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Author: Mielke, Alexander × Author: Mittnenzweig, Markus × End date: 2019 × Start date: 2019 × End date: 2019 × Start date: 2010 ×

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    Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities : dedicated to Peter Markowich on the occasion of his sixtieth birthday
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mielke, Alexander; Mittnenzweig, Markus
    We discuss how the recently developed energy-dissipation methods for reaction-diffusion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, with the important feature, that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.
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    An 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, Alexander
    We 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.