@misc{BornemannNettesheimSchuette, author = {Bornemann, Folkmar A. and Nettesheim, Peter and Sch{\"u}tte, Christof}, title = {Quantum-Classical Molecular Dynamics as an Approximation to Full Quantum Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1922}, number = {SC-95-26}, abstract = {This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a {\em partial} classical limit of the full Schr{\"o}dinger equation. This limit is achieved in two steps: separation of the full wavefunction and short wave asymptotics for its ``classical'' part. Both steps can be rigorously justified under certain smallness assumptions. Moreover, the results imply that neither the time-dependent self-consistent field method nor mixed quantum-semi-classical models lead to better approximations than QCMD since they depend on the separation step, too. On the other hand, the theory leads to a characterization of the critical situations in which the models are in danger of largely deviating from the solution of the full Schr{\"o}dinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of an Argon atom with a harmonic quantum oscillator.}, language = {en} } @misc{BornemannSchuette, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {A Mathematical Approach to Smoothed Molecular Dynamics: Correcting Potentials for Freezing Bond Angles}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1960}, number = {SC-95-30}, abstract = {The interaction potential of molecular systems which are typically used in molecular dynamics can be split into two parts of essentially different stiffness. The strong part of the potential forces the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles). However, the naive way of doing this via holonomic constraints is bound to produce incorrect results. The paper presents a mathematically rigorous discussion of the limit situation in which the stiffness of the strong part of the potential is increased to infinity. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given and its significant contribution is demonstrated in an illustrative example. It appears that this correcting potential is definitely not identical with the Fixman-potential as was repeatedly assumed in the literature.}, language = {en} } @misc{SchuetteDinandZumbuschetal., author = {Sch{\"u}tte, Christof and Dinand, Manfred and Zumbusch, Gerhard and Brinkmann, Ralf}, title = {Dynamics of Erbium-doped Waveguide Lasers: Modelling, Reliable Simulation, and Comparison with Experiments}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1856}, number = {SC-95-19}, abstract = {A theoretical investigation of the dynamic properties of integrated optical Er--doped waveguide lasers is presented. It includes the construction of a physical model and of numerical techniques which allow reliable simulations of the dynamical behaviour of the laser signal depending on essential parameters of the laser device and on its external, time--dependent pump radiation. Therefore, a physical theory is developed which describes the propagation of light and its interaction with the active substrate in the laser cavity. This is realized in two steps. First, a {\em fundamental model} based on Maxwell's equations and on rate equations for the transitions in the active medium is constructed. Since this turns out to prohibit reliable simulations, it is, in a second step, reformulated via averaging in time and space which suppresses the fluctuations on the fastest time scales but represents them correctly. For this {\em reduced model} reliable and efficient simulation techniques using adaptive control schemes are designed and implemented. We apply the linear--implicit Euler discretization with extrapolation in time and a multilevel quadrature scheme in space. Finally, the model is justified in comparison with experimental observations in four cases of technological relevance.}, language = {en} } @misc{Schuette, author = {Sch{\"u}tte, Christof}, title = {Smoothed Molecular Dynamics for Thermally Embedded Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1809}, number = {SC-95-14}, abstract = {This paper makes use of statistical mechanics in order to construct effective potentials for Molecular Dynamics for systems with nonstationary thermal embedding. The usual approach requires the computation of a statistical ensemble of trajectories. In the context of the new model the evaluation of only one single trajectory is sufficient for the determination of all interesting quantities, which leads to an enormous reduction of computational effort. This single trajectory is the solution to a corrected Hamiltonian system with a new potential \$\tilde{V}\$. It turns out that \$\tilde{V}\$ can be defined as spatial average of the original potential \$V\$. Therefore, the Hamiltonian dynamics defined by \$\tilde{V}\$ is smoother than that effected by \$V\$, i.e. a numerical integration of its evolution in time allows larger stepsizes. Thus, the presented approach introduces a Molecular Dynamics with smoothed trajectories originating from spatial averaging. This is deeply connected to time--averaging in Molecular Dynamics. These two types of {\em smoothed Molecular Dynamics} share advantages (gain in efficiency, reduction of error amplification, increased stability) and problems (necessity of closing relations and adaptive control schemes) which will be explained in detail.}, language = {en} } @misc{BornemannSchuette, author = {Bornemann, Folkmar A. and Sch{\"u}tte, Christof}, title = {Homogenization of Highly Oscillatory Hamiltonian Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2050}, number = {SC-95-39}, abstract = {The paper studies Hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale. In particular, we are interested in the limit situation where the size \$\epsilon\$ of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of order \${\rm O}(1)\$. The process of establishing an effective initial value problem for the limit positions will be called {\em homogenization} of the Hamiltonian system. This problem occurs in mechanics as the problem of realization of holonomic constraints, in plasma physics as the problem of guiding center motion, in the simulation of biomolecules as the so called smoothing problem. We suggest the systematic use of the notion of {\em weak convergence} in order to approach this problem. This methodology helps to establish unified and short proofs of the known results which throw light on the inherent structure of the problem. Moreover, we give a careful and critical review of the literature.}, language = {en} } @article{SchuetteZumbuschBrinkmann1995, author = {Sch{\"u}tte, Christof and Zumbusch, Gerhard and Brinkmann, Ralf}, title = {Dynamics of Erbium-doped Waveguide Lasers}, series = {preprint}, journal = {preprint}, year = {1995}, language = {en} } @article{SchuetteDinand1995, author = {Sch{\"u}tte, Christof and Dinand, Manfred}, title = {Theoretical Modeling of Relaxation Oscillations in Er-Doped Wave-Guide Lasers}, series = {J. Lightw. Techn.}, volume = {13}, journal = {J. Lightw. Techn.}, number = {1}, pages = {14 -- 23}, year = {1995}, language = {en} } @misc{Schuette1995, author = {Sch{\"u}tte, Christof}, title = {Smoothed Molecular Dynamics for Thermally Embedded Systems}, year = {1995}, language = {en} }