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This paper proposes an extended Petri net formalism as a suitable language for composing optimal scheduling problems of industrial production processes with real and binary decision variables. The proposed approach is modular and scalable, as the overall process dynamics and constraints can be collected by parsing of all atomic elements of the net graph. To conclude, we demonstrate the use of this framework for modeling the moulding sand preparation process of a real foundry plant., Dieser Beitrag schlägt einen erweiterten Petrinetz-Formalismus vor, der für die Zusammenstellung optimaler Planungsprobleme von vielfältigen industriellen Produktionsprozessen mit realen und binären Entscheidungsvariablen geeignet ist. Der vorgeschlagene Ansatz ist modular und skalierbar, da die Gesamtprozessdynamik und -beschränkungen durch Analyse und Betrachtung aller atomaren Elemente des Netzgraphen aufgestellt werden können. Der Ansatz wird abschließend für die Komposition des Planungsproblems des Formsandaufbereitungsprozesses einer realen Gießereianlage eingesetzt.

This is the material presented during the(first) IEA Wind Task 52 Lunch Seminar Series, which was held online onbetween 13-16 February2023 (all days 13-14 CET). The seminars were opened with an introduction by the Operating Agent, see20230215_IEA-Wind-Task52_lunch-seminar-INTRO_GotJul (1), followed by these presentations: Day 1 -- focus on Lidar-Assisted Control (LAC), organized by David Schlipf (20230213_LAC_WG) Feng Guo on "Improved modeling of lidar wind preview for wind turbine control"Feng_Guo_LunchSeminar_v2n Axel Schild on "Lidar-assisted model predictive turbine control"Schild_Challenges_on_the_way_to_indstrial_reality Day 2 -- focus on Complex Terrain, organized by Alexander Stoekl Andrew Hastings Black on "Point and lidar wind field reconstruction sensitivities in complex flow"IEATask52_LunchWebinarSeries_ComplexFlowWFR_AndrewBlack_14FEB2023 Johannes Becker on "Scanning LiDAR in Complex Terrain for Wind Ressource Assessment"GEO-NET_IEA-Task52-Lunch-Session_EWINO-Approach-Scanning-LiDAR-in-WRA_20230214 Day 3 -- focus on Scanning Lidar Offshore, organized by Andrew Oldroyd Yuko Ueda on "Validation of near-shore wind measurements using a dual scanning LiDAR"230215_task52_lunch_seminar_ueda Paula Gómez, Elliot Simon, Mike Courtney on "Guidelines for the usage of scanning lidars for power curve verification"20230215_IEA52_webinar Day 4-- focus on Floating Lidar, organized by Julia Gottschall Introductory presentation "Floating lidar within the IEA Wind 'Lidar' Task so far ..."20230216_IEA-Wind-Task52_Lunch-seminar_GotJul Peter Clive on "Floating lidar standards and guidance, an update centred around the upcoming technical specification IEC 61400-50-4"FLS Standard - update - 2023-02-16 Please note that each document or presentation may have its own license. If in doubt about reusing or citing any of the material presented here, please contact the authors.

These are the minutes of the IEA Wind Task 32 General Meeting, held online on November 16-18, 2021. The meeting is documented in T32GM2021.pdf (download link). This repository includes the overall minutes as well as some material presented in the meeting: Tobias Klaas-Witt (Fraunhofer IEE) are in T32GM2021-LoTarProjectOverview-KlaasWitt.pdf. Alexander Stoekl (energiewerkstatt) are in T32GM2021-MeasurementsInComplexTerrain-Stoekl.pdf Marc Defossez (Nergica) are in T32GM2021-WindLidarinColdClimates-Defossez.pdf A poster about "Mastering the challenges of industrial lidar assisted model predictive turbine control" from Axel Schild (IAV GmbH Ingenieurgesellschaft Auto und Verkehr). This presentation is in T32GM2021-PosterMCPforLAC-Schild.pdf Some material presented in the meeting is stored elsewhere: Information about the ITN LIKE Early Stage Researchers can be found at DOI: 10.5281/zenodo.5726899. A poster about "The Business Case for wind resource assessment using scanning Lidar" from Pedro Santos (Fraunhofer IWES) is at DOI: 10.5281/zenodo.5722362, Information about the new Task 32 round-robin on turbulence from forward-looking wind lidar can be found at DOI: 10.5281/zenodo.5713972. Information about the future direction of Task 32 can found at DOI: 10.5281/zenodo.5163487. Please note that each document or presentation may have its own license. If in doubt about reusing or citing any of the material presented here, please contact the authors., Feedback about this review should be directed to https://github.com/IEA-Wind-Task-32/2021-general-meeting-minutes/issues or to the Task 32 Operating Agents at ieawind.task32@ifb.uni-stuttgart.de.

