The subatomic level, underlines molecular interactions. Understand different approaches to reduce the problem complexity, for instance by averaging out the molecules by establishing the fluid as a continuous field. The continuum is then discretised spatially and temporally Wrap it all up with proper boundary treatment: Velocity, Pressure and Boundary Pressure. Our own simulator, can enable us to actively manipulate the physics and see how phenomena are linked to it. ---- [[Macroscopic Perspective Fluid Simulations]] Microscopic foundations really justify all of the macroscopic implementations of fluid simulations. ### Fluids You could think of fluids as: - Discrete Colliding Particles - Continuous Flowing Medium ![[Pasted image 20220110201352.png]] > The central theme underlying simulating fluids is reducing information in every possible way. Because we can't just process all the raw information that is out there. Before we simulate anything, the multitude of information we are dealing with: For instance, if we are going to represent all the molecules that make up a fluid on a computer, 2 things show up immediately: 1. Simply too many of them, overwhelmingly too much. ![[Pasted image 20220110201904.png]] 2. There is information that we would like to know, but rather statistically or may be not at all. For instance, there will probably not be a significant offset in lift if a particular molecule is rotated differently. Or the locations of molecules, what matters in the end are the average states of larger portions of them. Given all that, modelling subatomic mechanisms on a quantum level for each molecule would be off the charts, especially when we are interested in quantities of a more global character like net lift force for instance. ![[Pasted image 20220110202200.png]] We need to get rid of alot of this information.. by emphasizing mathematically what's relevant to us and averaging out the rest we don't care about. Building a simulation in a modular way to give structure. We need each layer of abstraction because of the problems of everything underneath The problem quantum mechanics tries to approach is so severe, that simulating even small fluid systems at a quantum level is practically impossible. To see what's going on, we look at the fundamental properties of nature #### Thinking about measuring For a macroscopic object like a ball, tracking it's path or trajectory is an easy task. You just look at it or just use a device that can look for you ![[Pasted image 20220111005423.png]] How do you track the path of smaller things like elementary particle. This could be an electron that moves around the nucleus of a hydrogen atom. We have to **rethink what looking means.** Here, looking means interacting, as when we are trying to gain information about the original location of electrons within an atom. By doing so you influence the future significantly. The specimen - Electron (of a hydrogen atom) and the measuring device - photon (UV radiation) **are closer on the energy scale** as compared to any macroscopic composition. So while you measure, you exert influence. Tracking the undisturbed trajectory of **small things appears impossible.** ![[Pasted image 20220111010024.png]] Another fundamental limit on how certain one can ever be in terms of location and momentum of a particle, regardless of the measurement process --> [[Heisenberg Uncertainty Principle]] ![[Pasted image 20220111010548.png]] Even theoretically, talking about trajectories of such things loses meaning. ![[Pasted image 20220111010636.png]] However, what does not lose meaning is the averaging of different observations to yield a different kind of information - a probabilistic perspective. Use a [[probability]] distribution to determine how likely it would be to detect an electron in a certain region of an atom. ![[Pasted image 20220111010951.png]] Depending on properties like electron energy or strength of the electric field, these probability distributions come in many different shapes. If there are multiple electrons around the atom, the picture is even more complicated, since electron motion is correlated ![[Pasted image 20220111011336.png]] ![[Pasted image 20220111011412.png]] Peculiar about elementary particles is that they behave not only particle like, which means there is an integer number of them and once measured you observe their impact at a certain distinct location. But they also behave wave-like, meaning they go around corners and superimpose, diffract and interfere, as long as you don't observe them. Nature chooses to behave this way, we want to then derive models upon this basis. To make sense of these probabilities and wave like properties we developed a mathematical tool, that reflects the indeterminate nature, seen in all experiments, in its core called Quantum Mechanics. ---- Timetable: ----------------- 00:00 - What We Build 02:06 - Guiding Principle - Information Reduction 04:03 - Measurement of Small Things 07:09 - Quantum Mechanics and Wave Functions 13:45 - Model Order Reduction 16:48 - Molecular Dynamics and Classical Mechanics 26:50 - Kinetic Theory of Gases 30:23 - Recap Selected Papers and Learning Resources: ------------------------------------------------------------------- 05:03 Experiment: Paper: "Stodolna, Aneta S., et al. Hydrogen atoms under magnification: direct observation of the nodal structure of stark states. Physical review letters 110.21 (2013): 213001." 06:49 Atomic- and Molecular Orbitals Tomography: Paper: "Vozzi, Caterina, et al. Generalized molecular orbital tomography. Nature Physics 7.10 (2011): 822-826." Paper: "Itatani, Jiro, et al. Tomographic imaging of molecular orbitals. Nature 432.7019 (2004): 867-871." 07:24 Matter Waves, Double Slit Experiment: Paper: "Jönsson, Claus. Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten. Zeitschrift für Physik 161.4 (1961): 454-474." Paper: "Jönsson, Claus. Electron diffraction at multiple slits. American Journal of Physics 42.1 (1974): 4-11." 07:57 Quantum Mechanics Overview, Wave packets, Standing Waves, Eigenstates: E-Book: "Mathur, Samir D. https://www.asc.ohio-state.edu/mathur..." E-Book: "van Dommelen, Leon. Quantum mechanics for engineers. https://web1.eng.famu.fsu.edu/~dommel... (2004)" 16:01 Model Order Reduction, Modes: Lecture Notes: from "Farhat, Charbel. Model Reduction. https://web.stanford.edu/group/frg/co..." to "CA-CME345-Ch9" 17:17 Classical Mechanics, Phase Space: Lecture Notes: "Cerfon, Antoine. Mechanics (Classical and Quantum). https://www.math.nyu.edu/~cerfon/mech..." 18:15 Molecular Dynamics, Born-Oppenheimer Approximation, Potential Energy Surface, and Non-Quantized Approximation of Energy Levels: Lecture Notes: "Allen, Michael P. Introduction to molecular dynamics simulation. Computational soft matter: from synthetic polymers to proteins 23.1 (2004): 1-28." Paper: "Parker, J. G. Rotational and vibrational relaxation in diatomic gases. The Physics of Fluids 2.4 (1959): 449-462." Paper: "Valentini, Paolo, et al. Direct molecular simulation of nitrogen dissociation based on an ab initio potential energy surface. Physics of Fluids 27.8 (2015): 086102." E-Book Chapter: "van Dommelen, Leon. Quantum mechanics for engineers. 9.2 The Born-Oppenheimer Approximation. https://web1.eng.famu.fsu.edu/~dommel... (2004)" 20:11 One-Dimensional Hydrogen Atom Approximation for the Coulomb Potential as opposed to the "true" One-Dimensional Coulomb Potential: "Loudon, Rodney. One-dimensional hydrogen atom. American journal of physics 27.9 (1959): 649-655." "Loudon, Rodney. One-dimensional hydrogen atom. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472.2185 (2016): 20150534." 27:06 Kinetic Theory of Gases, (Variable) Hard Sphere Approximation of Molecules: Lecture Notes: from "http://volkov.eng.ua.edu/ME591_491_NE..." to "NEGD-06"