We examine the properties of steady states in systems which interact at the
boundary with a nonequilibrium environment. The examination is based on a
nonlinear Fokker-Planck equation, the structure of which is determined by the
fact that it also governs the time evolution of the equilibrium fluctuations of the
system. The nonlinearities in the Fokker-Planck equation may have two origins:
thermodynamic nonlinearities which arise if the thermodynamic potential is not
a bilinear function of the state variables, and nonlinear mode coupling which
arises if the transport coefficients depend on the state. While these nonlinearities
have only a small effect on the equilibrium fluctuations of a system away from
critical points, they are shown to be important for the determination of fluctua-
tions about nonequilibrium steady states. In particular the state dependence of
the transport coefficients may lead to deviations from local equilibrium and to a
breakdown of detail balance. An explicit formula for the time correlations of
fluctuations about the nonequilibrium steady state is obtained. The formula
leads to long-range correlations in fluids in the presence of a temperature
gradient. The result is compared with earlier approaches to the same problem.
Finally, we study the linear response to external forces and obtain a generaliza-
tion of the fluctuation-dissipation formula relating the response functions with
the nonequilibrium correlation functions.