[P1, Arnold] focuses on the large-time analysis of dissipative classical and quantum systems and it is based on the construction of problem-specific Lyapunov functionals via entropy and spectral methods.
(1) The entropy method (or Gamma_2-calculus of Bakry-Emery) is based on differential inequalities between a suitable entropy functional and its first or second time derivative. But for degenerate, hypocoercive diffusion equations the entropy decays as a non-convex function of time. This requires to construct modified Lyapunov functionals of entropy (-dissipation) type. The focus of this project is to compare, optimize and unify the recent first approaches, and to extend their applicability (including nonlinear models and the pathwise analysis of stochastic processes).
(2) The evolution of dissipative (or open) quantum systems is typically described by a Lindblad equation for the density matrix (in the space of trace class operators). Rather abstract, operator theoretic criteria for the existence of steady states and the large-time convergence were given in the quantum probability community, and applied to the quantum-kinetic Wigner-Fokker-Planck equation, so far without exponential decay rates. Hence, this project aims at developing an entropy method for Lindblad equations. Moreover we shall investigate the possible gradient flow structure and hypercontractivity properties of Lindblad equations.
[P1, Arnold] clusters all SFB activities on entropy methods for continuous systems of (non)linear Fokker-Planck type as well as Lindblad equations. Its discrete analog is the following Project Part [P2, Jüngel], including entropic discretizations; and the large-time behavior of kinetic equations including mean-field models is the main topic of [P9, Schmeiser]. On top of these two natural cooperation partners, the pathwise analysis of stochastic processes will be carried out with W. Schachermayer. The investigation of Lindblad equations will be jointly with [P3, Maas].
Moreover, [P1, Arnold] will benefit from his already established international cooperations with E. Carlen (Rutgers), J. Carrillo (Imperial College), B. Jourdain (Paris), and F. Fagnola (Milan).