How to measure heat in stochastic systems

Heat is a complex quantity to measure in stochastic systems because it requires extremely small sampling timesteps. Unfortunately this is not always possible in real experiments, mainly because experimental setups have technical limits. To overcome this difficulty a Simpson-like quadrature scheme was suggested in [\emph{Phil. Trans. R. Soc. A 2017 375}] as a tool to compute the heat in stochastic systems. In this paper we study this new quadrature scheme. In particular, we first give a qualitative proof of the Simpson-like quadrature with the help of Riemann-Stieltjes integrals and we then perform supplementary numerical simulations to confirm our observations. Our main finding is that the Simpson-like quadrature yields errors that are much smaller than the ones obtained with the Stratonovich quadrature. This opens the possibility to design extremely sensitive experiments on stochastic systems without state-of-the-art sampling techniques.