Monte Carlo methods in physics

Three independent simulations, written to understand how random sampling converges on an answer and how to put an honest error bar on it.

Each one has a known result to check against. That constraint is the point: a Monte Carlo simulation will always produce a number, and without something to compare it to there is no way to tell a correct implementation from a confident wrong one.

Random walk#

Measuring the diffusion law — that mean squared displacement grows linearly with step count. The fitted slope came out at 1.002 ± 0.011 against a theoretical 1.

The error bar is the interesting part. The obvious approach, fitting all the points and taking the standard error, is wrong here: successive points on the same walk are not independent, because the displacement at step 100 contains the displacement at step 99. Treating correlated points as independent understates the uncertainty. Instead the error comes from ten independent groups of walks, which gives a spread that means something.

Radioactive decay#

Simulating a decaying population and recovering its half-life: 68.91 steps against a true 68.97.

The useful experiment was varying the starting population. The spread in the recovered half-life falls from 11.1% to 3.7% to 1.0% for 100, 1,000 and 10,000 initial nuclei — a factor of about ten in sample size for a factor of about three in precision, which is the $1/\sqrt{N}$ scaling that governs how expensive every Monte Carlo result is.

Bump hunt#

A toy version of the problem of finding a particle: a Gaussian signal at 125 GeV sitting on a falling background, fitted to recover the peak position.

Signal events	Fitted mass (GeV)
500	124.85 ± 0.22
200	125.33 ± 0.53
50	not recoverable

The last row is the one that taught me something. With 50 signal events the peak is still there in the data, and the fit still returns a number — it is simply indistinguishable from a fluctuation in the background. Knowing where that threshold sits for your own analysis is most of the discipline.