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Analysis of Multiscale Systems Driven by Functionals

Final Report Summary - ANAMULTISCALE (Analysis of Multiscale Systems Driven by Functionals)

The project AnaMultiScale was devoted to mathematical methods for evolutionary systems that are motivated by thermodynamical principles such as Hamiltonian dynamics or dissipative gradient flows. The aim was to develop new analytical methods to study such models, in particular in the presence of multiple scales. The work was organized in the following three, strongly interlinked tasks:

Task 1: Analysis of systems driven by energy or entropy

Task 2: Multiscale limits and Gamma convergence

Task 3: Applications in material modeling and optoelectronics

In Taks 1 we generalized the modern theory of metric gradient systems in a two-fold way. First, in our aim to study nontrivial material models like elastoplasticity, we studied generalized gradient systems where the dissipation potential is nonquadratic, thus allowing for activation thresholds like yield stresses. A new existence result for viscoplasticity in the geometrically nonlinear setting of finite-strain materials was derived. Second, the notion of balanced-viscosity solutions was developed to give a precise meaning and to replace previously developed ad hoc notions of solutions for the limit of vanishing viscosity (relative to the loading speed). For this, a new rescaling of the energy-dissipation principle (EDP), which was advocated by De Giorgi in the 1980, was developed to analyze the behavior of solutions at rapidly occurring jumps. Third, we generalized ideas from optimal transport theory to systems with reaction leading to the so-called Hellinger-Kantorovich distance between measures of arbitrary mass. A complete characterization of the distance and the geodesic curves were obtained. This work will be the basis for treating chemical reaction-diffusion systems as well as novel applications in image processing.

In Task 2 we studied several notions of evolutionary Gamma convergence for gradient systems and introduced the new concept of EDP convergence that builds on the strong relation between gradient structures and large-deviation principles for stochastic many-particle systems. We provide a few first positive cases for a general conjecture concerning EDP convergence and limit passages in the large-deviation context. In particular, we were able to understand the generation of reaction processes as limit of diffusion along a reaction coordinate that provides a suitable potential barrier between the substrate and the product phase. A introductory survey on the general methodology of evolutionary Gamma convergence was published, and a series of nontrivial applications was derived, including the passage from discrete to continuous Markov processes relating to a finite-volume discretization of Fokker-Planck equations. The chemical master equation for a finite set of species and reversible reactions was analyzed in the limit of large volume, where the detailed-balance condition was used to generate an entropic gradient structure.

The applications in Task 3 involve evolutionary models for microstructures in solids, where a surprising limit passage for a rate-independent two-phase model was possible by exploiting hierarchical H-measures. Further applications involve mathematical models for elastoplastic materials including damage, fracture and surface friction. The rate-independent case was summarized in an extensive monograph with T. Roubicek that covers the abstract theory of rate-independent systems and applications in material science, including finite-strain elasticity. A major breakthrough occurred in the thermodynamically consistent, dissipative coupling of quantum systems to classical models. For this, we derived an entropic gradient structure for the quantum master equations (Lindblad equations), where again a suitable detailed-balance condition was essential. This work allows us to prove that the entropy-production rate is non-negative under a large variety of couplings that can be described within the GENERIC framework. This theory provides a flexible basis for the modeling of quantum-dot lasers as well as as single-photon emitters.