We discuss the relevance of information contained in cross correlations among different degrees of freedom, which is crucial in nonequilibrium systems. In particular we consider a stochastic system where two degrees of freedom $X_1$ and $X_2$—in contact with two different thermostats—are coupled together. The production of entropy and the violation of equilibrium fluctuation-dissipation theorem (FDT) are both related to the cross correlation between $X_1$ and $X_2$. Information about such cross correlation may be lost when single-variable reduced models for $X_1$ are considered. Two different procedures are typically applied: (a) one totally ignores the coupling with $X_2$; and (b) one models the effect of $X_2$ as an average memory effect, obtaining a generalized Langevin equation. In case (a) discrepancies between the system and the model appear both in entropy production and linear response; the latter can be exploited to define effective temperatures, but those are meaningful only when time scales are well separated. In case (b) linear response of the model well reproduces that of the system; however the loss of information is reflected in a loss of entropy production. When only linear forces are present, such a reduction is dramatic and makes the average entropy production vanish, posing problems in interpreting FDT violations.

• Received 22 February 2012

DOI:https://doi.org/10.1103/PhysRevE.85.061127