@misc{HellingKleinWoitkeetal., author = {Helling, Christiane and Klein, Rupert and Woitke, Peter and Sedlmayr, Erwin}, title = {Dust formation in brown dwarf atmospheres under conditions of driven turbulence}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7294}, number = {03-07}, abstract = {Based on the knowledge gained from direct numerical simulations which are only possible in the microscale regime, a concept of driven turbulence is presented which allows to enter the mesoscopic scale regime. Here, dust formation under stochastic hydro- and thermodynamic conditions is studied: constructively superimposed stochastic waves initiate dust formation by the creation of singular nucleation events. It, hence, results a varying mean grain size and dust density in space and time. The newly formed dust changes the thermodynamic behavior from almost isotherm to adiabatic and chemically depletes the gas phase.}, language = {en} } @misc{RuprechtKleinMajda, author = {Ruprecht, Daniel and Klein, Rupert and Majda, Andrew J.}, title = {Moisture-Gravity Wave Interactions in a Multiscale Environment}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11358}, number = {09-21}, abstract = {Starting from the conservation laws for mass, momentum and energy together with a three species, bulk microphysic model, a model for the interaction of internal gravity waves and deep convective hot towers is derived by using multiscale asymptotic techniques. From the resulting leading order equations, a closed model is obtained by applying weighted averages to the smallscale hot towers without requiring further closure approximations. The resulting model is an extension of the linear, anelastic equations, into which moisture enters as the area fraction of saturated regions on the microscale with two way coupling between the large and small scale. Moisture reduces the effective stability in the model and defines a potential temperature sourceterm related to the net effect of latent heat release or consumption by microscale up- and downdrafts. The dispersion relation and group velocity of the system is analyzed and moisture is found to have several effects: It reduces energy transport by waves, increases the vertical wavenumber but decreases the slope at which wave packets travel and it introduces a lower horizontal cutoff wavenumber, below which modes turn into evanescent. Further, moisture can cause critical layers. Numerical examples for steadystate and timedependent mountain waves are shown and the effects of moisture on these waves are investigated.}, language = {en} } @misc{CarqueOwinohKleinetal., author = {Carqu{\´e}, Gunter and Owinoh, Antony Z. and Klein, Rupert and Majda, Andrew J.}, title = {Asymptotic Scale Analysis of Precipitating Clouds}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10548}, number = {08-03}, abstract = {Asymptotic analyses of the three dimensional compressible flow equations coupled with transport equations for the mixing ratios of water vapour, cloud water and rain water are described. We obtain reduced systems of equations for two particular regimes of length and time scales: Models for the long time evolution of deep convective columns and for the short time evolution of shallow convective layers. The asymptotic deep convective column model is anelastic, yet the vertical motion is pressure free, i.e., it evolves freely in interaction with buoyancy while the horizontal divergence adjusts to fullfil the anelastic constraint. The perturbation pressure guaranteeing compliance with the horizontal divergence constraint obeys a Poisson-type equation. Surprisingly, the vertical velocity plays an important role in the horizontal dynamics through the Coriolis term. The vertical acceleration in a saturated column is directly determined by the buoyancy induced by potential temperature differences relative to the background stratification. This potential temperature deviation is a conserved quantity. Evaporation is the only important microphysical process in the undersaturated regime. The evaporation rate depends on the saturation deficit and the amount of rain water present and determines the (downward) vertical velocity and the distribution of water vapour. To connect the deep convective column solutions to top and bottom boundary conditions, a different flow regime needs to be accounted for. Within shallow layers whose depth is comparable to the column diameters, adjustment to physical boundary conditions can take place. This is the second regime considered in this report. The shallow convective layer regime is shown to be asymptotically described by Boussinesq-type equations. These equations are closed by evolution equations which show that, in the saturated regime, the distributions of potential temperature and cloud water are determined by a condensation rate that is directly proportional to the vertical velocity. In the undersaturated regime, the potential temperature distribution is determined by the amount of rain present, since the water vapour in this case is shown to be a conserved quantity. In both regimes the distribution of rain water depends on the rain water flux.