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You will want to plot the exponent on a linear scale: the log scale is more appropriate for the difference graph should you choose to use it. The graphs for the exponent from differences in q and p individually may be misleading. You may also want to increase the time scale 100-fold so the phase space parameters of the single oscillators sync up a couple ...


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Enzo is fundamentally a grid-based finite-volume hydrodynamics code. That is, the domain is divided into cells, each is assigned various fluid properties (density, velocity, etc.), and at each timestep fluxes of those quantities across the interfaces between cells are used to update the quantities in the cells. It has a choice of particular methods for ...


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In situations like this, it is a good idea to adjust the time step based on the gradient of the force - because the whole concept of numerical integration is that "things don't change too much from now until the next time step", and that assumption is violated when you move rapidly through a region with fast-changing force. This has a risky side-effect: if ...


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You're getting particles that get 0 distance from each other, and therefore have infinite acceleration. The computer cannot resolve this, and so it explodes according to how close the step gets to zero. I'm not sure what "physical" behaviour would be, given that zero distance point masses are difficult to study. But if you added in a y coordinate and gave ...


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What matters here is how the value of $c$ compares to the value of $k$. Let us choose a $\zeta = \frac{c}{2\sqrt{mk}}$ One can show that when $\zeta =1 $ the system is critically damped, and will not exhibited any oscillations and will return to the origin in the shortest possible time interval. When $\zeta > 1 $ the system is over damped and will take ...


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For any PIC simulation, you are necessarily tying yourself to particles, particles that experience forces. Thus, we have the generic force law: \begin{align} m_i\frac{d\mathbf v_i}{dt}&=\mathbf F_i \tag{1a}\\ \frac{d\mathbf x_i}{dt}&=\mathbf v_i \tag{1b} \end{align} In the case of PIC, you are often considering the electromagnetic (Lorentz) force, ...


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I think you're overcomplicating things here. The whole point of the hole-model is to simplify the picture... instead of trying to model an entire sea of electrons, you model the hole itself. For example, consider a bubble rising in a glass of cola... yes, you could create a complicated model of all the liquid moving within the glass, but it's much easier ...


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Angular momentum operator in quantum mechanics is defined as: $$\hat{L}=-i\hbar[r\times\nabla]$$ You just need to insert this definition of $\hat{L}$ to $\langle \psi|\hat{L}|\psi\rangle$ (or integral) to calculate. You need to use the integral form for the average of $\hat{L}$ and use numerical methods to evaluate. Also, angular momentum is not of the ...


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You need the integration step to be much shorter than the impact time. For most real world collisions, 0.1 second will be much too long. The force will change during the collision - and given the very simplistic integration method you use, you have to integrate over sufficiently short steps during which the force doesn't change much. You could do this by ...


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Sorry, I would like to comment but don't have enough reputation. Are you trying to model gas or liquid? I actually wrote an iPhone game that simulates liquid so I have a decent amount of experience with this if you are trying to simulate liquid.


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For an incompressible fluid $\dot\rho=0$. Then the continuity equation implies $$ \nabla\cdot u = 0 . $$ We can now take the divergence of the Navier-Stokes equation and get $$ -\nabla^2 P = \rho\nabla_j(u_i\nabla_i u_j). $$ This means that the pressure is instantaneously determined by the velocity field (the pressure is no longer an independent ...


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This means that the universe cannot be simulated by a Turing Machine, it's not related to being able to write down th equations of physics. If the laws of physics are non-computable, you then also lift the restrictions any computing device is subject to and expand that to whatever the non-computable laws of physics would allow. E.g. classical physics is ...



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