theories [12] (the interested reader may wish to consult [22] for VISCOUS QUANTUM HYDRODYNAMICS OF DOEBNER-GOLDIN MODELS 519 a discussion and the derivation of the quantum Navier{Stokes equation from the Wigner{Fokker{Planck equation under a reinterpretation of the velocity moment operators). Contrary to the development experienced in the context of the Wigner equa- tion, this methodology has not found a consolidate basis in the framework of the Schr¨odingerequation yet. One of the main reasons that contribute to explain this lack of information relies on the ongoing discussion regarding the adequate mod- eling of viscous phenomena via nonlinear terms in the Schr¨odingerformulation of quantum mechanics. A large number of the nonlinear Schr¨odingerequations pro- posed in the literature involve several (real and complex) nonlinearities describing different phenomenologies in astrophysics, quantum gravity or condensed matter physics. In particular, the family of so{called Doebner{Goldin equations was intro- duced in [8] as the most general class of Schr¨odingertype equations compatible with the Fokker{Planck continuity equation for the probability density. Deep down in these equations there is the challenging idea consisting of how to model quantum{ mechanical open systems from a wavefunction approach, in the scope of a manifest increasing interest in a better understanding of how dissipative interactions between a quantum ensemble and a phonon bath in thermodynamic equilibrium should be mathematically described. In this spirit, our main purpose throughout the paper is to extend the ambit of the already known hydrodynamic derivations to that of the Doebner{Goldin{Schr¨odingermodels. The structure of the paper is as follows: In Section 2 we present the Doebner{ Goldin equations and establish some of their most remarkable properties. Section 3 is devoted to derive in full generality and interpret in the framework of viscous fluid mechanics the macroscopic equations governing the evolution of the local and cur- rent densities. Then, we perform the osmotic change of variables u 7! v to simplify the resulting equations and compare them with the viscous quantum hydrodynamic systems established in (1){(2) and (5){(6). Finally, Section 4 concerns the explo- ration of the quantum hydrodynamics associated with various particular models of Doebner{Goldin type to which some specific physical interest is attributed in the literature. 2. Generalities of the Doebner{Goldin equations. The class of Schr¨odinger{ Doebner{Goldin equations for the wavefunction is written as 2 0 2 i~D ∆xn m D jJj i~@t = H0 + + µ3 2 n ~ n2 r · J J · r n + mD0 µ x + µ x 1 n 4 n2 ∆ n jr nj2 + D0 µ x + µ x ; (9) ~ 2 n 5 n2 where n := j j2 ; nu := J = ~ Im r ; (10) m x denote the probability and current densities, respectively. Here, H0 stands for 2 ~ the Hamiltonian H0 := − 2m ∆x + V , while Im(z) and z denote the imaginary part and the conjugate of the complex number z, respectively. Also, D and D0 520 JOSE´ LUIS LOPEZ´ AND JESUS´ MONTEJO-GAMEZ´ represent diffusion coefficients while the parameters µ1; : : : ; µ5 are adimensional real constants. The nonlinearities participating in the Doebner{Goldin class were derived in the algebraic frame of group theory, more precisely from the representation analysis of the quantum kinematical group of diffeomorphisms Diff(R3), and proved to be phys- ically motivated in many different situations comprising diffusive quantum mechan- ics. Some of the most important properties of this family of equations are collected in the following list (see [9, 10] and references therein, where several studies con- cerning symmetries and explicit time{dependent solutions of the Doebner{Goldin equations (9) can also be found): (i) The total charge R n(t; x) dx is preserved along the time evolution for all values of the coefficients µj. (ii) If is a solution to Eq. (9), so is λψ for any λ 2 C. (iii) For initially uncorrelated N{particle systems, noninteracting subsystems re- main uncorrelated. k·x−!t (iv) Plane waves φ(t; x) = A0e always solve Eq. (9) in the free{particle regime (V ≡ 0 in the Hamiltonian operator). (v) Solutions to Eq. (9) are time{translation and Euclidean invariant. (vi) If the external potential V is chosen to be stationary, then there exist sta- tionary state solutions to Eq.