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- ItemAn entropic gradient structure for Lindblad equations and GENERIC for quantum systems coupled to macroscopic models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mittnenzweig, Markus; Mielke, AlexanderWe show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems.
- ItemMathematical modeling of semiconductors: From quantum mechanics to devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Kantner, Markus; Mielke, Alexander; Mittnenzweig, Markus; Rotundo, NellaWe discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of so-called GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.