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Assume the following, simple, 1D system: a particle on a spring, next to a wall, in a heat bath. Now, let's say that we implement a Monte Carlo scheme (Metropolis or other) for the position of the particle. The wall is hard, so whenever there is an attempt to pass it, the attempt will be rejected. My question is, is there a way to compute the average force exerted on the wall by the particle? My guess is that the force is related to the frequency of hitting the wall (in MC timestep units) but can't precisely see how.

here

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The average force is equal to the total impulse divided by time. Thus, it would be sufficient to calculate the total impulse delivered to the wall over some period of time.

One could do this by summing the impulse delivered at each collision. Suppose the particle goes into the collision with velocity $v$ (maybe this can be calculated by looking at how far the particle travels in this step). If the wall is inelastic, then the impulse delivered is $mv$. If the wall is elastic, then the impulse delivered is $2mv$.

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  • $\begingroup$ thank you for the response. The problem is, that in this type of MC there is no notion of velocity (velocities are independent of the position so those degrees of freedom can be integrated out of the partition function/position probability distribution). $\endgroup$
    – Botond
    Commented Sep 4, 2020 at 16:20
  • $\begingroup$ Maybe you could just use the average value of $v$ then. $\endgroup$
    – invjac
    Commented Sep 4, 2020 at 21:35
  • $\begingroup$ Hmm... Maybe a better bet is to just use some repulsive potential (exponential or alike) for the wall. $\endgroup$
    – Botond
    Commented Sep 7, 2020 at 17:22
  • $\begingroup$ Yes, I think that would work as well. Then, the average force exerted by the particle on the wall is the same as the average force exerted by the wall on the particle. $\endgroup$
    – invjac
    Commented Sep 7, 2020 at 23:52
  • $\begingroup$ Correct, and all you need to do is to compute its average position because you have the analytical form of the potential and derivating it gives the force. $\endgroup$
    – Botond
    Commented Sep 8, 2020 at 2:55

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