There are different ways to obtain an exact one-electron theory for a many-electron system, and the exact electron factorization (EEF) is one of them. In the EEF, the Schr\"odinger equation for one electron in the environment of other electrons is constructed. The environment provides the potentials that appear in this equation: A scalar potential $v^{\rm H}$ representing the energy of the environment and another scalar potential $v^{\rm G}$ as well as a vector potential that have geometric meaning. By replacing the interacting many-electron system with the non-interacting Kohn-Sham (KS) system, we show how the EEF is related to density functional theory (DFT) and we interpret the Hartree-exchange-correlation potential as well as the Pauli potential in terms of the EEF. In particular, we show that from the EEF viewpoint, the Pauli potential does not represent the difference between a fermionic and a bosonic non-interacting system, but that it corresponds to $v^{\rm G}$ and partly to $v^{\rm H}$ for the (fermionic) KS system. We then study the meaning of $v^{\rm G}$ in detail: Its geometric origin as a metric measuring the change of the environment is presented. Additionally, its behavior for a simple model of a homo- and heteronucler diatomic is investigated and interpreted with the help of a two-state model. In this way, we provide a physical interpretation for the one-electron potentials that appear in the EEF and in DFT., Comment: arXiv admin note: substantial text overlap with arXiv:2010.14885

Quantum trajectory calculations for electrons are a useful tool in the field of molecular dynamics, e.g. to understand processes in ultrafast spectroscopy. They have, however, two limitation: On the one hand, such calculations are typically based on the Born-Oppenheimer approximation (BOA) and the electron dynamics for stationary nuclei is considered, thus neglecting quantum effects of the nuclei. On the other hand, even if the quantum nuclear motion would be taken into account, a BOA dynamics on a single potential energy surface would not provide any electron trajectories because the electronic part is treated as a stationary problem. By using the exact factorization method, we overcome these limitations and generalize the theory of electronic quantum trajectories to a fully quantum-mechanical treatment of the nuclei. After reviewing the time-dependent theory of quantum hydrodynamics and quantum trajectories, we show that the nuclei can be viewed as a quantum clock for the electronic motion and we develop a fully quantum-mechanical clock-dependent version of quantum hydrodynamics. This theory is used to obtain electronic trajectories for quantum nuclei, as is exemplified for a model system of a proton-coupled electron transfer dynamics. Our work generalizes the concept of quantum trajectories and lays the foundations for the development of trajectory-based simulation methods of electron dynamics beyond the BOA., arXiv admin note: substantial text overlap with arXiv:1911.13033

A quantum system Q is characterized by a single potential v and its eigenstates. While v is usually postulated for a given physical problem, it represents the interaction with an implicit environment E. We use the exact factorization to show how v emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem S = Q boolean OR E corresponds to a different potential v(j) and state chi(j) of Q. Such a state chi(j) typically has no nodes and is the ground state of v(j), even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting Q is much smaller than for exciting E, the potentials v(j) are similar in shape and differ only by sharp spikes. An excitation of Q can then be viewed as an excitation of Q with its environment being unaffected, and Q is approximately described by a single spikeless potential v and its eigenstates. Second, chi(j) can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of chi(j) being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems., Physical Review Research, 3 (3), ISSN:2643-1564

