Possible bug in InternalEnergyDensitySquared?

[evanberkowitz]

On N=5 κ=0.5 W=1 I've run many many streams and InternalEnergyDensitySquared seems to consistently differ between the Worldline and Villain formulation by ~2σ. Not just once, but over ~10 independently-generated pairs of streams.

An example is below, which shows the discrepancy. Other observables are included to demonstrate that it is unlikely to be a sampling problem. I've run streams up to 1M samples for these parameters and the observables didn't converge.

[evanberkowitz]

I don't know. I tried N=11 and it's compatible. But there really shouldn't be any finite-volume effect; the observables ought to match exactly.

[evanberkowitz]

This is indeed a bug. The code implemented a formula for the InternalEnergyDensitySquared observable that contained a blunder.

The InternalEnergyDensitySquared worldline documentation previously read

The issue of course is that $\partial_\kappa (\kappa^{-2}) = -2 \kappa^{-3}$ and not ${}^{-1}$. The correct formula is

with a different coefficient for the first term in $\partial^2S$.