}, language = {en} } @misc{CarqueSchmidtStevensetal., author = {Carqu{\´e}, Gunter and Schmidt, Heiko and Stevens, Bjorn and Klein, Rupert}, title = {Plausibility Check of an Asymptotic Column Model for Deep Convective Clouds}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10967}, number = {08-44}, abstract = {By use of asymptotic analysis Carqu{\´e} et al. [ZIB-Report 08-03] derived an asymptotic column model for deep convective clouds based on the three dimensional compressible flow equations and a bulk microphysics parameterization. In the present study we check the plausibility of the reduced model equations by comparing implications of the model for the scaling of various terms in the governing equations with those extracted from large eddy simulation data based on the computational model UCLA-LES1.1. This code solves an anelastic system of equations with complete droplet based microphysics and LES closures. We observe that the simulation data corroborate the basic assumptions of the asymptotic analysis and the main conclusions implied by the asymptotically reduced model. The code output reflects the scales of space and time: The deep convective clouds show an anisotropic structure where the horizontal scale is considerably narrower than the vertical scale; with a period of about 20 min, from emergence to breakup, the life cycle of one particular deep convective cloud corresponds exactly to the reference time of the reduced model. The characteristic properties of dynamics as predicted by the reduced model are also reflected in the simulation data: The horizontal flow is controlled by the pressure field; the vertical velocity develops freely independent of pressure over the depth of the convective column; the vertical velocity is directly determined by the buoyancy induced by the potential temperature deviation relative to the background stratification. With respect to grid resolution we observe that refining the spatial step size of the equidistant computational grid from 125 m to 62.5 m does not influence the results: Even with the coarser grid the relevant physical phenomena are sufficiently resolved. Somewhat surprisingly, the Coriolis term involving vertical velocity and acting on the horizontal (east-west) velocity component appears at leading order in the asymptotics. Accordingly, we expected to find a nontrivial impact of this Coriolis effect on the horizontal flow velocity components within columns of updrafts. However, switching the term on and off in subsequent simulations did not sizeably affect the results.}, language = {en} } @misc{OevermannScharfenbergKlein, author = {Oevermann, Michael and Scharfenberg, Carsten and Klein, Rupert}, title = {A sharp interface finite volume method for elliptic equations on Cartesian grids}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10900}, number = {08-38}, abstract = {We present a second order sharp interface finite volume method for the solution of the three-dimensional poisson equation with variable coefficients on Cartesian grids. In particular, we focus on interface problems with discontinuities in the coefficient, the source term, the solution, and the fluxes across the interface. The method uses standard piecewiese trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27-point stencil. Singularities associated with vanishing partial volumes of intersected grid cells are removed by a two-term asymptotic approach. In contrast to the 2D method presented by two of the authors in [M.~Oevermann, R.~Klein: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J.~Comp.~Phys.~219 (2006)] we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz. This simplifies the treatment of the different cut-cell types and avoids additional special operations for degenerated interface topologies. The resulting set of linear equations has been solved with a BiCGSTAB solver preconditioned with an algebraic multigrid. In various testcases -- including large coefficient ratios and non-smooth interfaces -- the method achieves second order of accuracy in the L_inf and L_2 norm.}, language = {en} } @misc{VaterKlein, author = {Vater, Stefan and Klein, Rupert}, title = {Stability of a Cartesian Grid Projection Method for Zero Froude Number Shallow Water Flows}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9562}, number = {07-13}, abstract = {In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown incompressible height and piecewise constant test functions. The stability of the projection is proved using the theory of generalized mixed finite elements, which goes back to Nicola{\"i}des (1982). In order to do so, the validity of three different inf-sup conditions has to be shown. Since the zero Froude number shallow water equations have the same mathematical structure as the incompressible Euler equations of isentropic gas dynamics, the method can be easily transfered to the computation of incompressible variable density flow problems.