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You don't say what simulation technique you are using, but clearly errors are adding energy to the system. A simple one would be to take a starting position $(x,y)$ and velocity $(v_x,v_y)$. Note that if your velocity is not exactly right for a circular orbit, you should just get an ellipse that is close to the circle you are after, so that is not your ...

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The thermal diffusivity can indeed be spatially dependent--consider the case you present: an iron bar fixed to a cool copper bar with one end being heated, clearly there is a disjoint in the value at the joining point. Now extend that idea to say 100 alternating blocks of iron & copper and you have a nice clear spatially dependent coefficient. ...

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The best way to numerically work with continuous phase transitions is to study observables that have a vanishing length dimension (or mass dimension in the language of QFT). Take for example the Binder's cumulant ($\langle m^4\rangle/\langle m^2\rangle^2$ modulo factors of 3 and constants, where $m$ is the order parameter) or the correlation length scaled by ...

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This is certainly unexpected, for as Mark Mitchison commented the $J=0$ Heisenberg model is equivalent to free fermions in one dimension. Moreover I suspect that something is amiss even before that, for the $J=-1$ is the ferromagnetic model and $J=+1$ the antifferomagnetic one, and certainly they have different ground state energies. In fact, for $J\leq -1$ ...

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I did the 1st example in Schneider's online book both ways and got the same graphs, so I'm convinced the answer is that they are equivalent but that Yee's algorithm is faster by a factor of 4 (it uses 1/2 as many points and 1/2 as many time steps).

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You don't have to discretize your problem (XY model). For each step, just take some value as the new $\theta$, and calculate the transition rate accordingly. Of course, when choosing the new value of $\theta$, better don't do it in a completely random way, otherwise your transition rate might be usually too small and you are just wasting time. Having said ...

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A few off the top of my head: Double pendulum is a nice example of chaotic motion and quite simple to model. Ising model of a magnet can provide a good introduction to modelling quasi-random processes. Solving the heat equation with various combinations of sinks/sources. Lots you could do with optics from very simple to very complex. Also lots you could do ...

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