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- ItemModeling of drift-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Stephan, HolgerWe derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or
- ItemThermal effects in gravitational Hartree systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Aki, Gonca L.; Dolbeault, Jean; Sparber, ChristofWe consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb-Newton interaction. For a given mass $M>0$, we construct stationary states with non-zero temperature T by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold T^*>0 (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature T_c in (0, T^*) above which mixed states appear.
- ItemUniform estimate of the relative free energy by the dissipation rate for finite volume discretized reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Fiebach, André; Glitzky, AnnegretWe prove a uniform Poincaré-like estimate of the relative free energy by the dissipation rate for implicit Euler, finite volume discretized reaction-diffusion systems. This result is proven indirectly and ensures the exponential decay of the relative free energy with a unified decay rate for admissible finite volume meshes.