# Mini-symposium Mathematics and Materials and defence Michiel Renger

On February 21, 2013, the PhD Defence of Michiel Renger will take place, and the lectures of:

Alexander Mielke (Weierstrass Institute for Applied Analysis and Stochastics, Institut für Mathematik Humboldt-Universität zu Berlin<

13.15 hrs, in Ceres 0.31

Title: “Using gradient structures for modeling semiconductors”

Johannes Zimmer (Department of Mathematical Sciences, University of Bath)

14.00 hrs, in Ceres 0.31

Title: “Martensitic phase transitions: from microscopic waves to a macroscopic description”

PhD Defence Michiel Renger

16.00 hrs, in Auditorium

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Abstract Alexander Mielke:

“Using gradient structures for modeling semiconductors”

Many dissipative physical systems can be modeled via the Onsager principle which states that the rate of the state of a system is given in terms of a symmetric and positive definite matrix or operator applied to the associated thermodynamic driving force, which is obtained as derivative of the corresponding energy or entropy functional. In mathematical terms this leads to so-called gradient systems defined in terms of a functional and a dissipation potential.

We show that classical energy-drift-diffusion systems used for modeling of semiconductor devices can be written as gradient systems in the above sense. In the isothermal case the state is defined in terms of the densities of the relevant species and the functional is the relative entropy plus the electrostatic energy. In the temperature-dependent case the negative entropy acts as the functional, which is most easily described in as a function of the densities and the internal energy density. The Onsager operator consists of a Wasserstein part for the diffusion and a local part for the reactions.

We highlight how this modeling approach gives new understanding of critical issues in semiconductor physics, e.g. (i) proper modeling of different statistics (ii) drift enhancement (iii) including temperature effects (iv) modeling of active interfaces.

A. Mielke. A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity, 24: 1329-1346, 2011.

A.Glitzky and A. Mielke. A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces. Zeits. angew. Math. Physik, 2012, in press. DOI: 10.1007/s00033-012-0207-y.

Abstract Johannes Zimmer:

“Martensitic phase transitions: from microscopic waves to a macroscopic description”

The equations of elasticity in one space dimension become ill-posed if the potential energy density is nonconvex, or equivalently if the stress-strain relationship is non-monotone. This complication necessarily arises in the theory of so-called martensitic phase transitions, which are diffusionless solid-solid transformations where several stable phases can coexist.

Different regularizations of this ill-posed problem have been proposed; but one line of thought is to impose instead selection criteria. In mechanical terms, such criteria relate the velocity of a moving interface to a driving force, and are often called kinetic relations. Phenomenological kinetic relations have been proposed, but a natural question is whether they can in simple situations be derived from first principles, namely atomistic considerations, thus from nonlocal models.

To investigate this question, we study a one-dimensional chain model of martensitic materials, where neighboring atoms are coupled by a spring characterised by a nonconvex potential. We present existence results for the associated nonlocal travelling wave equation and discuss non-uniqueness of microscopic solutions. This non-uniqueness will be discussed in light of the macroscopic kinetic relation.

Alexander Mielke (Weierstrass Institute for Applied Analysis and Stochastics, Institut für Mathematik Humboldt-Universität zu Berlin<

__http://www.hu-berlin.de/__> <__http://www.hu-berlin.de/__)13.15 hrs, in Ceres 0.31

Title: “Using gradient structures for modeling semiconductors”

Johannes Zimmer (Department of Mathematical Sciences, University of Bath)

14.00 hrs, in Ceres 0.31

Title: “Martensitic phase transitions: from microscopic waves to a macroscopic description”

PhD Defence Michiel Renger

16.00 hrs, in Auditorium

========================================================================

Abstract Alexander Mielke:

“Using gradient structures for modeling semiconductors”

Many dissipative physical systems can be modeled via the Onsager principle which states that the rate of the state of a system is given in terms of a symmetric and positive definite matrix or operator applied to the associated thermodynamic driving force, which is obtained as derivative of the corresponding energy or entropy functional. In mathematical terms this leads to so-called gradient systems defined in terms of a functional and a dissipation potential.

We show that classical energy-drift-diffusion systems used for modeling of semiconductor devices can be written as gradient systems in the above sense. In the isothermal case the state is defined in terms of the densities of the relevant species and the functional is the relative entropy plus the electrostatic energy. In the temperature-dependent case the negative entropy acts as the functional, which is most easily described in as a function of the densities and the internal energy density. The Onsager operator consists of a Wasserstein part for the diffusion and a local part for the reactions.

We highlight how this modeling approach gives new understanding of critical issues in semiconductor physics, e.g. (i) proper modeling of different statistics (ii) drift enhancement (iii) including temperature effects (iv) modeling of active interfaces.

A. Mielke. A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity, 24: 1329-1346, 2011.

A.Glitzky and A. Mielke. A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces. Zeits. angew. Math. Physik, 2012, in press. DOI: 10.1007/s00033-012-0207-y.

Abstract Johannes Zimmer:

“Martensitic phase transitions: from microscopic waves to a macroscopic description”

The equations of elasticity in one space dimension become ill-posed if the potential energy density is nonconvex, or equivalently if the stress-strain relationship is non-monotone. This complication necessarily arises in the theory of so-called martensitic phase transitions, which are diffusionless solid-solid transformations where several stable phases can coexist.

Different regularizations of this ill-posed problem have been proposed; but one line of thought is to impose instead selection criteria. In mechanical terms, such criteria relate the velocity of a moving interface to a driving force, and are often called kinetic relations. Phenomenological kinetic relations have been proposed, but a natural question is whether they can in simple situations be derived from first principles, namely atomistic considerations, thus from nonlocal models.

To investigate this question, we study a one-dimensional chain model of martensitic materials, where neighboring atoms are coupled by a spring characterised by a nonconvex potential. We present existence results for the associated nonlocal travelling wave equation and discuss non-uniqueness of microscopic solutions. This non-uniqueness will be discussed in light of the macroscopic kinetic relation.