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The Boltzmann distribution is derived from the canonical ensemble, which in turn is derived by assuming your system has ergodic Hamiltonian dynamics on phase space. That in turn is true for a wide variety of real-world systems. If you discard all of this structure and simply postulate an arbitrary time evolution, you shouldn't expect the Boltzmann ...

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Temperature and pressure are ultimately statistical properties of a large number of particles. Asking for the temperature - or pressure - of a single atom is meaningless. In the case of an ideal gas, it turns out that temperature can be related to the average kinetic energy of the gas molecules via the equipartition theorem, which yields ${KE}_{av} = \frac{... 2 Assuming that it's a horizontal mass-spring system, then the Newtonian equation of motion is: $$ma=-kx$$ Or: $$\ddot{x}+\frac{k}{m} x=0$$ With: $$\omega^2=\frac{k}{m}$$ the solution is: $$x=A\sin(\omega t+\varphi)$$ We have two initial conditions: $$x(0)=x_0\text{ and }\dot{x}(0)=0$$ so that: $$\dot{x}(0)=A\omega\cos(0+\varphi)=0$$ $$\Rightarrow \cos\varphi=... 2 It sounds like you're running into the problem of ODE stiffness, you have a system in which vastly different timescales need to be considered. There is an entire field of applied mathematics dedicated to the study of numerical ODE solvers, and it's not possible to discern exactly which method would suit you best from the information you've provided. Your ... 1 If the goal is to obtain the movement equations in cartesian coordinates a good procedure is to use the lagrarangian formulation. Thus calling p=(x,y) we have$$ L = \frac 12 m \dot p\cdot\dot p - m g (y-l_0)+\lambda(x^2+y^2-l_0^2) $$The movement equations give$$ \cases{ m\ddot x - 2\lambda x = 0\\ m\ddot y -2\lambda y + m g = 0\\ x^2+y^2-l_0^2=0 }$\$ ...

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If electrons were placed randomly in a closed shape, would they accumulate on the surface of the shape as shown in my program? Yes, if that shape is enclosing the surface of a conductor. However, since you did not simulate a conducting material, I think this is a coincidence. Your simulation, by looking at the initial configuration in the second gif, begins ...

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