}, language = {en} } @misc{OevermannKlein, author = {Oevermann, Michael and Klein, Rupert}, title = {A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8996}, number = {06-05}, abstract = {We present a finite volume method for the solution of the two-dimensional Poisson equation \$ \nabla\cdot( \beta( {\mbox{\boldmath \$x\$}}) \nabla u({\mbox{\boldmath \$x\$}})) = f(\mbox{\boldmath \$x\$}) \$ with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution \$u({\mbox{\boldmath \$x\$})\$ resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the \$L^\infty\$ and \$L^2\$ norm.}, language = {en} } @misc{SchneiderBottaGeratzetal., author = {Schneider, Thomas and Botta, Nicola and Geratz, Karl Josef and Klein, Rupert}, title = {Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3749}, number = {SC-98-31}, abstract = {When attempting to compute unsteady, variable density flows at very small or zero Mach number using a standard finite volume compressible flow solver one faces at least the following difficulties: (i) Spatial pressure variations vanish as the Mach number \$M \rightarrow 0\$, but they do affect the velocity field at leading order; (ii) the resulting spatial homogeneity of the leading order pressure implies an elliptic divergence constraint for the energy flux; (iii) violation of this constraint would crucially affect the transport of mass, thereby disabling a code to properly advect even a constant density distribution. A previous companion paper derived the above observations from a single time - multiple length scale asymptotic analysis for \$M \ll 1\$, applied to the conservation form of the governing equations and assuming an ideal gas with constant specific heats. The paper then restricted to weakly compressible one-dimensional flows and introduced a semi-implicit extension of a compressible flow solver, designed to handle the interaction of long wavelength acoustics with small scale, large amplitude density fluctuations. In the present paper we concentrate on the limit of zero Mach number for multi-dimensional, variable density flows. The construction of numerical fluxes for all conserved quantities involves: An explicit upwind step (1) yielding predictions for the nonlinear convective flux components. This procedure still neglects the influence of pressure gradients on the convective fluxes during the time step. Suitable corrections are applied in step (2), which guarantees compliance of the convective fluxes with the divergence constraint. This step requires the solution of a Poisson-type equation to obtain the relevant pressure gradients. Step (3), which requires the solution of a second Poisson-type equation, yields the yet unknown (non-convective) pressure contribution to the total flux of momentum. The final, cell centered velocity field exactly satisfies a discrete divergence constraint consistent with the asymptotic limit. Notice that step (1) can be done by any standard finite volume compressible flow solver and that the input to steps (2) and (3) involves solely the fluxes from step (1), but is independent on how these were obtained. Thus, we claim that our approach allows any such solver to be extended to simulate incompressible flows. Extensions to the weakly compressible regime \$0 < M \ll 1\$, reactive flows and more complex equations of state will be addressed in follow-up publications.}, language = {en} } @misc{OevermannKleinBergeretal., author = {Oevermann, Michael and Klein, Rupert and Berger, Marsha and Goodman, Jonathan}, title = {A Projection Method for Two-Phase Incompressible Flow with Surface Tension and Sharp Interface Resolution}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5851}, number = {00-17}, abstract = {We present a fully second order projection method for the simulation of two-phase incompressible flow with surface tension. The Navier-Stokes equations are solved with a projection method on a fixed Cartesian grid. The free interface between the two fluids is tracked with a level set approach. The conditions at the interface for the pressure, the pressure gradient, and the velocity are explicitly incorporated into the scheme leading to a sharp representation of the pressure discontinuity and the interfacial force. The scheme in the presented form does not introduce additional points in the standard finite difference stencils. Computational results are compared with analytic solutions for a static round bubble, damped surface waves, and Rayleigh-Taylor instabilities.