A quantum system Q is characterized by a single potential v and its eigenstates. While v is usually postulated for a given physical problem, it represents the interaction with an implicit environment E. We use the exact factorization to show how v emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem S = Q boolean OR E corresponds to a different potential v(j) and state chi(j) of Q. Such a state chi(j) typically has no nodes and is the ground state of v(j), even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting Q is much smaller than for exciting E, the potentials v(j) are similar in shape and differ only by sharp spikes. An excitation of Q can then be viewed as an excitation of Q with its environment being unaffected, and Q is approximately described by a single spikeless potential v and its eigenstates. Second, chi(j) can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of chi(j) being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems. ISSN:2643-1564

In a recent comment, Ruth Signorell raises a number of issues that she considers to question the validity of our approach to determine mean free paths for electron scattering in liquid water and our comparison with the results on amorphous ice by Michaud, Wen, and Sanche. Here, we show that these critiques are unjustified, being either unfounded or based on misconceptions by the author of the comment. We nevertheless welcome the opportunity to further clarify certain aspects of our work that we did not discuss in detail in our letter., The comment which is reply is referring to was rejected for publication, hence this reply will also not be published

The theoretical and computational description of materials properties is a task of utmost scientific and technological importance. A first-principles description of electron-electron interactions poses an immense challenge that is usually approached by converting the many-electron problem to an effective one-electron problem. There are different ways to obtain an exact one-electron theory for a many-electron system. An emergent method is the exact electron factorization (EEF) -- one of the branches of the Exact Factorization approach to many-body systems. In the EEF, the Schr\"odinger equation for one electron, in the environment of all other electrons, is formulated. The influence of the environment is reflected in the potential $v^{\rm H}$, which represents the energy of the environment, and in a potential $v^{\rm G}$, which has a geometrical meaning. In this paper, we focus on $v^{\rm G}$ and study its properties in detail. We investigate the geometric origin of $v^{\rm G}$ as a metric measuring the change of the environment, exemplify how translation and scaling of the state of the environment are reflected in $v^{\rm G}$, and explain its shape for homo- and heteronuclear diatomic model systems. Based on the close connection between the EEF and density functional theory, we also use $v^{\rm G}$ to provide an alternative interpretation to the Pauli potential in orbital-free density functional theory.

If one-electron observables of a many-electron system are of interest, a many-electron dynamics can be represented exactly by a one-electron dynamics with effective potentials. The formalism for this reduction is provided by the exact electron factorization (EEF). We study the time-dependent features of the EEF effective potentials for a model of an atom ionized by an ultrastrong and ultrashort laser pulse, with the aim of understanding what is needed to develop computationally feasible approximations. It is found that the simplest approximation, the so-called time-independent conditional amplitude (TICA) approximation, is complementary to single-active electron approaches as it reproduces the exact dynamics well for high-photon frequencies of the laser field or large Keldysh parameter. For relatively low frequencies of the laser field or for smaller Keldysh parameters, we find that excited-state dynamics in the core region of the atom leads to a time-dependent ionization barrier in the EEF potential. The time dependence of the barrier needs to be described accurately to correctly model many-electron effects, and we conclude that a multistate extension of the TICA approximation is a possible route as to how this can be achieved. In general, our study sheds a different light on one-electron pictures of strong-field ionization and shows that many-electron effects for such processes may be included by solving a one-electron Schrödinger equation, provided the core dynamics can be modeled successfully., Physical Review Research, 2 (4), ISSN:2643-1564

For localized and oriented vibrationally excited molecules, the qualitative features of the one-body probability density of the nuclei (one-nucleus density) are investigated. Like the familiar and widely used one-electron density that represents the probability of finding an electron at a given location in space, the one-nucleus density represents the probability of finding a nucleus at a given position in space independent of the location of the other nuclei and independent of their type. In contrast to the electrons, however, the nuclei are comparably localized. Due to this localization of the individual nuclei, the one-nucleus density provides a quantum-mechanical representation of the “chemical picture” of the molecule as an object that can largely be understood in a three-dimensional space, even though its full nuclear probability density is defined on the high-dimensional configuration space of all the nuclei. We study how the nodal structure of the wavefunctions of vibrationally excited states translates to the one-nucleus density. It is found that nodes do not necessarily lead to visible changes in the one-nucleus density: Already for relatively small molecules, only certain vibrational excitations change the one-nucleus density qualitatively compared to the ground state. It turns out that there are simple rules for predicting the shape of the one-nucleus density from the normal mode coordinates. A Python module for the computation of the one-nucleus density is provided at https://gitlab.com/axelschild/mQNMc ISSN:2296-2646

