  Departures of observables from their thermal equilibrium expectation values
are studied under heat flow in steady-state non-equilibrium environments. The
relation between the spatial and temperature dependence of these
non-equilibrium behaviors and the underlying statistical properties are
clarified from general considerations. The predictions are then confirmed in
direct numerical simulations within the FPU-$\beta$ model. Non-equilibrium
momentum distribution functions are also examined and characterized through
their cumulants and the properties of higher order cumulants are discussed.
