  The absence of a simple fluctuation-dissipation theorem is a major obstacle
for studying systems that are not in thermodynamic equilibrium. We show that
for a fluid in a non-equilibrium steady state characterized by a constant
temperature gradient the commutator correlation functions are still related to
response functions; however, the relation is to the bilinear response of
products of two observables, rather than to a single linear response function
as is the case in equilibrium. This modified fluctuation-response relation
holds for both quantum and classical systems. It is both motivated and informed
by the long-range correlations that exist in such a steady state and allows for
probing them via response experiments. This is of particular interest in
quantum fluids, where the direct observation of fluctuations by light
scattering would be difficult. In classical fluids it is known that the
coupling of the temperature gradient to the diffusive shear velocity leads to
correlations of various observables, in particular temperature fluctuations,
that do not decay as a function of distance, but rather extend over the entire
system. We investigate the nature of these correlations in a fermionic quantum
fluid and show that the crucial coupling between the temperature gradient and
velocity fluctuations is the same as in the classical case. Accordingly, the
nature of the long-ranged correlations in the hydrodynamic regime also is the
same. However, as one enters the collisionless regime in the low-temperature
limit the nature of the velocity fluctuations changes: they become ballistic
rather than diffusive. As a result, correlations of the temperature and other
observables are still singular in the long-wavelength limit, but the
singularity is weaker than in the hydrodynamic regime.