}, language = {en} } @article{EbrahimiViandHoeflingKleinetal., author = {Ebrahimi Viand, Roya and H{\"o}fling, Felix and Klein, Rupert and Delle Site, Luigi}, title = {Theory and simulation of open systems out of equilibrium}, series = {The Journal of Chemical Physics}, volume = {153}, journal = {The Journal of Chemical Physics}, doi = {10.1063/5.0014065}, pages = {101102}, abstract = {We consider the theoretical model of Bergmann and Lebowitz for open systems out of equilibrium and translate its principles in the adaptive resolution simulation molecular dynamics technique. We simulate Lennard-Jones fluids with open boundaries in a thermal gradient and find excellent agreement of the stationary responses with the results obtained from the simulation of a larger locally forced closed system. The encouraging results pave the way for a computational treatment of open systems far from equilibrium framed in a well-established theoretical model that avoids possible numerical artifacts and physical misinterpretations.}, language = {en} } @article{DelleSiteKrekelerWhittakeretal., author = {Delle Site, Luigi and Krekeler, Christian and Whittaker, John and Agarwal, Animesh and Klein, Rupert and H{\"o}fling, Felix}, title = {Molecular Dynamics of Open Systems: Construction of a Mean-Field Particle Reservoir}, series = {Advanced Theory and Simulations}, volume = {2}, journal = {Advanced Theory and Simulations}, doi = {10.1002/adts.201900014}, pages = {1900014}, abstract = {The simulation of open molecular systems requires explicit or implicit reservoirs of energy and particles. Whereas full atomistic resolution is desired in the region of interest, there is some freedom in the implementation of the reservoirs. Here, a combined, explicit reservoir is constructed by interfacing the atomistic region with regions of point-like, non-interacting particles (tracers) embedded in a thermodynamic mean field. The tracer molecules acquire atomistic resolution upon entering the atomistic region and equilibrate with this environment, while atomistic molecules become tracers governed by an effective mean-field potential after crossing the atomistic boundary. The approach is extensively tested on thermodynamic, structural, and dynamic properties of liquid water. Conceptual and numerical advantages of the procedure as well as new perspectives are highlighted and discussed.}, language = {en} } @article{DoerffelPapkeKleinetal., author = {Doerffel, Tom and Papke, Ariane and Klein, Rupert and Ernst, Natalia and Smolarkiewicz, Piotr K.}, title = {Dynamics of tilted atmospheric vortices under asymmetric diabatic heating}, series = {Theoretical and Computational Fluid Dynamics}, volume = {35}, journal = {Theoretical and Computational Fluid Dynamics}, number = {6}, doi = {10.1007/s00162-021-00591-x}, pages = {831 -- 873}, abstract = {P{\"a}schke et al. (J Fluid Mech, 2012) studied the nonlinear dynamics of strongly tilted vortices subject to asymmetric diabatic heating by asymptotic methods. They found, inter alia, that an azimuthal Fourier mode 1 heating pattern can intensify or attenuate such a vortex depending on the relative orientation of the tilt and the heating asymmetries. The theory originally addressed the gradient wind regime which, asymptotically speaking, corresponds to vortex Rossby numbers of order unity in the limit. Formally, this restricts the applicability of the theory to rather weak vortices. It is shown below that said theory is, in contrast, uniformly valid for vanishing Coriolis parameter and thus applicable to vortices up to low hurricane strengths. An extended discussion of the asymptotics as regards their physical interpretation and their implications for the overall vortex dynamics is also provided in this context. The paper's second contribution is a series of three-dimensional numerical simulations examining the effect of different orientations of dipolar diabatic heating on idealized tropical cyclones. Comparisons with numerical solutions of the asymptotic equations yield evidence that supports the original theoretical predictions of P{\"a}schke et al. In addition, the influence of asymmetric diabatic heating on the time evolution of the vortex centerline is further analyzed, and a steering mechanism that depends on the orientation of the heating dipole is revealed. Finally, the steering mechanism is traced back to the correlation of dipolar perturbations of potential temperature, induced by the vortex tilt, and vertical velocity, for which diabatic heating not necessarily needs to be responsible, but which may have other origins.}, language = {en} } @article{vonLindheimHarikrishnanDoerffeletal., author = {von Lindheim, Johannes and Harikrishnan, Abhishek and D{\"o}rffel, Tom and Klein, Rupert and Koltai, Peter and Mikula, Natalia and M{\"u}ller, Annette and N{\´e}vir, Peter and Pacey, George and Polzin, Robert and Vercauteren, Nikki}, title = {Definition, detection and tracking of persistent structures in atmospheric flows}, series = {arXiv}, journal = {arXiv}, abstract = {Long-lived flow patterns in the atmosphere such as weather fronts, mid-latitude blockings or tropical cyclones often induce extreme weather conditions. As a consequence, their description, detection, and tracking has received increasing attention in recent years. Similar objectives also arise in diverse fields such as turbulence and combustion research, image analysis, and medical diagnostics under the headlines of "feature tracking", "coherent structure detection" or "image registration" - to name just a few. A host of different approaches to addressing the underlying, often very similar, tasks have been developed and successfully used. Here, several typical examples of such approaches are summarized, further developed and applied to meteorological data sets. Common abstract operational steps form the basis for a unifying framework for the specification of "persistent structures" involving the definition of the physical state of a system, the features of interest, and means of measuring their persistence.