Quantum hydrodynamics is a formulation of quantum mechanics based on the probability density and flux (current) density of a quantum system. It can be used to define trajectories which allow for a particle-based interpretation of quantum mechanics, commonly known as Bohmian mechanics. However, quantum hydrodynamics rests on the usual time-dependent formulation of quantum mechanics where time appears as a parameter. This parameter describes the correlation of the state of the quantum system with an external system -- a clock -- which behaves according to classical mechanics. With the Exact Factorization of a quantum system into a marginal and a conditional system, quantum mechanics and hence quantum hydrodynamics can be generalized for quantum clocks. In this article, the theory is developed and it is shown that trajectories for the quantum system can still be defined, and that these trajectories depend conditionally on the trajectory of the clock. Such trajectories are not only interesting from a fundamental point of view, but they can also find practical applications whenever a dynamics relative to an external time parameter is composed of "fast" and "slow" degrees of freedom and the interest is in the fast ones, while quantum effects of the slow ones (like a branching of the wavepacket) cannot be neglected. As an illustration, time- and clock-dependent trajectories are calculated for a model system of a non-adiabatic dynamics, where an electron is the quantum system, a nucleus is the quantum clock, and an external time parameter is provided, e.g. via an interaction with a laser field that is not treated explicitly.

For localized and oriented vibrationally excited molecules, the qualitative features of the one-body probability density of the nuclei (one-nucleus density) are investigated. Like the familiar and widely used one-electron density that represents the probability of finding an electron at a given location in space, the one-nucleus density represents the probability of finding a nucleus at a given position in space independent of the location of the other nuclei and independent of their type. In contrast to the electrons, however, the nuclei are comparably localized. Due to this localization of the individual nuclei, the one-nucleus density provides a quantum-mechanical representation of the “chemical picture” of the molecule as an object that can largely be understood in a three-dimensional space, even though its full nuclear probability density is defined on the high-dimensional configuration space of all the nuclei. We study how the nodal structure of the wavefunctions of vibrationally excited states translates to the one-nucleus density. It is found that nodes do not necessarily lead to visible changes in the one-nucleus density: Already for relatively small molecules, only certain vibrational excitations change the one-nucleus density qualitatively compared to the ground state. It turns out that there are simple rules for predicting the shape of the one-nucleus density from the normal mode coordinates. A Python module for the computation of the one-nucleus density is provided at https://gitlab.com/axelschild/mQNMc, Frontiers in Chemistry, 7, ISSN:2296-2646

We present a novel ab-initio single-electron approach to correlated electron dynamics in strong laser fields. By writing the electronic wavefunction as a product of a marginal one-electron wavefunction and a conditional wavefunction, we show that the exact harmonic spectrum can be obtained from a single-electron Schr\"odinger equation. To obtain the one-electron potential in practice, we propose an adiabatic approximation, i.e. a potential is generated that depends only on the position of one electron. This potential, together with the laser interaction, is then used to obtain the dynamics of the system. For a model Helium atom in a laser field, we show that by using our approach, the high-order harmonic generation spectrum can be obtained to a good approximation.

