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in ODE textbooks there are several examples of mixing problems, such as water flowing into a tank at a given rate, with a certain concentration of salt, and then there is an outflow from the tank. One has to find a ODE which describes the problem, and find the solution as a function. Many of these examples lead to ODEs that are solved by integrating factor, and the procedure is rather straight-forward.

However, over some years of reading quantum mechanical literature, I have not come across a problem description which reflects the oscillation of an electron, other than that of solving the Schrödingers equation. Even more interesting, the case of ocean waves, or waves in a river, which experience turbulence, can be described by rather complex PDEs, with up to 6th order of partial differential operators. Rogue wave phenomena are even more obscure, and the more we get into wave behavior, the more the "mixing problems of ODEs" seem to be long way away in terms of their simplicity.

Are there, to the knowledge of readers of this post, any examples of PDE problems such as the above mentioned "mixing problem" which describe QM, wave mechanics or even rogue waves, as exercises for setting up PDEs? That is, can PDEs be simplified into descriptions such as the mixing problem, and if yes, which website or source presents this?

Thanks

Luthier415Hz
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    The standard Schrödinger equation is a linear ODE so no much things could happen, but when diving into real phenomena the models fastly becomes non-linear, where commonly perturbation theory is used. As example, with laser light through non-linear materials (optical fiber), Soliton's solutions could be found using Mawxwell's Equations, which is a kind of wave that is also found in the equations that haves Rouge waves solutions, so maybe, if you search about the quantum mechanics explanation of Solitons' occurrence in fiber optics you will find the kind of equations you are looking for. – Joako Jan 06 '22 at 16:15
  • This you describe is summarized quite well here, https://www.semanticscholar.org/paper/Mathematical-modeling-of-rogue-waves-%2C-a-survey-of-Manzetti/f50ae3ef48a2cb8d976c1cfe78f2a517c9a6aa23 but it does not, as other sources I have inquired on, explain how they set up these difficult PDEs. They seem to be either set up by pure guessing, trial , error, or by some hidden rationale. I would like to find a good explanation of this "hidden rationale". – Luthier415Hz Jan 07 '22 at 09:13
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    From my point of view, physicists do experiments, have results, and then thy try to find a theory that numerically explains what they see... not in the other way arround... reality have been show to be so complicated that nowaday still now way to summarize every observed phenomena in one mathematical framework.. so I don´t really see why been doing trial and error is wrong... think about this: every finite-duration phenomena can´t be modeled with a linear ODE or PDE (because it crashes with uniqueness of solutions at the boundaries), ... – Joako Jan 07 '22 at 19:08
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    (...) neither with a non-constant Power Series (because been of finite-duration will make it of compact support), so for been modeled "right" non-linear model will been required, but non-linear dynamics are a hell of difficult, and no general solution could be probably been found (and just in the deterministic case)... so to be practical, Perturbation Theory says expand the non-linearity as an Taylor Series, that will be just add a polynomial with unknown constants, and start adding terms one by one and set the constants so your phenomena is been explained bit by bit. – Joako Jan 07 '22 at 19:13
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    and in this way, they find something that can been use to study the characteristics of the phenomena in a quantitative way, and sometimes, if the model fits good, they can find things unobserved in reality that are forecast by the model (like the Poisson's Spot famous example)... later, physicists and mathematicians will try to fit this equations into a existent o new bigger model. I remember from university (long ago), a Book about non-linearity in fiber optics by Agrawal et al, they follow the discovering process, maybe there you can find what you are looking. – Joako Jan 07 '22 at 19:19
  • Does this mean that the heat equation was built based on measurement of temperature on several parts of a heated block of metal and not by an idea that heat travels through a medium at a specific "speed" depending on the heat capacity of the materials and its density, which in turn has an acceleration, and then they can derive the second order derivatives of the heat equation from that acceleration? Because the latter, would be a good example to explain how to "make the heat equation". – Luthier415Hz Jan 07 '22 at 20:22
