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It is standard practice to plot the probability density function (not the wavefunction), in the output of such simulations. In this case, it is more practical to use simulations in the statistical mechanics regime. For eg. to simulate particles in a one dimensional lattice chain, you may wish to use the Ising model and observe the number of particles at ...

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The difficulty with Reynolds number is that the length scale (and often times the velocity scale) are both completely subjective, as you have identified. In standard nomenclature, an airfoil (or in your case, an ellipse) would use the freestream velocity as the velocity scale and the chord (semi-major axis) as the length scale. This is the assumed standard ...

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For those who may read this later: you cannot find the liquid and vapor phases independently in this way. The reason is that for densities in between the liquid and vapor densities, the system will phase segregate into droplets within the simulation, and so your total density will include contributions from both phases. Consequently you can change the ...

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Once you have an understanding of fluid mechanics, the two best books for CFD specifically that I have used are: Computational Fluid Dynamics by John Anderson. I don't know if you have ever used any of Anderson's fluid dynamics books, but I highly recommend all of them. His books are all very readable and spend most of the text describing what to do rather ...

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The book I learned from decades ago was Sabersky and Acosta "Fluid Flow: a First Course in Fluid Mechanics". There are a number of basic concepts: Continuity: mass within a bounding volume, as a function of mass flows across the boundaries. Streamlines and the stream function. Equations of motion with and without viscosity. The Bernoulli equation relating ...

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Your question is unclear. If you add transverse dimensions, your problem is no longer 1-dimensional. If you mean adding transverse components of your interested quantities, then there is no change needed to the usual FDTD method. For example, if you are analysing the propagation of an electromagnetic plane wave in one-dimension, say the positive $z$ ...

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Although this is a late answer, I hope it could be helpful to others. I happened to have done some work in both Monte Carlo (simulated annealing) and Langevin dynamics of classical spin models. My experience is that both could give you the correct ground state. But the former could be faster, as you can use non-local cluster algorithms to (partially) ...

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A couple of comments on your implementation: You initialize your distribution functions in an unconventional way; usually, we want to convert known macroscopic 'lattice' quantities like $\rho$ and $\vec{u}$ into $f_i$. An easy way to do this is using the equilibrium distribution you define above: ...

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This is clearly completely hopeless. Many particle systems in QFT or even in non-relativistic QM are computationally intractable. Just to store the many-body Schroedinger wave function of more than a few particles is out of the question (it is a function of $3^N$ variables, where $N$ for your problem would be $O(10^{23}))$. In practice, lattice QCD is ...

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Well, let's do some thinking here. You didn't specify what the cubic meter is composed of, but let's for sake of argument say that it is a gas of single, non-interacting particles (like, say, neutrons or something) at standard temperature and pressure. So we know we have 1 mole of gas in that cubic meter. Let's further assume that we only needed a single, ...

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I've only ever used the Ewald sum, I've never implemented it myself. However, you mention that you're not converging as $\kappa$ increases nor are you converging to the correct value. It would seem that regardless of the problem, if your implementation is correct it should converge at some point. If you do reach convergence wrt $\kappa$; as to the point ...

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The expansion of a gas into the vacuum is not a diffusive process. Depending on the initial conditions (density and density gradient) the expansion is either described by the Euler/Navier-Stokes equation of fluid dynamics, or the Boltzmann equation of kinetic theory. For the parameters that you mention the gas is sufficiently dense that most of it is in ...

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Well, for the harmonic oscillator, you have the full closed answer, so you don't really need numerics. It is, as Groenewold discovered in his 1946 thesis (Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.), merely rigid rotation!: ...

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How could I analytically/numerically solve for the density as a function of time, and are there any free software packages I could use to solve for and plot this? Let's focus on the diffusion equation in one dimension, to see what happens here (the results can easily be modified to multiple dimensions). $\frac{\partial \phi(x,t)}{\partial t} = D ... 2 They open as text files after you unzip them. You can open them with excel to parse the data 0 I don't know if you are still interested in this question, but there is some work in the literature that might be very interesting for you: In this paper they propose an interpretation of the Wigner function as a particular wave function and in this other paper they propose a numerical method to compute the propagation, note that you might be interested in ... 4 I asked my local SDSS experts, and the answer is that the fourth and fifth arrays are both bitmasks. Some background: the spectra are often generated with separate exposures that are combined into one dataset. Each exposure comes with its own set of flags. When combining datasets, the question naturally arises "Do we set a flag if it appears in any of the ... 2 This is a fairly old question, but given the current answer, I feel a need to add my own here as well for future reference.$h$is indeed an adjustable parameter. The problem with your integral is that it assumes a 1 dimensional setting while your smoothing kernel assumes a 3 dimensional setting. assuming a spherical coordinate system, the correct integral ... 1 I assume you are not asking about how parallel tempering works as an algorithm for sampling from probability distributions, but that you are in fact interested in finding the global minimum of a target function$E(x)$. If that's not the case, please clarify what exactly you didn't understand from the Wikipedia page you linked and I'll be happy to edit my ... 1 This is a trick used to save time when doing the acctual computation by taking advantage of symmetry in the problem. Note that$k^2$is an even function (invariant under${\bf k}\to-{\bf k}$) and the norm of the fourier transform of the lattice function$|S({\bf k})|^2$is also an even function since$\$S({\bf k}) = \sum_i q_i e^{i{\bf k}\cdot {\bf r_i}} ...

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