Nonlinear Ginzburg-Landau-type approach to quantum dissipation

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 $O\left({ħ}^3\right)$ $\mathrm{hydrodynamic}/O\left({ħ}^4\right)$ 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.