Nonlinear Ginzburg-Landau-type approach to quantum dissipation

Abstract

We formally derive two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium. These models mainly consist of a quantum, dissipative O(Planck 3) hydrodynamic/O(Planck 4) stochastic correction to the frictional (Caldeira-Leggett-)Schrödinger equation. The main ingredient lies in the use of the hydrodynamic/stochastic fluid model approach associated with the quantum Fokker-Planck equation and the identification of the associated pressure field. Then, Madelung transformations set the problem in the Schrödinger picture of dissipative quantum mechanics. We also describe the stationary dynamics associated with both systems.

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

“…[57]) or to the methodical considerations in information theory (cf. [63][64][65][66][67][68]). One of the particularly encouraging recent mathematical results obtained by Babin and Figotin [103] indicates that in the semi-classical range, the combination of the logarithmic nonlinearities with the coupled-equation structures may prove unexpectedly productive.…”

“…One of the best known examples is provided by [16,17] in which the toy models are considered in the form of the nonlinear logarithmic Schrödinger Equation (5) with the wave-function solutions ψ ∈ L 2 ( d ) studied in an interval of time t ∈ (t 0 , t 1 ). This equation, along with its relativistic analogue, finds multiple applications in the physics of quantum fields and particles [49][50][51][52][53][54][55], extensions of quantum mechanics [16,56], optics and transport or diffusion phenomena [57][58][59][60], nuclear physics [61,62], the theory of dissipative systems and quantum information [63][64][65][66][67][68], the theory of superfluidity [69][70][71][72] and the effective models of the physical vacuum and classical and quantum gravity [73][74][75][76], where one can utilize the well-known fluid/gravity analogy between inviscid fluids and pseudo-Riemannian manifolds [77][78][79][80][81]. The relativistic analogue of Equation (5) is obtained by replacing the derivative part with the d'Alembert operator and is not considered here.…”

Quantum Transport and Boltzmann Operators

“…For open quantum systems, the analysis of dissipative transport equations with Fokker-Planck-type scattering mechanism was done in (1) by A. Arnold, J. L. López, P. A. Markowich and J. Soler in the Wigner function formalism (26) (level of the kinetic equation), see also (6) . At the level of the density operator the same problem has been recently studied by A. Arnold and C. Sparber in (2) and by J. L. López in (21) in the setting of (logarithmic) Schrödinger systems. All the dissipative quantum models studied in the previous citations rely on a priori assumptions or principles.…”

Asymptotic behaviors of solutions for dissipative quantum Zakharov equations

“…For example, the quantum hydrodynamical model for the charged particle systems was successfully used for the description of quantum dissipation [4] under the closure hypothesis adopted by Manfredi and Haas [5] . In the case of quantum ion-acoustic waves [6] , several features of the pure quantum origin were observed for the linear, weakly nonlinear, and fully nonlinear waves. The fully nonlinear quantum ion-acoustic waves can have a coherent and periodic pattern, which is not present in the classical case.…”