We introduce functional methods to study the non-equilibrium dynamics of a
quantum massless scalar field at finite temperature in a gravitational field.
We calculate the Close Time Path (CTP) effective action and, using its formal
equivalence with the influence functional, derive the noise and dissipation
kernels of the quantum open system in terms of quantities in thermodynamical
equilibrium. Using this fact, we formally prove the existence of a
Fluctuation-Dissipation Relation (FDR) at all temperatures between the quantum
fluctuations of the plasma in thermal equilibrium and the energy dissipated by
the external gravitational field. What is new is the identification of a
stochastic source (noise) term arising from the quantum and thermal
fluctuations in the plasma field, and the derivation of a Langevin-type
equation which describes the non-equilibrium dynamics of the gravitational
field influenced by the plasma. The back reaction of the plasma on the
gravitational field is embodied in the FDR. From the CTP effective action the
contribution of the quantum scalar field to the thermal graviton polarization
tensor can also be derived and it is shown to agree with other techniques, most
notably, Linear Response Theory (LRT). We show the connection between the LRT,
which is applicable for near-equilibrium conditions and the functional methods
used in this work which are useful for fully non-equilibrium conditions.Comment: Final version published in Phys. Rev.
