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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
- ItemA generalization of Lagrange's algebraic identity and connections with Jensen's inequality(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Niculescu, Constantin P.; Stephan, HolgerWe discuss a generalization of Lagrange's algebraic identity that provides valuable insights into the nature of Jensen's inequality and of many other inequalities of convexity.
- ItemMemory equations as reduced Markov processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Stephan, Artur; Stephan, HolgerA large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.
- ItemPositivity and time behavior of a general linear evolution system, non-local in space and time(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Khrabustovskyi, Andrii; Stephan, HolgerWe consider a general linear reaction-diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle, and positivity of the solution, and investigate its asymptotic behavior. Moreover, we give an explicite expression of the limit of the solution for large times. In order to obtain these results we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution of a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction-diffusion system what allows us to investigate its properties.
- ItemWIAS-TeSCA - Two-dimensional semi-conductor analysis package(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Gajewski, Herbert; Liero, Matthias; Nürnberg, Reiner; Stephan, HolgerWIAS-TeSCA (Two- and three-dimensional semiconductor analysis package) is a simulation tool for the numerical simulation of charge transfer processes in semiconductor structures, especially in semiconductor lasers. It is based on the drift-diffusion model and considers a multitude of additional physical effects, like optical radiation, temperature influences and the kinetics of deep impurities. Its efficiency is based on the analytic study of the strongly nonlinear system of partial differential equations – the van Roosbroeck system – which describes the electron and hole currents. Very efficient numerical procedures for both the stationary and transient simulation have been implemented. WIAS-TeSCA has been successfully used in the research and industrial development of new electronic and optoelectronic semiconductor devices such as transistors, diodes, sensors, detectors and lasers and has already proved its worth many times in the planning and optimization of these devices. It covers a broad spectrum of applications, from heterobipolar transistor (mobile telephone systems, computer networks) through high-voltage transistors (power electronics) and semiconductor laser diodes (fiber optic communication systems, medical technology) to radiation detectors (space research, high energy physics). WIAS-TeSCA is an efficient simulation tool for analyzing and designing modern semiconductor devices with a broad range of performance that has proved successful in solving many practical problems. Particularly, it offers the possibility to calculate self-consistently the interplay of electronic, optical and thermic effects.
- ItemReverse inequalities for slowly increasing sequences and functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Stephan, HolgerWe consider sharp inequalities involving slowly increasing sequences and functions, i.e., functions f(t) with f'(t) ≤1 and sequences (ai) with ai+1
- ItemGlobal weak solutions of the Navier-Stokes-Vlasov-Poisson system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Anoschenko, Olga; Khruslov, Evgeni; Stephan, HolgerWe consider the Navier-Stokes-Vlasov-Poisson system of partial differential equations, describing the motion of a viscous incompressible fluid with small solid charged particles therein. We prove the existence of a weak global solution of the initial boundary value problem for this system.
- ItemMillions of Perrin pseudoprimes including a few giants(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Stephan, HolgerThe calculation of many and large Perrin pseudoprimes is a challenge. This is mainly due to their rarity. Perrin pseudoprimes are one of the rarest known pseudoprimes. In order to calculate many such large numbers, one needs not only a fast algorithm but also an idea how most of them are structured to minimize the amount of numbers one have to test. We present a quick algorithm for testing Perrin pseudoprimes and develop some ideas on how Perrin pseudoprimes might be structured. This leads to some conjectures that still need to be proved. We think that we have found well over 90% of all 20-digit Perrin pseudoprimes. Overall, we have been able to calculate over 9 million Perrin pseudoprimes with our method, including some very large ones. The largest number found has 1436 digits. This seems to be a breakthrough, compared to the previously known just over 100,000 Perrin pseudoprimes, of which the largest have 20 digits. In addition, we propose two sequences that do not provide any pseudoprimes up to 1,000,000,000 at all.
- ItemA mathematical framework for general classical systems and time irreversibility as its consequence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Stephan, HolgerIt is well known that important models in statistical physics like the Fokker-Planck equation satisfy an H-theorem, i.e., have a decreasing Lyapunov function (or increasing entropy). This illustrates a symmetry break in time and reflects the second law of thermodynamics. In this paper, we show that any physically reasonable classical system has to have this property. For this purpose, we develop an abstract mathematical framework based on the theory of compact topological spaces and convex analysis. Precisely, we show: 1) Any statistical state space can be described as the convex hull of the image of the canonical embedding of the bidual space of its deterministic state space (a compact topological Hausdorff space). 2) The change of any statistical state is effected by the adjoint of a Markov operator acting in the space of observables. 3) Any Markov operator satisfies a wide class of inequalities, generated by arbitrary convex functions. As a corollary, these inequalities imply a time monotone behavior of the solution of the corresponding evolution equations. Moreover, due to the general abstract setting, the proof of the underlying inequalities is very simple and therefore illustrates, where time symmetry breaks: A model is time reversible for any states if and only if the corresponding Markov operator is a deterministic one with dense range. In addition, the proposed framework provides information about the structure of microscopic evolution equations, the choice of the best function spaces for their analysis and the derivation of macroscopic evolution equations.
- ItemPositivity and polynomial decay of energies for square-field operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Stephan, Artur; Stephan, HolgerWe show that for a general Markov generator the associated square-field (or carré du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the square-field operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operator-theoretic normality condition, the sequence of energies is log-convex. In particular, this implies polynomial decay in time for the energy functionals along solutions.