  We study the back reaction of a thermal field in a weak gravitational
background depicting the far-field limit of a black hole enclosed in a box by
the Close Time Path (CTP) effective action and the influence functional method.
We derive the noise and dissipation kernels of this system in terms of
quantities in quasi-equilibrium, and formally prove the existence of a
Fluctuation-Dissipation Relation (FDR) at all temperatures between the quantum
fluctuations of the thermal radiance and the dissipation of the gravitational
field. This dynamical self-consistent interplay between the quantum field and
the classical spacetime is, we believe, the correct way to treat back-reaction
problems. To emphasize this point we derive an Einstein-Langevin equation which
describes the non-equilibrium dynamics of the gravitational perturbations under
the influence of the thermal field. We show the connection between our method
and the linear response theory (LRT), and indicate how the functional method
can provide more accurate results than prior derivations of FDRs via LRT in the
test-field, static conditions. This method is in principle useful for treating
fully non-equilibrium cases such as back reaction in black hole collapse.