}, language = {en} } @article{KleinEbrahimiViandHoeflingetal., author = {Klein, Rupert and Ebrahimi Viand, Roya and H{\"o}fling, Felix and Delle Site, Luigi}, title = {Nonequilibrium induced by reservoirs: Physico-mathematical model and numerical tests}, series = {Advances Theory and Simulation}, volume = {4}, journal = {Advances Theory and Simulation}, doi = {10.1002/adts.202100071}, pages = {2100071}, language = {en} } @article{GholamiHoeflingKleinetal., author = {Gholami, Abbas and H{\"o}fling, Felix and Klein, Rupert and Delle Site, Luigi}, title = {Thermodynamic relations at the coupling boundary in adaptive resolution simulations for open systems}, series = {Advances Theory and Simulation}, volume = {4}, journal = {Advances Theory and Simulation}, doi = {10.1002/adts.202000303}, pages = {2000303}, language = {en} } @misc{HellingKleinWoitkeetal., author = {Helling, Christiane and Klein, Rupert and Woitke, Peter and Nowak, Ulrich and Sedlmayr, Erwin}, title = {Dust in Brown Dwarfs IV. Dust formation and driven turbulence on mesoscopic scales}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7555}, number = {03-33}, abstract = {Dust formation {in brown dwarf atmospheres} is studied by utilizing a model for driven turbulence in the mesoscopic scale regime. We apply a pseudo-spectral method where waves are created and superimposed {within} a {limited} wavenumber interval. The turbulent kinetic energy distribution follows the Kolmogoroff spectrum which is assumed to be the most likely value. Such superimposed, stochastic waves may occur in a convectively active environment. They cause nucleation fronts and nucleation events and thereby initiate the dust formation process which { continues until} all condensible material is consumed. Small disturbances {are found to} have a large impact on the dust forming system. An initially dust-hostile region, which may originally be optically thin, becomes optically thick in a patchy way showing considerable variations in the dust properties during the formation process. The dust appears in lanes and curls as a result of the interaction with waves, i.e. turbulence, which form larger and larger structures with time. Aiming on a physical understanding of the variability of brown dwarfs, related to structure formation in substellar atmospheres, we work out first necessary criteria for small-scale closure models to be applied in macroscopic simulations of dust forming astrophysical systems.}, language = {en} } @misc{HellingKleinSedlmayr, author = {Helling, Christiane and Klein, Rupert and Sedlmayr, Erwin}, title = {The multi-scale dust formation in substellar atmospheres}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7567}, number = {03-34}, abstract = {Substellar atmospheres are observed to be irregularly variable for which the formation of dust clouds is the most promising candidate explanation. The atmospheric gas is convectively unstable and, last but not least, colliding convective cells are seen as cause for a turbulent fluid field. Since dust formation depends on the local properties of the fluid, turbulence influences the dust formation process and may even allow the dust formation in an initially dust-hostile gas. A regime-wise investigation of dust forming substellar atmospheric situations reveals that the largest scales are determined by the interplay between gravitational settling and convective replenishment which results in a dust-stratified atmosphere. The regime of small scales is determined by the interaction of turbulent fluctuations. Resulting lane-like and curled dust distributions combine to larger and larger structures. We compile necessary criteria for a subgrid model in the frame of large scale simulations as result of our study on small scale turbulence in dust forming gases.}, language = {en} } @misc{DeuflhardKleinReinefeldetal., author = {Deuflhard, Peter and Klein, Rupert and Reinefeld, Alexander and St{\"u}ben, Hinnerk}, title = {Power-User und Supercomputer}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4070}, number = {SC-99-19}, abstract = {Abstract-Sammlung zum gleichnamigen Workshop am ZIB vom 19.