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We derive the effective equations for the out of equilibrium time evolutionof the order parameter and the fluctuations of a scalar field theory inspatially flat FRW cosmologies.The calculation is performed both to one-loopand in a non-perturbative, self-consistent Hartree approximation.The methodconsists of evolving an initial functional thermal density matrix in time andis suitable for studying phase transitions out of equilibrium. Therenormalization aspects are studied in detail and we find that the countertermsdepend on the initial state. We investigate the high temperature expansion andshow that it breaks down at long times. We also obtain the time evolution ofthe initial Boltzmann distribution functions, and argue that to one-loop orderor in the Hartree approximation, the time evolved state is a ``squeezed''state. We illustrate the departure from thermal equilibrium by numericallystudying the case of a free massive scalar field in de Sitter and radiationdominated cosmologies. It is found that a suitably defined non-equilibriumentropy per mode increases linearly with comoving time in a de Sittercosmology, whereas it is {\it not} a monotonically increasing function in theradiation dominated case.Comment: 29 pages, revtex 3.0, 11 figures available upon request, PITT-93-6;  LPTHE-93-52; CMU-HEP-93-2

Nonequilibrium evolution of scalar fields in FRW cosmologies