Reaction mechanisms in chemistry are often indicated by arrows in Lewis structures. While these arrows account for changes in the electronic structure from reactant to product, they do not represent the time-dependent electron rearrangement. Such a time-dependent reaction mechanism can be obtained by calculation of an electronic flux density, but for reactions in the electronic ground state the Born-Oppenheimer separation prohibits straightforward calculation of this quantity. In this work, the concept of the electronic flux density is reviewed and analyzed within the framework of the Born-Oppenheimer approximation. Then, time-dependent reaction mechanisms are determined for cyclic electron rearrangements by using electron fluxes into or out of suitably defined bond sectors. The investigated models, tunneling in cyclooctatetraene, motion of benzene along the Kekule mode, proton tunneling in malonaldehyde and double proton tunneling in the formic acid dimer, yield interesting insights regarding questions like: How many electrons move from bond to bond? In which direction? Do electrons of different chemical type (like pi-electrons) behave differently? It becomes apparent that already a lot can be learned from the simple model systems, but also that a lot still needs to be done until time- dependent reaction mechanisms can be routinely calculated., Chemische Reaktionsmechanismen werden häufig mit Pfeilen in Lewisstrukturen dargestellt. Diese Pfeile repräsentieren Anderungen in der elektronischen Struktur von Reaktanden zu Produkten, aber sie geben nicht den zeitabhänigen Mechanismus wieder. Solch ein zeitlicher Reaktionsmechanismus könnte durch Berechnung der elektronischen Flussdichte erhalten werden, aber für Reaktionen im elektronischen Grundzustand verhindert die Born-Oppenheimer-Näherung eine einfache Berechung dieser Größe. In dieser Arbeit wird das Konzept der elektronischen Flussdichte erläutert und in Hinblick auf die Born-Oppenheimer- Näherung analysiert. Weiterhin werden zeitabhänige Reaktionsmechanismen für zyklische Elektronenumlagerungen mittels elektronischer Flüsse in oder aus geeignet definierten Bindungssektoren bestimmt. Die betrachteten Modelle, Tunneln in Cyclooctatetraen, die Bewegung von Benzol entlang der Kekule-Mode, Protonentunneln in Malonaldehyd und Doppelprotonentunneln im Ameisensäuredimer, bringen interessante Ergebnisse zu Tage bezüglich Fragen wie: Wieviele Elektronen fließen von Bindung zu Bindung? In welche Richtung fließen die Elektronen? Verhalten sich Elektronen verschiedenen chemischen Typs (z.B. pi-Elektronen im Vergleich zu anderen Valenzelektronen) unterschiedlich? Es wird deutlich, dass viel von diesen einfachen Modellsystemen gelernt werden kann, aber dass bis zur standardmäßigen Berechnung von Reaktionsmechanismen noch ein weiter Weg zurückgelegt werden muss.

Physical Review A, 97 (6), ISSN:1094-1622, ISSN:0556-2791, ISSN:1050-2947

This thesis focuses on the study of the Exact Electron Factorization (EEF). The EEF comes from a broader family of methods originating from the Exact Factorization (EF). The central principle of the EF is the separation of a quantum system into two parts, a marginal and a conditional part. A central result of the EF is that, if the full quantum system is described by a Schrödinger equation, also the marginal part is described by a Schrödinger equation. The conditional part constitutes an environment that provides the potentials for the marginal system. In this way, the full quantum system is exactly reduced to the marginal quantum system alone. In the case of the EEF, the EF is used to obtain a one-electron representation of the many-electron problem. We study several aspects of the EEF with analytically or numerically solvable model systems: For time-independent systems, we connect the EEF to density functional theory (DFT) and explain the appearance of charge-transfer steps and of a geometric part in the one-electron EEF potential. Additionally, we study the origin of spikes in the potential that are a feature of the EF in general. In this way, we do not only contribute to the development of simulation methods based on the EEF (and the EF), but we also provide fresh ideas for applied DFT. For time-dependent systems, we focus on the dynamics of molecular systems in strong ultrashort laser fields. Such a dynamics is often modeled with one-electron approaches. We present and contrast the standard method (the single-active electron assumption) with a multi-state model and an approximation based on the EEF (the TICA approximation), and we study how they perform to simulate experimentally relevant quantities for different parameter regimes of the laser field. Also, we explain two prominent features of the one-electron EEF potential for time-dependent systems, i.e., time-dependent steps and spikes, and we show that those are of minor importance. It turns out that different methods are applicable for different field parameters, and our multi-state models as well as the TICA model are suitable candidates for the simulation of strong-field phenomena.