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    I believe that it is exactly how is happen, surely many people tried before Poisson to find a Heat model, and we know now the Poisson Equations because is the modern explanation, but surely there are others before... Science doesn´t starts with Calculus, it starts centuries before: think how Persians/Greeks know to forecast the positions of the planets, the sun, the moon, and eclipses a millennia before the born of Johannes Kepler, and the make mechanical clocks/calculators to do that (like the Antikythera mechanism).... – Joako Jan 07 '22 at 23:42
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    Science search the "truth" through mathematical "models", but models aren´t "truths": philosophically speaking scientific truth is unattainable, because you always can ask another "why" for what you are explaining - models on the other hand, are quantitative explanations that try to forecast the output of events, and they are made through assumptions that make them limited to how good it forecast can be: as engineer, the classic model of current as a fluid through wires is more than enough to making homes and buildings, as example, and the engineer also know they are a "lie"... – Joako Jan 07 '22 at 23:47
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    electrons doesn´t flow carrying the energy at speed light, they actually moves at speeds of the order of 4 cm/hr (2 in/hr), and charge actually flow in the opposite direction than the traditional model, and using a accurate model is nonsense: using quantum electrodynamics to find how many watts will consume an installation by tracking every electron is just impossible with current computers, and is a nonsense effort. So, you will find a lot of models, not truths, and always it will be the same. – Joako Jan 07 '22 at 23:52
  • Just a minor comment, before reading your text. It was Fourier who developed the Heat equation. Which PDEs did Poisson develop? – Luthier415Hz Jan 08 '22 at 08:40
  • What you write is in fact in agreement with one of the Dysthe papers on rogue waves. I don't remember which of those papers it was, but he and his colleague "found out" that "adding the 6th order differential operator on the equation, acting on the unknown function $u$ increased precision of the results, when comparing with the New Years Wave, which hit the Draupner oil field (-11m +16 meters=27 m high). They did not explain why they insert the 6th order derivative, and it seemed like a "lucky shot" or as written before, trial and error. But some PDEs must be more reasonably connected – Luthier415Hz Jan 08 '22 at 08:45
  • to some phenomenon. For instance, the heat equation describes acceleration of a wave with respect to change of time. It seems more like a Newtonian equation for a moving object than an "obscure" model such as the rogue wave equations. In fact, this goes further, the quantum mechanical relations are known to be derived from the Newtonian equations, if I am not wrong - see this link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qmoper.html , where kinetic energy known in classical physics $\frac{1}{2}mv^2$ is equivalent to $-\hbar\frac{1}{2m}\frac{\partial^2}{\partial x^2}\psi$ – Luthier415Hz Jan 08 '22 at 08:47
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    I made a mistake, you are right, its Fourier who find the heat equation... What would happen if their model was the actual Linear PDE equation but the phenomena wasn´t perfectly fit the results?... sometimes happens, and if a non-linear term rises is highly probable that the closed-form solution is not findable, so you will try approximations as the perturbation theory (are many more also)... remember that the Heat equation is a model, that works under assumptions, and not always work, as example, search for non-linear heat equations, like for non-constant heat capacity mediums. – Joako Jan 08 '22 at 14:13
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    About the oscillation of electrons, maybe thinking first in photons could give an idea of what you are looking for: I don´t precisely know if photons oscillates, but surely electric field does in light, being modeled precisely through wave equations when obtaining the dark and shinny fringes in the double slit experiment... actually when noticing that also electrons follows the same patterns, it rises the use of waves for matter particles: here technically each electron individually hits one spot, but its aggregated probabilities certainly show a wave behavior, so "something" is oscillating – Joako Jan 11 '22 at 04:07
  • Yes, this is a good point you give, and the fact that it is inevitably about probabilities, maybe you are in fact answering the question of this post, for the quantum physical part: there is no "mixing problem" example of quantum phenomena, because of its discrete nature. Hence, we are left to find out if we can find a mixing example for continuum phenomena, such as turbulence in a river or waves on the sea. But the former has unfortunately a lot in common with magnetic field turbulence and formation of charges in the electron gas and superfluids, so it is even more strange, the fact... – Luthier415Hz Jan 11 '22 at 08:22
  • .. that continuum phenomena for waves in turbulence can describe electromagnetism, which is of lower energy than quantum phenomena. – Luthier415Hz Jan 11 '22 at 08:23

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