--20. Mai 1999}, language = {de} } @article{HorenkoKleinDolaptchievetal.2008, author = {Horenko, Illia and Klein, Rupert and Dolaptchiev, S. and Sch{\"u}tte, Christof}, title = {Automated Generation of Reduced Stochastic Weather Models I}, series = {Mult. Mod. Sim.}, volume = {6}, journal = {Mult. Mod. Sim.}, number = {4}, doi = {10.1137/060670535}, pages = {1125 -- 1145}, year = {2008}, language = {en} } @article{RiedelGelssKleinetal., author = {Riedel, Jerome and Gelß, Patrick and Klein, Rupert and Schmidt, Burkhard}, title = {WaveTrain: a Python Package for Numerical Quantum Mechanics of Chain-like Systems Based on Tensor Trains}, series = {The Journal of Chemical Physics}, volume = {158}, journal = {The Journal of Chemical Physics}, number = {16}, doi = {10.1063/5.0147314}, pages = {164801}, abstract = {WaveTrain is an open-source software for numerical simulations of chain-like quantum systems with nearest-neighbor (NN) interactions only. The Python package is centered around tensor train (TT, or matrix product) format representations of Hamiltonian operators and (stationary or time-evolving) state vectors. It builds on the Python tensor train toolbox Scikit_tt, which provides efficient construction methods and storage schemes for the TT format. Its solvers for eigenvalue problems and linear differential equations are used in WaveTrain for the time-independent and time-dependent Schr{\"o}dinger equations, respectively. Employing efficient decompositions to construct low-rank representations, the tensor-train ranks of state vectors are often found to depend only marginally on the chain length N. This results in the computational effort growing only slightly more than linearly with N, thus mitigating the curse of dimensionality. As a complement to the classes for full quantum mechanics, WaveTrain also contains classes for fully classical and mixed quantum-classical (Ehrenfest or mean field) dynamics of bipartite systems. The graphical capabilities allow visualization of quantum dynamics "on the fly," with a choice of several different representations based on reduced density matrices. Even though developed for treating quasi-one-dimensional excitonic energy transport in molecular solids or conjugated organic polymers, including coupling to phonons, WaveTrain can be used for any kind of chain-like quantum systems, with or without periodic boundary conditions and with NN interactions only. The present work describes version 1.0 of our WaveTrain software, based on version 1.2 of scikit_tt, both of which are freely available from the GitHub platform where they will also be further developed. Moreover, WaveTrain is mirrored at SourceForge, within the framework of the WavePacket project for numerical quantum dynamics. Worked-out demonstration examples with complete input and output, including animated graphics, are available.}, language = {en} } @article{GelssKleinMateraetal., author = {Gelß, Patrick and Klein, Rupert and Matera, Sebastian and Schmidt, Burkhard}, title = {Solving the time-independent Schr{\"o}dinger equation for chains of coupled excitons and phonons using tensor trains}, series = {The Journal of Chemical Physics}, volume = {156}, journal = {The Journal of Chemical Physics}, number = {2}, doi = {10.1063/5.0074948}, pages = {024109}, abstract = {We demonstrate how to apply the tensor-train format to solve the time-independent Schr{\"o}dinger equation for quasi-one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fr{\"o}hlich-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low-rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach that directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in the case of homogeneous systems, the tensor-train ranks of state vectors only marginally depend on the chain length, which results in a linear growth of the storage consumption. However, the central processing unit time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton-phonon coupling strength, although only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fr{\"o}hlich-Holstein type Hamiltonians.}, language = {en} } @article{delRazoWinkelmannKleinetal., author = {del Razo, Mauricio and Winkelmann, Stefanie and Klein, Rupert and H{\"o}fling, Felix}, title = {Chemical diffusion master equation: formulations of reaction-diffusion processes on the molecular level}, series = {Journal of Mathematical Physics}, volume = {64}, journal = {Journal of Mathematical Physics}, number = {1}, doi = {10.1063/5.0129620}, abstract = {The chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction--diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation for reaction--diffusion processes. The CDME consists of an infinite ordered family of Fokker--Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators --linear operators representing chemical reactions. These operators change the number of particles in the system, and thus transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally we discuss applications to multiscale simulations of biochemical systems among other future prospects.}, language = {en} }