Uncertainties of numerical results

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SUMMARY

The discussion centers on determining uncertainties in results derived from Monte Carlo simulations, specifically in the context of particle diffusion through a small hole in a confined box. Participants agree that performing multiple simulations and calculating the standard deviation of the results is a reliable method for estimating uncertainty. Additionally, it is suggested that if computational resources are limited, one can conduct simulations with smaller sample sizes and use curve fitting to extrapolate uncertainty as a function of the sample size, following the principle that uncertainty decreases with the square root of the sample size (1/sqrt(N)).

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How do people generally determine uncertainties on results that are based on Monte Carlo simulations? Take this fictive example, just so we have something specific to talk about:

I look at 106 particles, confined to a box. There is a small hole in one of the walls, and at some time t0 I am interested in knowing how many particles N have diffused out of the box. This can be simulated by a Monte Carlo approach (Brownian motion).

This number N will vary each time I perform the simulation, but it will converge the larger I make the initial sample. Nonetheless, I guess an uncertainty is still present - how can we determine that in general?

Thanks for input in advance.
 
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I'm not sure if there's a more elegant method, but I'd repeat the simulation multiple times and take the standard deviation of the results. Perhaps if it's too computationally expensive to do this with a very high N, you can repeat it for a few low N and extrapolate the uncertainty as a function of N using a curve fit? I assume it would be 1/sqrt(N)?
 

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