Fisher information analysis of a Langevin System with Thermal and Athermal Noise

Fisher Information (FI) has proven to be a useful tool for estimating the parameters of a system when only statistical data are available. We apply this method to a system of Langevin equations analogous to the Brownian Gyrator, in which the coupling parameter $b$ and the athermal noise strength $D$ are treated as unknown quantities. Using the FI, we discuss the estimation of the coupling parameter between the degrees of freedom and the quantity $D$ associated with the external noise effects. We find that the diagonal components of the FI exhibit nontrivial behavior, whereas the off-diagonal components display a monotonic dependence on the parameters of interest. This distinction may contribute to an accurate estimation of $b$ and $D$.