Fluctuation-dissipation theorem, kinetic stochastic integral, and efficient simulations

Diffusive systems respecting the fluctuation-dissipation theorem with multiplicative noise have been studied on the level of stochastic differential equations. We propose an efficient simulation scheme motivated by the direct definition of the "kinetic stochastic integral", which differs from the better known Ito and the Stratonovich integrals. This simulation scheme is based on introducing the identity matrix, expressed in terms of the diffusion tensor and its inverse, in front of the noise term, and evaluating these factors at different times.