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Heat is a complex quantity to measure in stochastic systems because itrequires extremely small sampling timesteps. Unfortunately this is not alwayspossible in real experiments, mainly because experimental setups have technicallimits. To overcome this difficulty a Simpson-like quadrature scheme wassuggested in [\emph{Phil. Trans. R. Soc. A 2017 375}] as a tool to compute theheat in stochastic systems. In this paper we study this new quadrature scheme.In particular, we first give a qualitative proof of the Simpson-like quadraturewith the help of Riemann-Stieltjes integrals and we then perform supplementarynumerical simulations to confirm our observations. Our main finding is that theSimpson-like quadrature yields errors that are much smaller than the onesobtained with the Stratonovich quadrature. This opens the possibility to designextremely sensitive experiments on stochastic systems without state-of-the-artsampling techniques.

How to measure heat in stochastic systems